arXiv:1610.06161v2 [] 8 Feb 2017

FINITE LOCALITIES III

Andrew Chermak
Kansas State University
October 2016

Introduction
This paper is the third in a series on finite localities, whose earlier installments are
Part I [Ch2] and Part II [Ch3]. We continue the convention of referring to results in
earlier parts by prefixing a “I” or (now) a “II” to citations. For example, “II.2.4” refers
to the definition 2.4 of proper locality in [Ch3]. Familiarity with the earlier parts is
assumed, but we will provide a brief review of some of the core material here in section
1.
This Part III relies more heavily on the language of fusion systems than do Parts I
and II. In particular, if F is the fusion system of a proper locality (L, ∆, S), then the set
F q of F -quasicentric subgroups of S and, more importantly, the set F s of F -subcentric
subgroups of S will play a key role here in understanding the structure of L. (These
collections, along with the set F c of F -centric subgroups, and the set F cr of F -centric
radical subgroups of S were defined in II.1.8.)
Throughout this paper (L, ∆, S) will be a proper locality on F . Thus, (L, ∆, S) is
a locality, F = FS (L) is its fusion system, and it is assumed that F cr ⊆ ∆, and that
the normalizer subgroups NL (P ) for P ∈ ∆ are of characteristic p (whence ∆ ⊆ F s , by
II.2.8). The main theorems in Part II showed how one may alter the set ∆ to “restrict”
or “expand” L, while preserving F , and while preserving also the structure of the poset
of partial normal subgroups of L. These theorems provide the flexibility whereby ∆ may
be chosen to be whatever F -closed collection ∆ of subgroups of S is most convenient for
a particular analysis, subject only to the requirement F cr ⊆ ∆ ⊆ F s . The principal aim
of this paper is to show that there is such an F -closed collection of subgroups of S, to
be denoted δ(F ), which is in many ways preferential - and for at least the following two
inter-connected reasons.
(1) If ∆ = δ(F ) then each partial normal subgroup N E L is itself a proper locality.
Typeset by AMS-TEX

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(2) If ∆ = δ(F ) then, for each partial normal subgroup N E L, the set CL (N ) of all
g ∈ L such that xg is defined and is equal to x, for all x ∈ N , is a partial normal
subgroup of L.
For any locality (L, ∆, S) there is an action of S on L by conjugation, and we may
therefore speak of CS (X) for any non-empty subset X of L. We shall say that a partial
normal subgroup M E L is large if CS (M) ≤ M. One way to define the set δ(F ) is to
begin by expanding L to a proper locality (Ls , F s , S) on F whose set of objects is as large
as possible. That is, we may employ Theorem II.A1 in order to obtain the unique (up
to isomorphism) proper locality on F whose set of objects is the set F s of F -subcentric
subgroups of S. Then define F ∗ (Ls ) to be the intersection of the set of all large partial
normal subgroups M of Ls containing Op (F ).
F ∗ (Ls ) =

\

{M E Ls | Op (F )CS (M) ≤ M}.

Now define δ(F ) to be the set of all subgroups P of S such that P ∩ F ∗ (Ls ) ∈ F s . We
show in lemma 6.10 that F cr ⊆ δ(F ) ⊆ F s and that δ(F ) is F -closed. Thus δ(F ) is a
permissible choice for ∆; and we say that the proper locality (L, ∆, S) on F is regular if
∆ = δ(F ).
Theorem C. Let (L, ∆, S) be a regular locality on F , and let N E L be a partial normal
subgroup of L. Set T = S ∩ N and let E = FT (N ) be the fusion system on T generated
by the conjugation maps cg : V → T with g ∈ N and with V ≤ T . Define CL (N ) to be
the set of all g ∈ L such that, for all x ∈ N , xg is defined and is equal to x. Then the
following hold.
(1) δ(E) = {V ≤ T | CS (N )V ∈ δ(F )}, and N is itself a regular locality on E.
(2) CL (N ) E L, and CL (N ) = O p (CL (T ))CS (N ) (cf. section II.7). Moreover, for
all g ∈ CL (N ) and all x ∈ N the product [x, y] := x−1 g −1 xg is defined and is
equal to 1.
For any proper locality (L, ∆, S) on F , define F ∗ (L) to be the partial normal subgroup
L ∩ F ∗ (Ls ) of L corresponding to the partial normal subgroup F ∗ (Ls ) of Ls via Theorem
II.A2. A partial subgroup K of L is subnormal in L if there is a sequence (N0 , · · · , Nm ) of
partial subgroups of L with K = N0 E · · · E Nm = L. The partial subnormal subgroup K
of L is a component of L if K = O p (K) and K/Z(K) is simple (where Z(K) = CS (K)∩K))
).
For subsets X and Y of a partial group, write [X, Y ] = 1 to indicate that [x, y] is
defined and is equal to 1 for all x ∈ X and all y ∈ Y .
Theorem D. Let (L, ∆, S) be a regular locality on F and let (K1 , · · · , Kn ) be a nonredundant list of all of the components of L. Let E(L) be the smallest partial subgroup
of L containing all of the components Ki . Then E(L) E L, and
F ∗ (L) = Op (F )E(L) = Op (F )K1 · · · Kn ,
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where [Op (F ), E(L)] = 1, and where [Ki , Kj ] = 1 for all i and j with i 6= j. Further, we
have CS (F ∗ (L)) ≤ F ∗ (L), and L = F ∗ (L)H, where H := NL (S ∩ F ∗ (L)) is a subgroup
of L which acts on L by everywhere-defined conjugation.
The last of our main theorems concerns the category of regular localities as a full
subcategory of the category of partial groups.
Theorem E. Let Lδ be the full sub-category of the category of partial groups, whose
class of objects is the class of regular localities. Let (L, ∆, S) be a regular locality on F .
(a) If N E L is a partial normal subgroup of L then the inclusion map N → L is a
Lδ -homomorphism.
(b) If X ≤ S is fully normalized in F , and X ≤ F ∗ (L), then there is a unique regular
locality LX on NF (X) such that LX is a subset of L, and such that the inclusion
map LX → L is an Lδ -homomorphism.
(c) If Z ≤ CS (O p (L)) is a normal subgroup of L then the quotient locality L/Z is
regular, and the quotient map L → L/Z is an Lδ -homomorphism.
It is a defect of the category Lδ that, aside from some special cases (such as the one
given by point (c) of Theorem E), homomorphic images of regular localities need not be
regular - or even proper. One way in which to address this defect (but which will not be
pursued here) is as follows. Define a locality (L, ∆, Σ) to be semi-regular if there exists
a regular locality Le and a projection (cf. I.4.4) ρ : Le → L. Composites of projections
are projections, so the category Lσ of partial groups whose objects are semi-regular
localities is closed with respect to projections. Further, the Correspondence Theorem
I.4.7 shows that partial normal subgroups of semi-regular localities are themselves semiregular localities. What is then needed is a version of point (b) in Theorem E, in order
to obtain the beginnings of a satisfactory category.
Section 1: Localizable pairs
We assume that the reader has become comfortable with the basic notions introduced
in the first section of Part I; specifically partial groups, partial subgroups, partial normal
subgroups, and homomorphisms of partial groups. These notions will not be reviewed
here.
From section 2 of Part I:A partial group L is objective if there is a set ∆ of subgroups
of L (a set of objects) which defines the domain D of the product Π in L, and which
satisfies a pair of closure conditions as expressed in definition II.2.1. A (finite) locality is
a finite, objective partial group (L, ∆) such that ∆ is a set of subgroups of some S ∈ ∆,
S is a p-group for some prime p, and S is maximal in the poset of p-subgroups of L.
A fundamental property of localities is given by I.2.6 and I.2.7. Namely, let (L, ∆, S)
be a locality, and let w = (g1 , · · · , gn ) be a word in the free monoid W(L). Define Sw
to be the set of all elements x = x0 ∈ S such that x0 is conjugated to an element x1 of
S by g1 , x1 is conjugated to an element x2 of S by g2 , and so on. Then
(*) Sw is a subgroup of S, and w ∈ D if and only if Sw ∈ ∆.
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This result is involved in virtually every argument, and should require (and will receive)
no further reference. In connection with (*) one says of a word w ∈ W(L) that w is in
D via P if P ∈ ∆ and P ≤ Sw . A trivial consequence of (*) is:
(**) If w ∈ D then Sw ≤ SΠ(w) .
Part II introduced the relationship between a locality (L, ∆, S) and its fusion system
F = FS (L). Thus, F is the fusion system on S whose isomorphisms are compositions of
restrictions of conjugation maps cg : Sg → Sg −1 for g ∈ L.
Most readers will have at least some familiarity with general fusion systems, but
sections 1 and 6 of Part II provide all that will be needed here. The notion of “saturation”
appears in the proof of Therem II.6.1, but will play no further role. As mentioned in the
introduction, the notion of F -closed set of subgroups of S (F a fusion system on S), and
the definitions of the sets F cr , F c , F q , and F s appear early in section II.2. Theorem
II.6.2 (due to Henke [He2]) establishes the sequence of inclusions
F cr ⊆ F c ⊆ F q ⊆ F s ,
and establishes that F s is F -closed.
The locality (L, ∆, S) on F is defined to be proper (cf. II.2.4) if F cr ⊆ ∆, and if
each of the groups NL (P ) for P ∈ ∆ is of characteristic p. Lemma II.2.8 provides a
“dictionary”, translating back and forth between fusion-theoretic conditions on a given
P ∈ ∆ and properties of NL (P ). Other basic notions from Parts I and II will be recalled
as needed.
For any locality (L, ∆, S), define
Op (L) =

\

{Sw | w ∈ W(L)}.

Equivalently (by I.2.14) Op (L) is the largest subgroup of S which is a partial normal
subgroup of L. Then Op (L) ≤ Op (F ), where F is the fusion system of L. In the case
that L is proper one has the important equality Op (L) = Op (F ) (cf. II.2.3).
Definition 1.1. Let (L, ∆, S) be a locality and let H ≤ L be a partial subgroup of L.
Set T = S ∩ H, let Γ be a set of subgroups of T , and set
HΓ = {h ∈ H | Sh ∩ T ∈ Γ}.
Let E := FT (H, Γ) be the fusion system on T generated by the set of conjugation maps
ch : Sh ∩ T → T for h ∈ HΓ . Then (H, Γ) is a localizable pair in L if:
(1) Γ is E-closed.
(2) T is maximal in the poset of p-subgroups of H.
(3) w ∈ W(H) and Sw ∩ T ∈ Γ =⇒ w ∈ D(L).
4

Lemma 1.2. Let (L, ∆, S) be a locality, and let (H, Γ) be a localizable pair in L. Set
T = S ∩ H and E = FT (H, Γ). Further, set
HΓ = {h ∈ H | Sh ∩ T ∈ Γ},
and
D(HΓ ) = {w ∈ W(HΓ ) | Sw ∩ T ∈ Γ}.
Then HΓ is a partial group with respect to the restriction of the product Π : D(L) → L
to a mapping D(HΓ ) → HΓ , and with respect to the restriction to HΓ of the inversion in
L. Moreover, (HΓ , Γ, T ) is then a locality on E.
Proof. We must verify that the conditions (1) through (4) in the definition (I.1.1) of a
partial group are satisfied by HΓ and the given product and inversion. Thus HΓ is the
set of words of length 1 in D(HΓ ), and it is plain that if the word u ◦ v is in D(HΓ ) then
so are u and v. Thus the condition (1) holds. If w ∈ D(HΓ ) then w ∈ D, and Π(w) is in
H since H is a partial subgroup of L. Then Π(w) ∈ HΓ since Sw ≤ SΠ(w) , and thus Π
restricts to a mapping D(HΓ ) → HΓ . The conditions 1.1(2) and (3) are then inherited
from Π. For any h ∈ HΓ we have Sh−1 ∩ T = (Sh ∩ T )h , so HΓ is closed under the
inversion in L. The required condition (4), that Π(w−1 ◦ w) = 1 for w ∈ D(HΓ ), is then
immediate. Thus HΓ is a partial group.
The inclusion HΓ → L is a homomorphism since the product in HΓ is the restriction
of the product Π. Moreover it is for this same reason, and because Γ is E-closed, that
(HΓ , Γ) is an objective partial group. Notice that any subgroup of HΓ is also a subgroup
of H. Then 1.1(2) implies that T is maximal in the poset of p-subgroups of HΓ , and thus
(HΓ , Γ, T ) is a locality. By the definition of E we have E = FT (HΓ ). 
Remark. We have already, in II.2.11, encountered one sort of localizable pair. Namely,
if Γ is an F -closed subset of ∆ then (L, Γ) is a localizable pair, and (LΓ , Γ, S) is the
restriction L |Γ of L to Γ.
Lemma 1.3. Let the hypothesis and notation be as in 1.2, and let FT (H) be the fusion
system on T generated by the conjugation maps ch : Sh ∩ T → T for all h ∈ H. In order
for (HΓ , Γ, T ) to be a locality on FT (H) it suffices that Γ contain E cr , and that FT (H)
be Γ-generated.
Proof. Set E = FT (HΓ ) and E1 = FT (H). Then E is a fusion subsystem of E1 . Let E2
be the fusion subsystem of E generated by the set of all E-automorphisms of members
U ∈ (E1 )cr such that U is fully normalized in E1 and such that U = Op (NE1 (U )). If
(E1 )cr ⊆ Γ then E2 is a subsystem of E, and if E1 is Γ-generated then E1 = E2 . Thus
E = E1 if the two stated conditions are fulfilled. 
Lemma 1.4. Let (L, ∆, S) be a proper locality on F , and let V ≤ S be fully normalized
in F . Set
Γ = {X ∈ ∆ | V E X}, and Σ = {Y ≤ CS (V ) | V Y ∈ Γ}.
(a) If F c ⊆ ∆ then NF (V )cr ⊆ Γ and CF (V )cr ⊆ Σ.
5

(b) If NF (V )cr ⊆ Γ then (NL (V ), Γ) is a localizable pair and (NL (V )Γ , Γ, NS (V )) is
a proper locality on NF (V ).
(c) If CF (V )cr ⊆ Σ then (CL (V ), Σ) is a localizable pair and (CL (V )Σ , Σ, CS (V )) is
a proper locality on CF (V ).
Proof. Set H = NL (V ) and K = CL (V ). Then H is a partial subgroup of L, and K E H.
Set E = NF (V ) and D = CF (V ). Then E and D are (cr)-generated by II.6.1. Then HΓ
is a locality on E if Γ ⊆ ∆, and KΣ is a locality on D if XΣ ⊆ ∆, by 1.3.
It remains to show that the locality HΓ is proper if E cr ⊆ Γ, and that KΣ is proper
if D cr ⊆ Σ. Under these hypotheses on Γ and Σ, it is only necessary to show that the
normalizers of objects in HΓ or in KΣ are of characteristic p. As Γ ⊆ ∆, and since
NH (X) = NNL (X) (V )
for X ∈ Γ, it follows from II.2.7(b) that NH (X) is of characteristic p. For Y ∈ Σ we have
V Y ∈ ∆, and so
NK (Y ) = CNH (V Y ) (V ) E NH (V Y ).
Then NK (Y ) is of characteristic p by II.2.7(a), and the proof is complete. 
Corollary 1.5. Let (L, ∆, S) be a proper locality on F , and let T ≤ S be strongly closed
in F . Set LT = NL (T ) and CT = CL (T ).
(a) (LT , ∆, S) is a proper locality on NF (T ).
(b) Set Σ = {V ≤ CS (T ) | V T ∈ ∆}, and assume that CF (T )cr ⊆ Σ. Then
(CT , Σ, CS (T )) is a proper locality on CF (T ). Moreover, the condition CF (T )cr ⊆
Σ is fulfilled if F c ⊆ ∆.
+
+
(c) Let (L+ , ∆+ , S) be an expansion of L and set L+
T = NL+ (T ). Then (LT , ∆ , S)
is an expansion of (LT , ∆, S).
Proof. As T is strongly closed in F one observes that
NF (T )cr = {P ∈ F cr | T ≤ P },
and hence NF (T )cr ⊆ ∆. Then 1.4(a) yields (a). Point (b) is immediate from 1.4(b),
and point (c) is immediate from (a). 
Section 2: The basic setup
Most of the results to be proved in sections 2 through 5 will be concerned with a single
partial normal subgroup N of L. (The only exceptions are 2.10 and 2.11, which concern
pairs of partial normal subgroups.) The following notation will remain fixed.
2.1 (Basic setup). (L, ∆, S) is a proper locality on F , and N E L is a partial normal
subgroup of L. Set T = S ∩ N , and let E be the fusion system FT (N ) on T , generated
by the conjugation maps cg : Sg ∩ T → T for g ∈ N . Further, set LT = NL (T ) and
CT = CL (T ).
6

Recall from I.3.1 that T is strongly closed in F , and T is maximal in the poset of psubgroups of N . There is no reason to suppose that E should be inductive (see II.1.11),
or that E should remain invariant under the process N 7→ N + of expansion given by
Theorem II.A2. Indeed, this non-rigidity of E relative to F will be the source of most of
the technical difficulties that will be encountered.
Lemma 2.2. Assume the setup of 2.1, set Te = CS (T )T , and set H = NL (Te). Then the
following hold.
(a) H is a subgroup of L, and Op (H) = Op (NF (Te)) ∈ F cr .
(b) Te is strongly closed in F .
(c) FS (H) = NF (Te).
(d) LT = CT H.
e be the partial subgroup hN , CT i of L generated by
Proof. Set Te = CS (T )T , and let N
e E L and Te = S ∩ N
e , by I.5.5. Then (b) follows from I.3.1(a), with
N and CT . Then N
e in the role of N . Point (d) is immediate from the Frattini Lemma (I.3.11).
N
Clearly Te is F -centric. Set Q = Op (NF (Te)). Then Q E S, so Te E Op (NF (Q)), and
hence Q = Op (NF (Q)). Thus
(*)

Q ∈ F cr ,

so Q ∈ ∆, and NL (Q) is a subgroup of L.
By 1.5(a), (H, ∆, S) is a proper locality on NF (Te). This yields (c), and II.2.3 shows
that Op (H) = Q. Then H is the subgroup NNL (Q) (Te) of L. This result, together with
(*), completes the proof of (a). 
Lemma 2.3. Assume the setup of 2.1. Let H be a partial subgroup of LT having the
property that (h−1 , x, h) ∈ D for all x ∈ N and all h ∈ H, and let λ : H → Aut(T ) be
the homomorphism which sends h ∈ H to conjugation by h. Then Im(λ) ⊆ Aut(E).
Proof. Let φ : U → U ′ be an E-isomorphism. By definition, there is a sequence
φn

φ1

U = U0 −→ · · · −→ Un = U ′
of E-isomorphisms, such that each φi : Ui−1 → Ui is the restriction of a conjugation
map cxi : T ∩ Sgi → T , with xi ∈ N . Let h ∈ H. By hypothesis, (h−1 , xi , h) ∈ D
for all i, and then (xi )h ∈ N as N E L. As H ≤ NL (T ) we may define Vi := (Ui )h ,
and then define ψi : Vi1 → Vi to be given by conjugation by (xi )h . Then each ψi is
an E-isomorphism, and the composite ψ := ψ1 ◦ · · · ◦ ψn is given by ch−1 ◦ φ ◦ ch as
an E-isomorphism U h → (U ′ )h . This shows that the conjugation map ch : T → T is
E-fusion-preserving. That is, ch ∈ Aut(E). The verification that the map h 7→ ch is a
homomorphism H → Aut(E) is a straightforward application of I.2.3(c). 
7

Corollary 2.4. Suppose that xh is defined for all x ∈ N and all h ∈ H := NL (CS (T )T ).
Let Γ be a set of subgroups of subgroups of T which is both E-invariant and Aut(E)invariant. Then Γ is F -invariant.
Proof. Let g ∈ L, and let U ∈ Γ with U ≤ Sg ∩ T . Let φ : U → U g be the conjugation
map. Define LT and CT as in 1.4. By the Splitting Lemma (I.3.12) we may write g = xf
where x ∈ N , f ∈ LT , and Sg = S(x,f ) . As CT E LT we may also write f = yh where
y ∈ CT , h ∈ H, and Sf = S(y,h) . Then Sg = S(x,y,h) , and φ is then the composition of
the E-isomorphism cx followed by ch . Then U g ∈ Γ by 2.3. Then Γ is F -invariant since
F is generated by the conjugation maps cg with g ∈ L. 
Lemma 2.5. Assume the setup of 2.1, and let Q ≤ T be a subgroup of T such that
Q is fully normalized in E. Suppose that Op (L)Q ∈ ∆, and set K = NL (Q). Then
K is a normal subgroup of the group NL (Q), NT (Q) is a Sylow subgroup of K, and
NE (Q) = FNT (Q) (K).
Proof. Set M = NL (Op (L)Q). As Op (L)Q ∈ ∆, M is a subgroup of L. Then K =
N ∩ NM (Q) = N ∩ NL (Q) is a normal subgroup of the group NL (Q) by I.1.8.
Set X = NT (V ) and let Y be a Sylow p-subgroup of K containing X. By I.2.11 there
exists g ∈ L with Y g ≤ S, and then Y g ≤ T since T is strongly closed in F . Employ the
splitting lemma (I.3.12) to obtain g = f h with f ∈ N , h ∈ NL (T ), and with Sg = S(f,h) .
Then Y f ≤ T , and so Y f ≤ NT (V f ). As V is fully normalized in E, we conclude that
X f = Y f , and thus X ∈ Sylp (K).
Next, let A and B be subgroups of X containing V , such that A and B are conjugate
in NE (V ). Let γ : A → B be an NE (V )-isomorphism. By the definition of E = FT (N ),
this means that there exists w = (f1 , · · · fn ) ∈ W(N ) such that A ≤ Sw , and such that
γ is given by composing the conjugation maps cfi . As Op (L)V ≤ Sw we have w ∈ D,
and then Π(w) ∈ N . Then Π(w) ∈ K, and thus NE (Q) = FX (K). 
Recall from II.1.8 that E cr is defined to be the set of all U ∈ E c such that there exists an
E-conjugate V of U such that V is fully normalized in E, and such that V = Op (NE (V )).
It has already been remarked that this definition does not quite agree with the usual
one (see [AKO], for example. The following lemma provides the justification for this
discrepancy. Namely, points (b) and (c) of the lemma establish a “descent” from F cr to
E cr which relies on our definition, and which would otherwise be lacking.
Lemma 2.6. Assume the setup of 2.1.
(a) Let U ≤ T be a subgroup of T . Assume that U is fully normalized in F and that
CT (U ) ≤ U . Then U F ⊆ E c .
(b) Let Q ∈ F cr and set U = Q ∩ T . Suppose that U is fully normalized in F . Then
U F ⊆ E c , U is fully normalized in E, and U = Op (NE (U )).
(c) Let M E L be a partial normal subgroup of L containing N . Set R = S ∩ M, set
D = FR (M), and suppose that D is inductive. Let Q ∈ D cr , and set U = Q ∩ T .
Assume that Op (L)Q ∈ ∆ and that U is fully normalized in D. Then U D ⊆ E c ,
U is fully normalized in E, and U = Op (NE (U )).
8

Proof. Let U be chosen as in (a), and let U ′ be an F -conjugate of U . As F is inductive
by II.6.1, there exists an F -homomorphism φ : NS (U ′ ) → NS (U ) with U ′ φ = U . As
T is strongly closed in F we obtain CT (U ′ )φ ≤ CT (U ). As CT (U ) ≤ U it follows that
CT (U ′ ) ≤ U ′ . Since E-conjugates of F -conjugates of U are F -conjugates of U , we obtain
U F ⊆ E c , and thus (a) holds.
Point (b) is the special case of (c) where D = F , so it remains only to prove (c). Let
M, R, D, Q, and U be as stated in (c). Then U is fully normalized in E by II.1.17.
Set K = NM (Q). By 2.5, with M in the role of N , we have NR (Q) ∈ Sylp (K) and
ND (Q) = FNR (Q) (K). Also by 2.5, K is a normal subgroup of the group NL (Q), where
NL (Q) is a local subgroup of NL (Op (L)Q). Then K is of characteristic p by II.2.7, and
hence Op (K) = Op (ND (Q)). As Q ∈ D cr , where D is inductive, it follows from II.1.14
that Q = Op (K). Set D = NCT (U) (Q). Then
[Q, D] ≤ CQ∩T (V ) = Z(U ),
and thus D centralizes the chain Q ≥ U ≥ 1 of normal subgroups of K. Then D ≤ Q by
II.2.7(c), and thus CT (U ) ≤ Q. Then CT (U ) ≤ Q ∩ T = U . A straightforward variation
on the proof of (a) then yields U D ⊆ E c .
Set X = Op (NE (U )). Then U ≤ X ≤ T . By 1.5 the elements of S act as automorphisms of E by conjugation, so Q acts on NE (X), and thus X is Q-invariant. By
definition, each E-automorphism of U extends to an E-automorphism of X, and this
translates into the following statement.
(*) Let β be an E-automorphism of U , let β be an extension of β to an E-automorphism
of X, and let x ∈ X. For any a ∈ X, let ca be the automorphism of U given by
conjugation by a. Then β −1 ◦ ca ◦ β = caβ . In particular, AutX (U ) is a normal
subgroup of AutE (U ).
Set K0 = K ∩ N and set A = NX (Q). Then K0 is a normal subgroup of K, while
(*) yields AutA (U ) E AutK0 (U ). As U ∈ E c , CK0 (U ) is the direct product of Z(U )
with a normal p′ -subgroup of K0 . Here Op′ (K0 ) = 1 as K is of characteristic p, and so
CK0 (U ) = Z(U ). There is then a natural isomorphism of AutK0 (U ) with K0 /Z(U ), from
which it follows that A/Z(U ) E K0 /Z(U ). Then A E K0 , so A ≤ Op (K0 ) ≤ Op (K) = Q.
Then X ≤ Q, and since also X ≤ T we arrive at X ≤ U . Thus X = U , completing the
proof of (c). 
For any partial subgroup H ≤ L define Z(H) to be the set of all z ∈ H such that, for
all h ∈ H, hz is defined and is equal to h.
Lemma 2.7. Assume the setup of 2.1. Then Z(N ) = CT (N ).
Proof. As tz is defined and is equal to t for all t ∈ T and all z ∈ Z(N ), we have
Z(N ) ≤ NN (T ). Set H = NL (CS (T )T ). As L is proper, I.3.5 yields NN (T ) ≤ H. As
H is a subgroup of L by 2.2(a), Z(N ) is then a normal abelian subgroup of H. As
H is of characteristic p, II.2.7 implies that Z(N ) is of characteristic p. Thus Z(N ) is
a p-subgroup of NL (T ). As T is a maximal p-subgroup of N by I.3.1(a), the lemma
follows. 
9

For any fusion system F on a p-group S, and any non-empty set Γ of subgroups of S,
define the F -closure of Γ to be the smallest F -closed set of subgroups of S containing
Γ. Thus, the F -closure of Γ is the set of subgroups X of S such that X contains an
F -conjugate of a member of Γ.
For any pair Γ and Σ of non-empty sets of subgroups of S, write ΓΣ for the set of all
products XY with X ∈ Γ and Y ∈ Σ. If Γ = {X} is a singleton we may write XΣ for
{X}Σ.
Lemma 2.8. Assume the setup of 2.1, and set X = Op (L)CS (N ). Let Γ be an E-closed
set of subgroups of T containing E cr , and let ∆0 be the F -closure of XΓ. Assume that
∆0 ⊆ ∆ and that E is Γ-generated. Then (N , Γ) is a localizable pair, and (NΓ , Γ, T ) is a
proper locality on E. Moreover, if Op (L)Γ ⊆ ∆, then the following hold.
(a) F cr ⊆ ∆0 ⊆ ∆, the restriction L0 of L to ∆0 is a proper locality (L0 , ∆0 , S) on
F , and NΓ = N ∩ L0 .
(b) NL (T ) is a subgroup of L0 , and xg is defined for all x ∈ L0 and all g ∈ NL (XT ).
Moreover, the mapping g → cg which sends g ∈ NL (XT ) to the conjugation map
cg : L0 → L0 is a homomorphism λ : NL (XT ) → Aut(L0 ).
(c) For each g ∈ NL (T ), the restriction of cg to T is an automorphism of E, and one
obtains in this way a homomorphism λT : NL (T ) → Aut(E).
Proof. We check that (N , Γ) is a localizable pair by verifying the two conditions in
definition 1.1. The second of these - that T is maximal in the poset of p-subgroups of N
- is given by I.3.1(c).
Let w ∈ W(N ) with Sw ∩ T ∈ Γ. Then X(Sw ∩ T ) ∈ ∆ by hypothesis, so Sw ∈ ∆
and w ∈ D. Thus the condition 1.1(1) obtains, and (N , Γ) is a localizable pair. By 1.2
(NΓ , Γ, T ) is a locality. Since E cr ⊆ Γ and E is Γ-generated by hypothesis, 1.3 yields
E = FT (NΓ ).
Let U ∈ Γ and set P = XU . Then P ∈ ∆, so NL (P ) is a group of characteristic p.
Then NN (P ) is of characteristic p by II.2.7(a), and then NN (U ) is of characteristic p by
II.2.7(b). Thus (NΓ , Γ, T ) is a proper locality on E.
We assume for the remainder of the proof that Op (L)Γ ⊆ ∆. Let R ∈ F cr and, via
II.1.18, let R′ be an F -conjugate of R such that both R′ and R′ ∩ T are fully normalized
in F , and then R′ ∩ T ∈ E cr by 2.6(b). Then Op (N ) ≤ R′ . Thus R′ ∈ ∆0 , and then also
R ∈ ∆0 . Thus F cr ⊆ ∆0 , and the restriction (L0 , ∆0 , T ) of L is then given by II.2.11 as
a proper locality on F . The equality NΓ = N ∩ L0 is immediate from the definitions of
NΓ and of L0 . Thus (a) holds.
Set H = NL (T ). As Op (L)T = XT ∈ ∆0 , H is a subgroup of L0 . Let x ∈ L0 ,
let h ∈ H, and set P = Sx ∩ Op (L)T . Then P ∈ ∆ (by the definition of L0 ), and
(h−1 , x, h) ∈ D via P h . Points (b) and (c) now follow from 2.3. 
The remainder of this section involves the following variation on the setup of 2.1.
2.9 (Product setup). (L, ∆, S) is a proper locality on F , and N1 and N2 are partial
normal subgroups of L. Set Ti = S ∩ Ni , and set Ei = FTi (Ni ). Assume:
(*)

S ∩ Ni ≤ CS (N3−i ) (i = 1, 2).
10

Set M = N1 N2 (a partial normal subgroup of L by I.5.1, or by [He]), and set R = S ∩ M
and D = FR (M).
Lemma 2.10. Assume the setup of 2.9. Assume further that D cr ⊆ ∆ and that D is
D cr -generated. Set Γ = {P ∈ ∆ | P ≤ R}, and set
Γi = {P T3−i ∩ Ti | P ∈ Γ}, (i = 1, 2).
Let ∆0 be the overgroup-closure of Γ1 Γ2 in S, and let ∆+
0 be the overgroup-closure of Γ
in S. Then the following hold.
cr
(a) ∆0 and ∆+
⊆ ∆ 0 ⊆ ∆+
0 are F -closed subset of ∆, and F
0.
(b) (M, Γ) and (Ni , Γi ) (i = 1, 2) are localizable pairs, (MΓ , Γ, R) is a proper locality
on D, and each ((Ni )Γi , Γi , Ti ) is a proper locality on Ei . Moreover, (Ni )Γi =
Ni ∩ M Γ .
(c) R ∈ ∆0 , and the group NL (R) acts by conjugation on all three of the localities in
(b). Indeed, conjugation by h ∈ NL (R) is an automorphism of D which restricts
to an automorphism of each Ei .
(d) D cr = (E1 )cr (E2 )cr .
(e) Let Y ≤ R be a subgroup of R, set Yi = Y ∩ Ti , and suppose that Y = Y1 Y2 . Then
Y ∈ D c if and only if each Yi is in (Ei )c , and Y ∈ D s if and only if Yi ∈ (Ei )s
(i = 1, 2).
Proof. By definition, D cr is D-invariant, so Γ is D-closed. Since D is D cr -generated and
D cr ⊆ ∆, by hypothesis, we may appeal to 2.8 with M in the role of N . Thus (MΓ , Γ)
is a localizable pair, and (MΓ , Γ, R) is a proper locality on D.
We have R = R1 R2 = T1 T2 by I.5.1, so R ∈ ∆0 . For any P ∈ Γ set Pi = P T3−i ∩ Ti .
Then P ≤ P1 P2 , and thus Γ1 Γ2 ⊆ Γ. As Γ ⊆ ∆ we then have ∆0 ⊆ ∆+
0 ⊆ ∆. Clearly Γ
+
is F -invariant, so ∆0 is F -closed. For any P = P1 P2 ∈ Γ1 Γ2 and any g ∈ L with P ≤ Sg
we have P g = (P1 )g (P2 )g ∈ Γ1 Γ2 , and it follows that ∆0 is F -closed.
Set H = NL (R). Then H is a subgroup of L as R ∈ Γ1 Γ2 . Evidently Γ is H-invariant,
so H acts on MΓ , and then H acts on D by 2.3. Thus (b) and (c) hold insofar as these
points refer only to M and not to Ei .
Let A and B be D-conjugate subgroups of R, and let φ : A → B be a D-isomorphism.
By definition, this means that there exists w = (g1 , · · · , gn ) ∈ W(M) such that φ may
be written as a composition cw of conjugation maps:
(*)

cg

cg

1
n
A1 −
→ ··· −
→ An−1 −−→
An = B.
A = A0 −−→

By I.5.2 each gk is a product gk = xk yk with xk ∈ N1 , yk ∈ N2 , and with Sgk = S(xk ,yk ) .
Set
w′ = (x1 , y1 , · · · , xk , yk ), u = (x1 , · · · , xk ), and v = (y1 , · · · , yk ).
Then φ = cw′ , and then φ = cu ◦ cv since Ti ≤ CS (N3−1 ). Thus:
(1) Each D-isomorphism may be factored as a composition ψ1 ◦ ψ2 , where ψi is a
composition of conjugation maps by elements of Ni (i = 1, 2).
11

As a consequence:
(2) Let Xi ≤ Ti (i = 1, 2), and set X = X1 X2 . Then X D = (X1 )E1 )(X2 )E2 . In
particular, X is fully normalized in D if and only if each Xi is fully normalized
in Ei .
Let Ui ∈ (Ei )cr (i = 1, 2), and set X = U1 U2 . There then exists an Ei -conjugate Vi of Ui
such that Vi is fully normalized in Ei , and such that Vi = Op (NEi (Vi )). Set Y = V1 V2 .
Then (2) shows that Y ∈ X D and that Y is fully normalized in D. We compute:
CR (V1 V2 ) = CR (V1 ) ∩ CR (V2 ) = CT1 (V1 )T2 ∩ CT2 (V2 )T1 ≤ V1 T2 ∩ V2 T1
= V1 (T2 ∩ V2 T1 ) = V1 V2 (T1 ∩ T2 ) ≤ V1 V2 ,
since T1 ∩ T2 ≤ Z(T1 ) ≤ V1 . Thus V1 V2 ∈ D c . In fact, we have shown:
(3) (E1 )c (E2 )c ⊆ D c .
Let φ1 : A → B be an NE1 (V1 )-isomorphism, where A and B are subgroups of NT1 (V1 )
containing V1 . Then φ1 is a D-isomorphism, and evidently φ extends to a D-isomorphism
γ : AV2 → BV2 whose restriction to V2 is the identity map. Set P = Op (ND (Y )). As
γ is in fact a ND (Y )-isomorphism, γ extends to a D-isomorphism ψ : AP → BP which
leaves P invariant.
Write ψ = ψ1 ◦ ψ2 as in (1). Thus,
ψ1

ψ2

AP −→ CQ −→ BP,
where C = Aψ1 and Q = P ψ1 = P ψ2−1 . Then
Q ≤ AP T1 ∩ P T2 = P (T1 ∩ T2 ) = P,
since
T1 ∩ T2 ≤ Z(T1 ) ∩ Z(T2 ) ≤ V1 ∩ V2 ≤ V1 V2 ≤ P.
This shows that ψ1 : AP → BP is an extension of φ1 . There is an obvious further
extension of ψ1 to a D-isomorphism λ1 : AP T2 → BP T2 , given in the same way as ψ1 as
a composite of conjugations by elements of N1 . Thus φ1 extends to an E1 -homomorphism
A(P T2 ∩T1 ) → B(P T2 ∩T1 ). As V1 = Op (NE1 (V )) we conclude that V1 = P T2 ∩T1 . That
is, V1 is the projection of P into T1 relative to the decomposition R = T1 T2 . Similarly,
V2 is the projection of P into T2 , and thus P = V1 V2 . With (3), this shows:
(4) ∆cr ⊇ (E1 )cr (E2 )cr .
Let P now be an arbitrary member of D cr . In order to show that P ∈ (E1 )cr (E2 )cr , it
suffices to consider the case where P is fully normalized in D, by (2). Set M = NM (P ).
Then M is a normal subgroup of the group NL (P ). Set Pi = P T3−i ∩ Ti , and set
Ki = NNi (Pi ). Then P ≤ P1 P2 E Ki , and P1 P2 ∈ ∆. Thus Ki is a normal subgroup
of NM (P1 P2 ). We again quote I.5.2 in order to write an arbitrary g ∈ M as a product
g = x1 x2 with xi ∈ Ni and with Sg = S(x1 ,x2 ) . Then (Pi )xi = Pi , and thus M ≤ K1 K2 .
12

As MΓ is a proper locality, II.2.3 shows that P = Op (M ). Then NPi (P ) ≤ P , and thus
P = P1 P2 . This completes the proof of (d).
In order to now show that (Ni , Γi ) is a localizable pair, three points have to be verified.
First: Ti is maximal in the poset of p-subgroups of Ni (I.3.1(c)). Second: Γi is an Ei closed set of subgroups of Ti (true, since Γi is F -invariant). Third: w ∈ D for each
w ∈ W(Ni ) such that Sw contains some U ∈ Γi . This third point is a consequence of
(1) and (d), since Sw ≥ U T3−i ∈ ∆. So then, (Ni , Γi ) is a localizable pair, and then
((Ni )Γi , Γi , Ti ) is a locality by 1.2.
Set Ki = Ni ∩ MΓ and set D1 = FT1 (K1 ). Then Ki = (Ni )Γi , and (E1 )cr = (D1 )cr ,
by (d). Thus (E1 )cr ⊆ Γi . As D is D cr -generated, it follows from (1) and (d) that E1 is
(E1 )cr -generated. As D1 is a fusion subsystem of E1 , we conclude that E1 = D1 . Since
MΓ -normalizers of members of Γ are of characteristic p, the normalizer in Ki of any
member of Γi is of characteristic p, so (Ni )Γi is a proper locality on Ei . Thus (b) holds.
Evidently H acts on MΓ by conjugation, and this action restricts to an action on Ki
for each i. The proof of (c) is then completed by 2.8(c).
In order to complete the proof of (a) it remains to show that F cr ⊆ ∆0 . Let P ∈ F cr
and let Q be an F -conjugate of P such that both Q and Q ∩ R are fully normalized in
F . Then Q ∩ R ∈ D cr by 2.6(b). Then (d) shows that Q ∩ R ∈ ∆0 , so Q ∈ ∆0 . As D0 is
F -invariant we obtain P ∈ ∆0 , so (a) holds.
It now remains to prove (e). Let Y = Y1 Y2 be as stated in (e). By (2) we may proceed
under the assumption that Y is fully normalized in F and that each Yi is fully normalized
in Ei . Suppose that Y is D-centric. Then CTi (Yi ) ≤ CS (Y ) ∩ Ti ≤ Y ∩ Ti = Yi , and thus
Yi ∈ (Ei )c . With (3), we then have the first “if and only if” in (e).
Set Qi = Op (NEi (Yi ) and set Q = Op (ND (Y )). Then Q1 Q2 = Q, as follows from (1).
Further, by II.1.16, Q is fully centralized in D, and Qi is fully centralized in Ei . Since
Q1 Q2 ∈ D c if and only if each Qi is centric in Ei , we conclude that Y ∈ D s if and only
if each Qi ∈ (Ei )s . 
Corollary 2.11. Assume the setup of 2.9, with N1 N2 = L. Set Γi = {P T3−i ∩ Ti | P ∈
∆} (i = 1, 2). Then (Ni , Γi , Ti ) is a proper locality on Ei , Γ1 Γ2 is an F -closed subset of
∆, and L is the same partial group as its restriction L |Γ1 Γ2 to Γ1 Γ2 .
Proof. Since F is (cr)-generated by II.2.3, and since F cr ⊆ ∆, we may apply 2.10 with
M = L and with Γ = ∆. Then (Ni , Γi , Ti ) is a proper locality on Ei , by 2.10(b), and
LΓ = L. Set ∆0 = Γ1 Γ2 . Then ∆0 is F -closed, and F cr ⊆ ∆0 ⊆ ∆ by 2.10(a). Let
g ∈ L and write g = g1 g2 with gi ∈ Ni and with Sg = S(g1 ,g2 ) (via I.5.2). Then
Sg = (Sg1 ∩ T1 )(Sg2 ∩ T2 ) ∈ ∆0 , so L∆0 = L. 
Section 3: Alperin-Goldschmidt variations
For any finite group G let Γp (G) be the graph whose vertices are the Sylow p-subgroups
of G, and whose edges are the pairs {X, Y } of distinct Sylow p-subgroups such that
X ∩ Y 6= 1. By Sylow’s theorem, the action of G on Γp (G) by conjugation is transitive
on vertices.
13

Let S be a fixed Sylow p-subgroup of G, let Σ be the connected component of Γp (G)
containing the vertex S, and let H be the set-wise stabilizer of Σ in G. Then g ∈ H
for all g ∈ G such that p divides |H ∩ H g |, and hence H = G if and only if Γp (G) is
connected. We shall say that G is p-disconnected if Γp (G) is disconnected. Otherwise, G
is p-connected.
Remark. In the standard terminology, one says that a proper subgroup X of G such
that p divides |X| and such that g ∈ K whenever p divides |X ∩X g | is strongly p-embedded
in G. One easily deduces in that case, that Sylp (X) contains a connected component
Σ of Γp (G), and that X contains the set-wise stabilizer H of Σ. Thus G has a strongly
p-embedded subgroup if and only if Γp (G) is disconnected.
The following result is well known. Since it is not so easy to find a reference for it, we
provide a proof.
Lemma 3.1. Let X be a p′ -group and let A be an elementary abelian group of order p2
such that A acts on X. Then X = hCX (a) | 1 6= a ∈ Ai.
Proof. Let G be the semi-direct product XA formed via the action of A on X. Let q
be a prime dividing |X| and let Y be a Sylow q-subgroup of X. Then G = NG (Y )X,
so we may choose Y to be A-invariant. Then A acts on the elementary abelian q-group
V := Y /Φ(Y ). By Maschke’s Theorem V is a direct sum of irreducible A-submodules.
By Schur’s Lemma CA (W ) contains a maximal subgroup of A for each irreducible Asubmodule W of V . As A is assumed to be non-cyclic we thereby obtain V = hCV (a) |
1 6= a ∈ Ai. Here CV (a) = CY (a)Φ(Y )/Φ(Y ) by coprime action (cf. II.2.7(c)), so
Y = hCY (a) | 1 6= a ∈ AiΦ(Y ). It is a basic property of the Frattini subgroup that if Y0
is a subgroup of Y such that Y = Y0 Φ(Y ) then Y = Y0 . Thus we have the lemma in the
case that X = Y . We conclude that hCX (a) | 1 6= a ∈ Ai contains a Sylow subgroup of
X for each prime divisor of |X|, and this completes the proof. 
Lemma 3.2. Let G be a p-disconnected finite group, and let K be the poset (via inclusion)
of normal subgroups K of G such that p divides |K/Op (G)| and such that Op (G) ≤ K.
Then there exists a unique minimal K ∈ K.
Proof. We may assume that Op (G) = 1. Fix a Sylow p-subgroup S of G and a strongly
p-embedded subgroup H of G containing S. Suppose that there exist K, K ′ minimal in
K with K 6= K ′ , and set X = K ∩ K ′ . Then X is a p′ -group, and S ∩ KK ′ contains
an elementary abelian subgroup A of order p2 . By 3.1, X is then generated by its
subgroups CX (a) as a varies over the set of non-identity elements of S ∩ KK ′ . Thus
X ≤ H. Since [K, S ∩ K ′ ] ≤ X we have K ≤ NG (S ∩ K ′ )X, and so K ≤ H. Then
G = NG (S ∩ K)K ≤ H, which is contrary to H being a proper subgroup of G. Thus K
has a unique minimal member K. 
In what follows we shall refer to the group K in the preceding lemma as the p-socle
of G.
14

Notation 3.3. Let (L, ∆, S) be a proper locality on F , and let T ≤ S be strongly closed
in F . Denote by A(F ) the set of all P ∈ F cr such that:
(1) NS (P ) ∈ Sylp (NL (P ), and
(2) either P = S or NL (P )/Op (NL (P )) is p-disconnected.
Let AT (F ) be the set of all P ∈ A(F ) such that also:
(3) P ∩ T is fully normalized in F .
Note that the condition (1) is equivalent to the statement that P is fully normalized in
F , and that condition (2) is equivalent to the statement that either P = S or OutF (P )
is p-disconnected. Thus the sets A(F ) and AT (F ) depend only on F and T .
Definition 3.4. An element g ∈ L is AT (F )-decomposable if there exists w ∈ D and a
sequence σ of members of AT (F ):
w = (g1 , · · · gn ),

σ = (P1 , · · · , Pn ),

such that the following hold.
(1) Sg = Sw and g = Π(w).
(2) Pi = Sgi for all i.
(3) Either Pi = S or gi ∈ O p (Ki ), where Ki is the p-socle of NL (Ri ).
We say also that (w, σ) is an AT (F )-decomposition of g if (1) through (3) hold.
Since condition (2) in 3.4 implies that the sequence σ is determined by w, there is
some redundancy in the definition. For that reason we shall also speak of the AT (F )decomposition w and its auxiliary sequence σ.
The following result is a version of the Alperin-Goldschmidt fusion theorem [Gold].
Theorem 3.5. Let (L, ∆, S) be a proper locality on F , and let T be strongly closed in
F . Then every element of L is AT (F )-decomposable.
Proof. Set A = AT (F ). The following point is then immediate from definition 3.4.
(1) Let u = (x1 , · · · , xk ) ∈ D with Su = SΠ(u) . Suppose that each xi is Adecomposable. Then Π(u) is A-decomposable.
Among all g ∈ L such that g is not A-decomposable, choose g with |Sg | as large as
possible, and set P = Sg . Then P 6= S, as otherwise w = (g) and σ = (S) provide an
A-decomposition for g. Set P ′ = P g .
As T is strongly closed in F , there exists an F -conjugate Q of P (and hence also of P ′ )
such that both Q and Q ∩ T are fully normalized in F , by II.1.15. As P, P ′ ∈ ∆, there
exist a, b ∈ L such that P a = Q, (P ′ )b = Q, and NS (P )a ≤ NS (Q) ≥ NS (P ′ )b . The
maximality of |P | implies that a and b are A-decomposable, and the same is then true of
a−1 and b−1 via the inverses of the words (and the reversals of the sequences of subgroups
of S) which yield A-decomposability for a and b. Set g ′ = a−1 gb and set M = NL (Q).
Then g ′ ∈ M , (a, g ′ , b−1 ) ∈ D via Q, and ag ′ b−1 = g. If g ′ has an A-decomposition then
so does g, by (1). Thus we may assume that g = g ′ , whence P = Q = P ′ .
15

Set R = NS (Q). Then R ∈ Sylp (M ) by II.2.1, so Op (M ) ≤ S, and then Op (M ) = Q.
Let K be a normal subgroup of M which is minimal subject to Q ≤ K and Q ∈
/ Sylp (K).
p
By the Frattini lemma we may write g = f h, where f ∈ O (K) and h ∈ NM (S∩K). Then
h is A-decomposable, and S(f,h) = Sg , so (1) implies that f has no A-decomposition.
Thus we may assume that g ∈ O p (K). If M/Q is p-disconnected then K is the p-socle
of M , so Q ∈ A, and ((g), (Q)) is a A-decomposition of g. Thus M/Q is p-connected.
Let Γ be the graph Γp (M/Q), and let R = R0 , · · · , Rm = Rg be a sequence of Sylow
p-subgroups of M such that (R0 /Q, · · · , Rm /Q) is a geodesic path in Γ from R/Q to
Rg /Q. We may assume that, among all g ∈ K having no A-decomposition, g has been
chosen so that the distance m from R/Q to Rg /Q in Γ is as small as possible. Then m 6= 0
−1
as R  Q. Suppose m = 1. Then R ∩ Rg properly contains Q, and then (R ∩ Rg )g is
a subgroup of Sg which properly contains Q, contrary to Sg = Q. Thus m ≥ 2.
Let d ∈ M such that (Rm−1 )d = Rm , and set h = gd−1 . Then Rh = Rm−1 , and the
minimality of m implies that there exists an A-decomposition u of h. Since g = hd we
have Q = S(h,d) , and then (1) implies that d has no A-decomposition. Then Sd = Q by
the maximality of |Sg | in our choice of g. Set e = hdh−1 . Conjugation by h sends the
pair (R, Re ) to (Rm−1 , Rm ), so R/Q is adjacent to Re /Q in Γ. As m > 1 there then
exists an A-decomposition v for e. Set w = u−1 ◦ v ◦ u. Then Π(w) = h−1 eh = d, and
then w is an A-decomposition for d since Sw ≥ Q = Sa . But we have already determined
that d has no A-decomposition, and this contradiction completes the proof. 
We shall assume the setup of 2.1 until introducing a variation on that setup in 2.9.
Thus, for now, (L, ∆, S) is a proper locality on F , N E L is a partial normal subgroup,
T = N ∩ S, and E = FT (N ).
Lemma 3.6. Let P ∈ A(F ), set M = NL (P ), and set U = P ∩ T . If P 6= S, let K be
the p-socle of M .
(a) If P ∈ AT (F ) then U F ⊆ E c , and U = Op (NE (U )).
(b) Either T ≤ P or CS (U ) ≤ P ; and if T  P then K ≤ NN (P )P .
Proof. Point (a) is a direct application of 2.6(b). If If P = S then U = T , and then (b)
holds vacuously. Thus we may assume that P 6= S, and hence that M is p-disconnected.
Set K = NN (P ). Then K E M , and [P, K] ≤ P ∩ K ≤ U .
Set D = NCS (U) (P ) and set X = CS (U )P . Then NX (P ) = DP . Set H = hDM i.
Then [U, H] = 1, so H ∩ K centralizes the chain P ≥ U ≥ 1. As M is of characteristic p,
II.2.7(c) yields H ∩ K ≤ P . Thus HKP/P is the direct product of HP/P with KP/P .
As M is p-disconnected, either HP/P or KP/P is a p′ -group, so either NT (P ) ≤ P or
D ≤ P . This yields the first of the statements in (b). Now suppose that T  P . Then
NT (P )  P , and hence K ≤ hNT (P )M iP ≤ NN (P )P . Thus (b) holds. 
Set H = NL (CS (T )T ). Then H is a subgroup of L, by 2.2(a).
Definition 3.7. Denote by ΣT (F ) the set of all products Op (L)U V such that:
· U = (P ∩ T )h for some P ∈ AT (F ) and some h ∈ H, and
· V ∈ CF (T )cr .
16

Recall from 1.5 that (NL (T ), ∆, S) is a proper locality on NF (T ), and that CL (T ) E
NL (T ). We shall often write LT for the locality NL (T ), and CT for CL (T ).
Lemma 3.8. Let X ∈ ΣT (F ), and write X = Op (L)U V as in the preceding definition.
Then U = X ∩ T and V = CX (T ). Moreover, ΣT (F ) depends only on F and T , and not
on L.
Proof. Notice first of all that Op (L)T ≤ Op (LT ). As LT is a proper locality on NF (T )
we have Op (LT ) = Op (NF (T )), and then
Op (L)T ∩ CT ≤ Op (NF (T )) ∩ CT ≤ Op (FCS (T ) (CT )) ≤ Op (CF (T )),
since FCS (T ) (CT ) is a subsystem of CF (T ). Then
CX (T ) = COp (L)U (T )V ≤ Op (CF (T ))V = V,

(*)

since V ∈ CF (T )cr .
Let P ∈ AT (F ) and let h ∈ H with U = (P ∩ T )h . If T ≤ P then U = T = X ∩ T .
So assume that T  P . Then CS (T ) ≤ P by 3.6. As P ∈ F cr we have also Op (L) ≤ P .
Then
−1
−1
X h = Op (L)(P ∩ T )V h ≤ P,
−1

so X h ∩ T = P ∩ T , and X ∩ T = U .
Since Op (L) = Op (F ), and since H-conjugation on subgroups of T is the same as
AutF (CS (T )T )-conjugation (2.2(c)), FT (F ) depends only on F and T . 
Lemma 3.9. Assume ΣT (F ) ⊆ ∆. Then:
(a) FCS (T ) (CT ) = CF (T ).
(b) The image of H under the natural homomorphism H → Aut(CS (T )) is contained
in Aut(CF (T )).
(c) Every member of ΣCS (T ) (NF (T )) contains a member of ST (F ). In particular,
ΣCS (T ) (NF (T )) ⊆ ∆.
Proof. Set Ψ = {Y ≤ CS (T ) | Op (L)T Y ∈ ∆}. Since ΣT (F ) ⊆ ∆ we have CF (T )cr ⊆ Ψ.
We may then appeal to 1.4(c), with Op (L)T in the role of V and with Ψ in the role of
Σ. Thus (CT , Ψ) is a localizable pair and (CT )Ψ , Ψ, CS (T )) is a proper locality on CF (T ).
This yields (a).
Set FT = NF (T ) and let ∆0 be the FT -closure of ΣT (F ). Let R ∈ (FT )cr . As
FT = FS (LT ), FT is inductive by 6.1. By II.1.15 there is then an FT -conjugate R′ of
R such that R′ ∩ CS (T ) is fully normalized in FT . Then R′ ∩ CS (T ) ∈ CF (T )cr by
2.6(b). Since T ≤ R′ , we have thus shown that (FT )cr ⊆ ∆0 . Since ∆0 ⊆ ∆, we have the
restriction L0 of L to ∆0 (II.1.10), and plainly H is a subgroup of L0 . Then 2.3 applies
with (CT )Ψ in the role of N , and yields the desired action of H on CF (T ). That is, (b)
holds.
Set T ∗ = CS (CS (T )). By definition 3.8, ΣCS (T ) (LT ) is the set of products Op (LT )U V
such that:
(1) U = (Q ∩ CS (T ))h for some Q ∈ ACS (T ) (LT ) and some h ∈ NLT (T ∗ ), and
(2) V ∈ CFT (CS (T ))cr .
17

Let h, U , and V be as in (1) and (2). Then h ∈ H := NL (T ), and (2) yields T ≤ V . Set
U ′ = Q ∩ CS (T ). Then U ′ ∈ CF (T )cr by 2.6(b), and then (b) yields U ∈ CF (T )cr . Then
Op (L)T U ∈ ΣT (F ). Since T U ≤ V U we then obtain (c). 
Lemma 3.10. Let U ∈ E c and let V be a subgroup of CS (T ), such that Op (L)U V ∈ ∆.
Then NN (CS (T )U ) and NCT (V ) are subgroups of the group NL (U V ), and
[NN (CS (T )U ), NCT (V )] = [NN (U V ), O p (NCT (U V )] = 1.
Proof. Set R = U V and set M ∗ = NL (Op (L)R), M = NL (R), K = NN (R), K0 =
NN (CS (T )U ), and X = NCT (R). As Op (L)R ∈ ∆, M ∗ is a subgroup of L, and M =
NM ∗ (R) is a subgroup of L. Moreover, M and K are of characteristic p by II.2.6.
We have [CS (T ), K0 ] ≤ CS (T ) ∩ N = Z(T ) ≤ U , as U ∈ E c . Thus K0 ≤ K, while
clearly NCT (V ) ≤ X. As K is of characteristic p and U ∈ E c we have CK (U ) ≤ U . Then
[K, X] ≤ U , and so [K, O p (X)] = 1 by coprime action. 
Definition 3.11. Let g ∈ L, and let w = (a, g1 , · · · , gn ) ∈ D. Then w is an N decomposition of g if:
(1) Sw = Sg ,
(2) Π(w) = g,
(3) a ∈ NL (T ), and
(4) there exists a sequence (X1 , · · · , Xn ) of members of ΣT (F )∩∆ such that CS (T ) ≤
Xi and such that
′

gi ∈ O p (NN (Xi )) ∩ O p (NN (Xi )) (1 ≤ i ≤ n).
A word of caution regarding the above definition: Even though CS (T ) ≤ Xi for all i,
it cannot be concluded that CS (T ) ≤ Sw . Indeed, there are examples to the contrary.
The point is that CS (T ) need not be invariant under gi .
The following result may be thought of as a refinement of the splitting Lemma (I.3.12).
Theorem 3.12. Assume ΣT (F ) ⊆ ∆. Then each g ∈ L has an N -decomposition.
Moreover, if g ∈ N , and w = (a, g1, · · · , gn ) is an N -decomposition of g, then a ∈
NN (CS (T )T ).
e = {X ∈ ΣT (F ) | CS (T ) ≤ X}. Let g ∈ L, and let Φ be the set of all
Proof. Set Σ
words v = (x1 , · · · , xn ) ∈ W(L) such that Sw = Sg , Π(w) = g, and having the following
property.
(*) For each i, one of the following holds.
(i) xi ∈ H, and Sxi ∈ A(NF (T )).
(ii) CS (T )  Sxi ∈ A(NF (T )), and xi ∈ O p (Ki ) where Ki is the p-socle of
NL (Sxi ).
′
e
(iii) T  Sxi , and xi ∈ O p (NN (Xi )) ∩ O p (NN (Xi )) for some Xi ∈ Σ.
18

We shall see, first of all, that Φ contains the set of all AT (F )-decompositions of g. Let
u = (x1 , · · · , xn ) be an arbitrary such. By 3.6, (i) or (ii) holds for any index j such that
Sxj ∈ A(LT ). Let then i be an index such that Sxi ∈
/ A(LT ). Set x = xi , P = Sxi ,
p′
and let K be the p-socle of NL (P ). Then K = O (K) by definition, and K ≤ NN (P )P
e and D ≤ P (again by 3.6(b)). Then
by 3.6(b). Set D = (P ∩ T )CS (T ). Then D ∈ Σ,
D E K since [P, NN (P )] ≤ P ∩ T . Since x ∈ O p (K) by definition 3.4, we obtain (iii),
and thus u ∈ Φ.
Now let v be an arbitrary member of Φ, and suppose that there is segment (c, b) of v
e such that c ∈ NL (X) (and such that c and X
such that T  Sc and T ≤ Sb . Let X ∈ Σ
in the role of xi and Xi satisfy also the stronger condition given by (iii) in (*)).
Set Q = Sb , and suppose first that CS (T )  Q. Then b ∈ O p (L), where L is the
p-socle of NL (Q). Set E = NCS (T ) (Q). Then E  Q, so 2.2 yields L ≤ hE NL (Q) i,
e such that c ∈ NL (X) (and such that c
and hence L ≤ CL (T ) as T E L. Let X ∈ Σ
and X in the role of xi and Xi satisfy also the stronger condition given by (iii)). We
have CQ (T ) ∈ CF (T )cr by 2.6(b), so [K, O p (L)] = 1 by 3.10. Thus c and b commute.
Since c ∈ N and b ∈ NL (T ) we have S(c,b) = S(b,cb ) by I.3.2(a), and so S(c,b) = S(b,c) .
Write v = v1 ◦ (c, b) ◦ v2 , and set v ′ = v1 ◦ (b, c) ◦ v2 . Then Sv = Sv ′ (so v ′ ∈ D), and
Π(v ′ ) = Π(v). The condition (*) is evidently in place for v ′ , so v ′ ∈ Φ.
Suppose next that CS (T ) ≤ Q. Then CS (T )T E NL (Q), and so b ∈ H. We have
e T (F ), and X ∈ ∆ by hypothesis. Then
X = (X ∩ T )CS (T ) by the definition of Σ
−1
(b, b , c, b) ∈ D via X, and S(c,b) = S(b,cb ) as before. Conjugation by b is an isomorphism
′
from NL (X) to NL (X b ) (I.2.3(b)), so cb ∈ O p (NL (X b )) and cb ∈ O p (NL (X b )). Moreover
e since Σ
e is H-invariant. With v defined as in the preceding case, set
we have X b ∈ Σ,
v ′ = v1 ◦ (b, cb ) ◦ v2 . We conclude that v ′ ∈ Φ.
It follows from the preceding analysis that there exists v ∈ Φ such that v can be
written as v = v1 ◦ v2 , where (i) or (ii) holds for each entry of v1 , and where (iii) holds for
each entry of v2 . As v1 is a prefix of a member of D we have v1 ∈ D. Set set a = Π(v1 ).
Then a ∈ LT , and set w = (a) ◦ v2 . Then Π(w) = g, and since Sg ≤ Sw ≤ Sv = Sg we
obtain Sg = Sw , and w is an N -decomposition of g.
Suppose finally that g ∈ N . Then a ∈ N by cancellation (I.1.4(e)), and so a ∈ NN (T ).
As L is proper, it follows from I.3.5 that NN (T ) ≤ H. Thus a ∈ H, and the proof is
complete. 
Corollary 3.13. Assume that ΣT (F ) ⊆ ∆, and let Γ be the set of subgroups U of T of
the form (P ∩T )h , with P ∈ AT (F ) and with h ∈ H. Then Γ ⊆ E c , and E is Γ-generated.
Proof. The existence of an N -decomposition for g ∈ N shows that the conjugation map
cg : Sg ∩ T → (Sg ∩ T )g is a composition of E-homomorphisms between members of Γ.
Thus E is Γ-generated. That Γ is a subset of E c is given by 2.6(b). 
The following corollary to 3.12 answers a question that was left hanging from Part II.
Namely, in Theorem A (the union of Theorems A1 and A2) one has invariance of the
poset of partial normal subgroups of L under expansion of objects from ∆ to ∆+ , but
nothing is said about what becomes of the fusion systems of the various partial normal
19

subgroups of L under this process. One is told only (in the setup of Theorem A2) that
if N E L with T = S ∩ N , then also T = S ∩ N + .
Corollary 3.14. Assume that ΣT (F ) ⊆ ∆, and let ∆+ be an F -closed set of subgroups
of S such that F cr ⊆ ∆ ⊆ ∆+ ⊆ F s . Let (L+ , ∆+ , S) be the unique (in the sense of
Theorem I.A1) proper locality on F which contains L and such that the inclusion map
L → L+ is a homomorphism of partial groups. Let N + be the partial normal subgroup
of L+ (as in Theorem II.A2) generated as a partial subgroup of L+ by the set of all
L+ -conjugates of elements of N . Then:
(a) FT (N ) = FT (N + ), and
(b) N + is the partial subgroup hN i of L+ generated by N .
Proof. By 3.12, E is generated by the set of conjugation maps cg : Sg ∩ T → T such that
e T (F ). The same is true of E + , so (a)
g ∈ N and such that Sg contains a member of Σ
holds.
Let g ∈ N + , and let w be an N + -decomposition of g. Then w is a sequence of elements
of N , and so N + ≤ hN i. The reverse inclusion is immediate (as N + is a partial subgroup
of L+ ). Thus (b) holds. 
Lemma 3.15. Suppose that Op (L)E c ⊆ ∆. Then:
(a) ΣT (F ) ⊆ ∆.
(b) The image of the natural homomorphism LT → Aut(T ) is contained in Aut(E).
(c) (N , E c) is a localizable pair, and ((N )E c , E c , T ) is a proper locality on E.
Proof. Let P ∈ AT (F ), and set U = P ∩ T . Then U H ⊆ E c by 2.6(b), and then
Op (L)U H CF (T )cr ⊆ ∆ by hypothesis. This establishes (a). Set Γ = E c . Then 3.12
shows that E is Γ-generated.
Let V ∈ ∆ with P ≤ T , set M = NL (V ), and set K = NN (V ). Then K is of
characteristic p by II.2.6(a), and then U ∈ E c if and only if CK (V ) = Z(V ). This shows
that E c is F -invariant.
As Γ ⊆ ∆, and since T is maximal in the poset of p-subgroups of N , it is immediate
that (N , Γ) is a localizable pair. Then (NΓ , Γ, T ) is a locality on a fusion subsystem E0
of E, by 1.2. As E is Γ-generated we get E0 = E, and then (c) follows from 1.3. Point (b)
is given by 2.3. 
We end this section with an application of 3.5.
Theorem 3.16. Let (L, ∆, S) be a proper locality on F , and let K be the set
T of all partial
normal subgroups K E L such that L/K is an abelian group. Set [L, L] = K. Then:
(a) [L, L] = h[NL (P ), NL (P )] | P ∈ A(F )i ∈ K.
(b) CS ([L, L]) E L.
(c) Let ∆+ be an F -closed subset of F s containing ∆, and let L+ be the expansion of
L to ∆+ given by Theorem II.A1, and let [L, L]+ be the expansion of [L, L] to a
partial normal subgroup of L+ given by Theorem II.A2. Then [L, L]+ = [L+ , L+ ].
20

Proof. For brevity, write L′ for [L, L], and for any group G S
is write G′ for [G, G]. By
I.1.9 L′ is the union of its subsets L′k (k ≥ 0), where L′0 = {NL (P )′ | P ∈ ∆}, and
where L′k+1 is the set of all products Π(w) with w ∈ W(L′k ) ∩ D.
Let Y be the set of all g ∈ L for which there exists x ∈ L′ such that xg is defined and
g
x ∈
/ L′ . Assume Y 6= ∅ and choose g ∈ Y so that |Sg | as large as possible, and then so
that the minimal length of an A(L)-decomposition for g is as small as possible. Suppose
first that Sg = S, and let k be the least index for which there exists x ∈ L′k and g ∈ NL (S)
with xg ∈
/ L′k . Then k > 0 since, by I.2.3(b), conjugation by g induces an isomorphism
NL (P ) → NL (P g ) for any P ∈ ∆. Thus x = Π(w) for some w ∈ W(L′k−1 ) ∩ D. Set
w = (x1 , · · · , xn ) and set wg = (g −1 , x1 , g, · · · , g −1 , xn , g). Then wg ∈ D via (Sw )g , and
so xg = Π(wg ) = xg1 · · · xgn ∈ L′k . This contradiction (to g ∈ Y ) shows that Sg 6= S.
Let v be an A(L)-decomposition for g of minimal length. Then Sv = Sg , and hence
v −1 ◦ (x) ◦ v ∈ D via S(g −1 ,x,g) . The minimality of the length of v then implies that
v = (g) is of length 1, and thus g ∈ NL (Q) where Q = Sg ∈ A(L). Set H = NL (Q) and
R = NS (Q). Then g = f h for some f ∈ H ′ and some h ∈ NH (R), and we have Q = S(f,h) .
Set u = (h−1 , f −1 , x, f, h). Then Su = S(g −1 ,x,g) ≤ Q, and xg = Π(u) = (xf )h . Here
xf ∈ L′ since L′ is a partial group, and then (xf )h ∈ L′ since the maximality of Sg in
the choice of g yields h ∈
/ Y . Thus Y = ∅, which is to say that L′ E L.
The set ∆ of objects of the quotient locality L = L/L′ is the set of all P = P/(S ∩ L′ )
such that S ∩ L′ ≤ P ∈ ∆. For any such P we have NL (P ) ∼
= NL (P )/NL′ (P ), and
thus NL (P ) is abelian. Set Θ(P ) = Op′ (NL (P )). Then Θ(P ) centralizes NS (P ), and a
straightforward argument by induction shows that Θ(P ) ≤ Θ(S) for all P ∈ ∆. Thus
L = NL (S) is the group S×Θ(S). Since S E L, the “correspondence theorem” I.4.5 yields
L′ S E L, and then L = L′ NL (S) by the Frattini Lemma (I.3.11). Since CS (L′ ) E NL (S)
and CS (L′ ) E L′ S, it follows from I.3.13 that CS (L′ ) E L. Thus (a) holds.
Now let N E L be a partial normal subgroup of L such that L/N is an abelian group,
and let ρ : L → L/N be the canonical projection (see I.4.4). Then ρ maps subgroups
of L to subgroups of L/N by I.4.2; hence NL (P )′ ≤ N for all P ∈ ∆. Thus L′ ≤ N ,
completing the proof of (b).
Let ∆+ and L+ be given as in (c). For any partial normal subgroup N E L let N + be
the unique partial normal subgroup of L+ which intersects L in N , as given by Theorem
II.A2. In particular, taking N = (L+ )′ ∩ L, we have N + = (L+ )′ . Let ρ+ : L+ →
L+ /(L+ )′ be the canonical projection. Then N is the kernel of the restriction ρ of ρ+ to
L. Since Im(ρ+ ) is an abelian group, it follows that L′ ≤ N , and so (L′ )+ ≤ (L+ )′ . On
the other hand, we have L/L′ ∼
= L+ /(L′ )+ by II.5.6(a). As L/L′ is an abelian group we
+ ′
′ +
thereby obtain (L ) ≤ (L ) , which completes the proof of (c). 

Section 4: ΣT (F )
The preceding section indicates that the set ΣT (F ) introduced in 3.7 plays an important role, if it so happens that ΣT (F ) ⊆ ∆. But the only indication that has been given
so far, as to when this condition is met is the one given by lemma 3.15: in which there
21

is the rather strong assumption that Op (L)E c ⊆ ∆. One aim of this section is to show
that one indeed has ΣT (F ) ⊆ ∆ provided only that ∆ is large enough. For example,
it will suffice that ∆ be as large as possible, i.e. that ∆ be the set F s of F -subcentric
subgroups of S.
We continue the setup of 2.1. Thus (L, ∆, S) is a proper locality on F , N E L is a
fixed partial normal subgroup of L, T = S ∩ N , and E = FT (N ). We also continue the
notation: H = NL (CS (T )T ) (a subgroup of L by 2.2), LT = NL (T ), and CT = CL (T ).
Recall from 1.5 that (LT , ∆, S) is a proper locality on NF (T ), and that CT E LT .
Lemma 4.1. Assume that S = CS (T )T , and assume that F c ⊆ ∆.
(a) U V ∈ F c for all (U, V ) ∈ E c × CF (T )c .
(b) Let V be a subgroup of CS (T ) containing Z(T ), and with T V ∈ F c . Then V ∈
CF (T )c .
Proof. First, let U ≤ T be a subgroup of T , and let U ′ ∈ U F . Thus there exists
w = (g1 , · · · , gn ) ∈ W(L), and a sequence (U0 , · · · , Un ) of F -conjugates of U such that
U0 = U , Un = U ′ , and Uk = (Uk−1 )gk for all k with 1 ≤ k ≤ n. Write gk = xk yk where
xk ∈ N and yk ∈ CL (T ). We have NN (T ) ≤ NL (CS (T )T ) by I.3.5, and S(xk ,yk ) = Sgk
by I.3.10 and I.3.12. It follows that Uk = (Uk−1 )xk , and thus:
(1) U F = U E .
Assume now that (a) is false, and among all (U, V ) ∈ E c × CF (T )c with U V ∈
/ F c choose
(U, V ) so that |U ||V | is as large as possible. Set R = U V . Notice that NT (U ) ≤ NS (R)
and that NCS (T ) (V ) ≤ NS (R). As S = CS (T )T we have NS (R) = NT (U )NCS (T ) (V ), so
the maximality of |U ||V | yields NS (R) ∈ F c .
By II.1.15 there exists an F -conjugate R′ of R such that both R′ and R′ ∩ T are fully
normalized in F . There is then an F -homomorphism λ : NS (R) → NS (R′ ) such that
R′ = Rλ. As NS (R) ∈ ∆, λ is given by conjugation by some g ∈ L. We may employ
I.3.12 in order to write g = f y, with f ∈ NL (T ), y ∈ N , and with Sg = S(f,y) . Set
U1 = U f and V1 = V f . Then U1 ∈ U E by (1), and so U1 ∈ E c . Applying I.3.12 also to
LT and its partial normal subgroup CT , we obtain f = hx where h ∈ H, x ∈ CT , and
Sf = S(h,x) . Here H = NL (S) since S = CS (T )T , so conjugation by h is defined on all
of L, and then 2.3 shows that conjugation by h preserves CF (T )c . Thus V1 ∈ CF (T )c ,
and we may therefore assume that U = U1 and V = V1 , and that g = y. Then I.3.1(b)
yields:
(2) V g ≤ V T .
Set U ′ = U y and V ′ = V y . We now compute:
CS (R′ ) = CS (U ′ V ′ ) = CS (V ′ ) ∩ CS (U ′ ) = CS (V ′ ) ∩ CCS (T )T (U ′ )
= CS (V ′ ) ∩ CS (T )Z(U ′ ) (as U ′ is centric in E)
= CC (T ) (V ′ )Z(U ′ ) (as [U ′ , V ′ ] ∼
= [U, V ] = 1).
S

= CCS (T ) (V ′ T )Z(U ′ ) = CCS (T ) (V T )Z(U ′ ) (by (2))
= Z(V )Z(U ′ ).
22

Thus, in order to show that R′ , and hence R, is centric in F it suffices to show that
Z(V ) ≤ R′ . Since CS (R) = Z(V )Z(U ), it then suffices to show that |Z(V )Z(U ′ )| =
|Z(V )Z(U )|. As V ≤ CS (T ), and since U, U ′ ∈ E c we have:
Z(V ) ∩ Z(U ′ ) = Z(V ) ∩ U ′ = Z(V ) ∩ T = Z(V ) ∩ U = Z(V ) ∩ Z(U ).
Thus |Z(V )∩Z(U ′ )| = |Z(V )∩Z(U )|, and hence |Z(V )Z(U ′ )| = |Z(V )Z(U )|, as required.
Thus R ∈ F c , and this contradiction completes the proof of (a).
Now let V be a subgroup of CS (T ) containing Z(T ), and with T V ∈ F c , and suppose
that V ∈
/ CF (T )c . Let V ′ be a CF (T )-conjugate of V , with V ′ fully centralized in CF (T ).
Then CCS (T ) (V ′ )  V ′ by II.1.10. That is, we have CS (T V ′ )  V ′ . On the other hand,
since T V ′ ∩ T = T and T V ′ ∩ CS (T ) = Z(T )V ′ = V ′ , we have
CS (T V ′ ) ∩ T V ′ = Z(T V ′ ) = Z(T )Z(V ′ ) ≤ V ′ .
This shows that CS (T V ′ )  T V ′ , and so T V ′ ∈
/ F c . But CF (T ) = FCS (T ) (CT ) by 1.5(b),
and thus T V ′ is an F -conjugate of T V via the same sequence of conjugation maps by
elements of CT that sends V to V ′ . As T V ∈ F c and T V ′ ∈
/ F c , we have the contradiction
which proves (b). 
Corollary 4.2. If S = CS (T )T and F c ⊆ ∆, then ΣT (F ) ⊆ ∆.
Proof. Let X = Op (L)U V ∈ ΣT (F ), with U and V given as in definition 3.7. Then
U ∈ E c by 2.6(b), while V ∈ CF (T )c by definition. The preceding lemma then yields
U V ∈ F c , and hence X ∈ ∆. 
Lemma 4.3. Assume ΣT (F ) ⊆ ∆ and assume T ∈ F q .
(a) If ∆ = F s then (N , E c ) is a localizable pair, and (NE c , E c , T ) is a proper locality
on E.
(b) If ∆ = F s then the image of the natural homomorphism λ : LT → Aut(T ) is
contained in Aut(E).
(c) E c ⊆ F q , and E s ⊆ F s .
(d) E is (cr)-generated.
Proof. If points (c) and (d) hold in the case that ∆ = F s then they hold in general, by
3.14. Thus, we may assume throughout that ∆ = F s .
Set U c = {U ∈ E c | U F ⊆ E c }, and set U s = {U ∈ E s | U F ⊆ E s }. The strategy of the
proof is to first establish all three parts of the lemma under the following assumption.
(1) U c = E c ⊆ ∆.
Assume (1). Then 3.15 yields (a) and (b), and (d) then follows from (a) and from II.2.10.
Further, (b) implies that U s = E s . Pick U ∈ E s . In order to show that U ∈ F s we may
assume that U is fully normalized in F . Set X = Op (NE (U )). Then X ∈ E c , and so
X ∈ F s (and X ∈ ∆) by (1). Further, II.1.16 implies that X is fully centralized in F .
Set M = NL (X) and K = CL (X). Then K is of characteristic p, and X = Op (K) by
II.2.3.
23

Set Y = Op (NF (U )). Then X E Y , and then Y = Op (NM (U )). As X ≤ Y we may
compute:
CL (Y ) = CM (Y ) = CNM (U) (Y ).
As NM (U ) is of characteristic p by II.2.6(b), we conclude that CL (Y ) ≤ Y . Then Y ∈ F c
by II.2.8(b), and U ∈ F s . Thus E s ⊆ F s .
It now suffices to prove the following stronger version of (1).
(2a) U c ⊆ F q , and
(2b) U c = E c .
Assume that (2a) is false, and choose U ∈ U c − F q with |U | as large as possible. As
U ∈ U c we may assume that U is fully normalized in F . Set Γ = {X ∈ ∆ | U E X}
and Σ = {Y ≤ CS (U ) | U Y ∈ ∆}. By 1.4 (and since F c ⊆ ∆) both (NL (U ), Γ) and
(CL (U ), Σ) are localizable pairs. Write LU for NL (U )Γ , and CU for CL (U )Σ . Then, by
1.4, (LU , Γ, NS (U )) is a proper locality on NF (U ), (CU , Σ, CS (U )) is a proper locality on
CF (U ), and Evidently CU = CLU (U ), and so CU E LU .
Set B = NT (U ). Thus U is properly contained in B, and so B ∈ F q . Let V ∈ CF (U )cr
with V fully normalized in CF (U ), and set D = NCU (P ). Then V ∈ ∆ by 1.4(a), so D
is a subgroup of CU , and D = NCU (U V ). Set K = N ∩ D. Then B ≤ K E D, and then
(*)

[B, D] ≤ CK (U ) = Z(U ) × Op′ (K),

since B ∈ E c . As LU is a proper locality, NLU (U V ) is of characteristic p, so also D and K
are of characteristic p by II.2.6. Thus Op′ (K) = 1, and so (*) yields [B, D] ≤ Z(U ). Then
[B, O p (D)] = 1 by II.2.6(c). As B ∈ F q we conclude that O p (D) = 1. As V ∈ CF (V )cr
it follows that V = CS (U ) = CU . Then CF (U ) is the trivial fusion system on CS (U ),
and hence U ∈ F q . This completes the proof of (2a).
Finally, assume (2b) to be false, and among all U ∈ E c − U c , choose U so that |U |
is as large as possible. Then U 6= T , so U is a proper subgroup of NT (U ), and then
NT (U ) ∈ F q by (2a). Thus NT (U ) ∈ ∆, and we may then argue - in a perhaps familiar
way - as follows. Let U ′ be a fully normalized F -conjugate of U , and let φ : NS (U ) → S be
an F -homomorphism such that U φ = U ′ . Then φ is given by conjugation by an element
g ∈ L (as NT (U ) ∈ ∆), and g = xh for some x ∈ N and h ∈ LT with Sg = S(x,h)
(I.2.12). As E c is both E-invariant and LT -invariant, we then have U ′ ∈ E c , and a similar
argument then show that (U ′ )F ⊆ E c . Thus U ∈ U c after all. This completes the proof
of (2b), and thereby proves the lemma. 
Remark. It appears that point (c) of the preceding lemma cannot be improved upon
in any obvious way. For example, E cr need not be contained in F c , even if T ∈ F c .
For example, let G be a semi-direct product V ⋊ H where H = GL3 (2) and where V is
elementary abelian of order 16, with CV (H) = 1. Then G may be viewed as a proper
locality whose objects are the overgroups of V in a Sylow 2-subgroup S of G. Let N be
the subgroup of G of index 2 in G, and set T = S ∩ N . Then T is centric in FS (G), and
V ∩ T is centric radical in FT (N ); but V ∩ T is not centric in FS (G).
24

Our aim now is to use 4.1 and 4.3 to show that ΣT (F ) ⊆ F q if ∆ = F s . At the same
time, we want to obtain information about the special case where CS (N ) is contained in
N . The next result (a corollary of 3.11) prepares the way for these goals.
e T (F ) be the set of all X ∈ ΣT (F ) such
Lemma 4.4. Assume ΣT (F ) ⊆ ∆, and let Σ
that CS (T ) ≤ X. Then
(*)

CS (N ) =

\
e T (F )}.
{CS (NN (X)) | X ∈ Σ

Proof. Let R be the right-hand intersection in (*). Then CS (N ) ≤ R, so it remains to
prove the reverse inclusion. Let f ∈ N and let g ∈ R. As ΣT (F ) ⊆ ∆ by hypothesis,
there exists an N -decomposition w = (a, f1 , · · · , fn ) of f , and a ∈ N ∩ H, by 3.12. Set
w′ = (g −1 , a, g, g −1, f1 , g, · · · , g −1 , fn , g). Then w′ ∈ D via (Sw )g . By definition 3.11,
e T (F ), so (fi )g = fi . As CS (T )T ∈ Σ
e T (F ), also ag = a.
each fi normalizes a member of Σ
Thus f g = Π(w′ ) = Π(w) = g, and so g ∈ CS (N ). Thus R ≤ CS (N ), and the reverse
inclusion is obvious. 
Proposition 4.5. Assume ΣT (F ) ⊆ ∆, and assume that CS (N ) ≤ N . Set Γ = E s .
(a) E c ⊆ F q , and ΣT (F ) ⊆ F q .
(b) If ∆ = F s , then (N , E s ) is a localizable pair, and (NΓ , Γ, T ) is a proper locality
on E.
(c) If ∆ = E s then LT is a subgroup of L, and the image of LT in Aut(T ) is a
subgroup of Aut(NΓ ).
(d) If ∆ = F s then Op (E) = Op (NΓ ) = Op (N ) E L.
Proof. It follows from 3.14 that if (a) holds if it holds under the assumption that ∆ = F s .
We may therefore assume throughout that ∆ = F s .
The proof will have two parts. In the first we assume:
(*) ΣT (F ) ⊆ F q .
Given (*), we will show that (a) through (d) hold. Once that has been achieved, we will
then be able to show that the hypothesis (*) is redundant, and to thereby complete the
proof.
e T (F ), and set U = X ∩ T . Then X =
Assume (*). Define wΣT (F ) as in 4.4, let X ∈ Σ
cr
Op (L)CS (T )U . Let V ∈ CF (T ) with V fully normalized in CF (T ). Then Op (L)T V ∈
ΣT (F ), so Op (L)T V ∈ ∆. Then NCF (T ) (V ) is the fusion system of NCT (T V ) over
CCS (T ) (V ), by II.2.2. As L is proper, NL (T V ) is of characteristic p. Since NL (T V ) =
NLT (V ), and since CT E LT , the group K := O p (NCT (V )) is of characteristic p by
II.2.6(a). Set V0 = V ∩ K. Then V0 ≤ CS (N ), by 3.9 and 4.4. As CS (N ) ≤ T by
hypothesis, we then have V0 ≤ Z(K). Then [V, K] = 1 by II.2.6(c), and then K = 1
since V is centric in CF (T ). As V is also radical in CF (T ) we conclude that V = CS (T ).
Thus CF (T ) is the trivial fusion system on CS (T ). That is, we have T ∈ F q . Point (a)
is then given by 4.3(c) and by (*). Points (b) and (c) are given by the relevant points in
4.3, in conjunction with expansion from E c to E s via Theorem II.A.
25

Next, it follows from (b) and II.2.3 that Op (E) = Op (NΓ ), and from (c) that Op (E) is
H-invariant. As NΓ = N ∩ L0 we have NΓ E L0 . As L0 = N LT we obtain Op (E) E L0 ,
and then Op (E) E L by Theorem II.A2. This establishes (d). It now remains to remove
the hypothesis (*).
Let M be the partial subgroup hCT N i of L. Then M E L and S ∩ M = CS (T )T ,
by I.5.5. Set Te = CS (T )T and set D = FTe (M). Then Te ∈ F c , so 4.3(a) implies that
(M, D c ) is a localizable pair and that MDc is a proper locality on D. Then ΣT (F ) ⊆ D c
by 4.1(a). with M in the role of L. Since D c ⊆ F q by 4.3(c) we thereby obtain
ΣT (L) ⊆ F q . Thus (*) holds, and the proof is complete. 
Corollary 4.6. Assume the setup of 2.1. Then ΣT (F ) ⊆ F q .
Proof. By 3.8 the definition of ΣT (F ) depends only on F and on the strongly closed
subgroup T , and not on ∆. By Theorem II.A we may therefore assume that ∆ = F s . In
this case the proof is given by repeating - verbatim - the final paragraph in the proof of
4.5. 
Corollary 4.7. Assume the setup of 2.1, and assume that T is abelian. Then T E N ,
and N is a subgroup of NL (CS (T )T ). Moreover, if T ≤ Z(N ) then N = T .
Proof. As ΣT (F )F q ⊆ F s there exists an F -closed set ∆+ of subgroups of S such that
∆ ∪ ΣT (F ) ⊆ ∆+ ⊆ F s . Let L+ be the expansion of L to ∆+ via Theorem II.A1,
and let N + be the partial normal subgroup of L+ corresponding to N via Theorem
II.A2. Let g ∈ N . Then g ∈ N + , so g has an N + -decomposition w = (a, g1, · · · , gn ).
For each index i (if any) with 1 ≤ i ≤ n there exists Xi ∈ ΣT (F ) such that T  Xi ,
NT (Xi ) ∈ Sylp (NN (Xi )), and gi ∈ O p (NN (Xi )). As NT (Xi ) is abelian, these conditions
imply that O p (NN (Xi )) = 1, and so w = (a). Thus g ∈ NN (CS (T )T by 3.12.
Now N is a subgroup of the group H := NL (CS (T )T ). Assume that T ≤ Z(N ). Then
N = T × Op′ (N ), and then Op′ (N ) = 1 since H is of characteristic p. Thus N = T in
this case. 
Recall (cf. 2.7) that Z(N ) is the set of all z ∈ N such that g z is defined and is equal
to f for all g ∈ N .
Lemma 4.8. Suppose ΣT (F ) ⊆ ∆. Then CS (N ) E H, and Z(N ) = CT (N ) E L.
e T (F ) as in 4.4. Thus, each X ∈ Σ
e is of the form Op (L)CS (T )U where
Proof. Define Σ
e T (F ) is
U = (P ∩ T )h for some P ∈ AT (F ) and some h ∈ H. As CS (T ) E H, Σ
H-invariant. Since 4.4(*) can be expressed as
\
e T (F )}),
CS (N ) = CS (T ) ∩ ( {CS (NN (X)) | X ∈ Σ
it follows that CS (N ) E H. As Z(N ) = CT (N ) by 2.7, we obtain
Z(N ) = T ∩ CS (N ) E H.
As CT (N ) E CT , we may apply I.3.13 with LT and CT in the roles of L and N to obtain
Z(N ) E LT . Then apply I.3.13 to L and N to obtain Z(N ) E L. 
26

Section 5. N ⊥
We continue the setup and the notation of the preceding sections. Thus (L, ∆, S) is a
proper locality on F , N is a partial normal subgroup of L, T = S ∩ N , and E = FT (N ).
Further, we have the abbreviations LT = NL (T ), CT = CL (T ), and H = NL (CS (T )T ).
The collection ΣT (F ) of subgroups of S, defined in 3.7, will continue to play a key role.
In this section we will be taking a roundabout path to the structure of L and N
by way of the structure of LT and CT . Recall from 2.2 that H is a subgroup of L,
and from 1.5 that (LT , ∆, S) is a proper locality on NF (T ), that CT E LT , and that
CF (T ) = FCS (T ) (CT ) if ΣT (F ) ⊆ ∆.
Theorem 5.1. Assume Op (L)E cr ⊆ ∆, and assume that E is (cr)-generated. Let λ be
a mapping which assigns to each U ∈ E cr a group λ(U ), with
λ(U ) E NL (U ) and λ(U ) ≤ NN (U ).
Set M(λ) = hλ(U ) | U ∈ E cr i. Then λ(U )h and λ(U h ) are defined for all h ∈ LT , and
M (λ) E L if the following two conditions hold for all U ∈ E cr .
′

(1) O p (NN (U )) ∩ O p (NN (U )) ≤ λ(U ), and
(2) λ(U )h = λ(U h ) for all h ∈ LT .
Proof. Let Γ be the overgroup closure of E cr in T . Then Op (L)Γ ⊆ ∆, and by 2.8
shows that (N , Γ) is a localizable pair, that (NΓ , Γ, T ) is a proper locality on E, LT
is a subgroup of L which acts on NΓ by conjugation, and the image of the natural
homomorphism LT → Aut(T ) is contained in Aut(E). In particular, λ(U )h and λ(U h )
are defined for U ∈ E cr and h ∈ LT .
Let ∆0 be the overgroup closure of Op (L)Γ in S. Then F cr ⊆ ∆0 ⊆ ∆ by 2.6(b),
the restriction (L0 , ∆0 , S) of L to ∆0 is a proper locality on F , and NΓ = N ∩ L0 by
2.8(a). We shall write N0 for NΓ . Let M0 be the partial subgroup of L0 generated by
{λ(V ) | V ∈ E cr }.
Set U = {(P ∩ T )h | P ∈ AT (L), h ∈ NL (CS (T )T }. Then U ⊆ E cr by 2.6(b), and
so ΣT (F ) ⊆ ∆0 . Then every element of L0 has an N0 -decomposition, by 3.12. Let
f ∈ M0 and let g ∈ L0 such that f g is defined in L0 . Let u = (a, f1 , · · · , fm ) be
an N -decomposition of f , let v = (b, g1 , · · · , gn ) be an N -decomposition of g, and set
w = v −1 ◦ u ◦ v. Thus
w = (gn−1 , · · · , g1−1 , b−1 , a, f1 , · · · , fm , b, g1 , · · · , gn ),
and Sw = S(g −1 ,f,g) ∈ ∆0 . By (1) and definition 3.11, there exist elements Ui and Vj of U
(and hence of E cr ) such that fi ∈ λ(Ui ) and gj ∈ λ(Vj ). Further, we have a, b ∈ NL (T ).
As f ∈ M0 , and f = a(f1 · · · fm ) with f1 · · · fm ∈ M0 , the Dedekind lemma (I.1.10)
implies that a ∈ M0 , and so a ∈ NM0 (T ). We may employ the Frattini Calculus (I.3.4),
to obtain a word
(*)

′
w′ = (b−1 , a, b, gn′ , · · · , g1′ , f1′ , · · · , fm
, g1 , · · · , gn )
27

such that Sw = Sw′ , Π(w) = Π(w′ ), and (by (2)) with gj′ and fi′ in M0 for all i and j.
S
Set Y0 = {λ(V ) | V ∈ E cr , and for each k > 0 set
Yk = {Π(e
u) | u
e ∈ W(Yk−1 ) ∩ D(L0 )}.
Then M0 is (by I.1.9) the union of the sets Yk . We now show by induction on k that each
Yk is invariant under conjugation by LT . By (2), Y0 is LT -invariant. Suppose that Yk−1
is LT -invariant, let u
e = (y1 , · · · , yd ) ∈ Yk , and let h ∈ LT . Then Sue ∩ Op (L)T ∈ ∆0 , so
the word
u
eh = (h−1 , y1 , h, · · · , h−1 , yd , h)

is in D(L0 ). Since each of (y1 )h , · · · , (yd )h is in Yk−1 , and since Π(e
uh ) = Π(e
u)h (by
D(L0 )-associativity), the induction is complete. Thus LT acts on M0 by conjugation.
In particular, NM0 (T ) is a normal subgroup of LT , and so ab ∈ M0 . Now (*) yields
f g = ab f ′ for some f ′ ∈ M0 , and so f g ∈ M0 . Thus M0 E L0 .
Set M = M(λ), and let M+ be the partial subgroup of L generated by the set ML
of all L-conjugates of elements of M. Thus:
M0 ⊆ M ≤ M+ .
Since ∆0 is contained in ∆, we have ΣT (L) ⊆ ∆, and so each element of L has an N decomposition. Let f now be an arbitrary element of M+ , and let u = (a, f1 , · · · , fm )
be an N -decomposition (in L) of f . As we have already seen, (1) yields fi ∈ M0 for all
i. Then f1 · · · fm ∈ M, and a ∈ NM+ (T ). By Theorem II.A2, we have M+ E L, and
L0 ∩ M+ = M0 . Then
NM+ (T ) = NL (T ) ∩ M+ = NL (T ) ∩ M0 ,
as NL (T ) ≤ L0 . This shows that a ∈ M0 , and that f = Π(u) ∈ M. Thus M = M+ ,
and so M E L. 
Remark. The hypotheses of the preceding proposition are fulfilled trivially if N = L,
′
and by the mapping U 7→ O p (NN (U )) (and similarly for U 7→ O p (NN (U ))).
p
By II.7.2 there is a smallest partial normal subgroup K := OL
(N ) of L such that
′
p
′
KT = N , and a smallest partial normal subgroup K = OL (N ) of L such that T ≤ K.
In the following arguments K will play a decisive role, and it will be useful to know that
the defining properties of K are invariant under change of objects. Thus, by II.7.3, if
∆ ⊆ ∆+ ⊆ F s and ∆+ is F -closed, then
p
p
+
(N )+
OL
) = OL
+ (N
′

p
(N )).
(and the analogous property holds for OL
28

Corollary 5.2. Assume Op (L)E cr ⊆ ∆, and define mappings λ and λ′ on E cr by
λ(U ) = O p (NN (U ))

and

′

λ′ (U ) = O p (NN (U )).

′

p
p
Then OL
(N ) = M(λ), and OL
(N ) = M(λ′ ).

Proof. Let µ be either of the mappings λ or λ′ . Then µ(U ) is a subgroup of NN (U )
which is normal in NL (Op (L)U ). The conditions (1) and (2) of 5.1 are immediate from
I.2.3(b), so 5.1 yields M(µ) E L.
Let NΓ and L0 be defined as in the first two paragraphs of the proof of 5.1, and set
M0 = M(µ) ∩ NΓ . Then M0 E L0 , and M0 is the partial subgroup of L0 generated (in
L0 ) by {µ(U ) | U ∈ E c }. As NΓ is a locality, we have the quotient locality NΓ /M0 , and
the canonical projection ρ : NΓ → NG . As NΓ is a proper locality on E, each element
g ∈ NΓ has an AT (E)-decomposition w = (g1 , · · · , gn ) (cf. 3.4). By definition, either
′
gi ∈ O p (NN (Ri )) ∩ O p (NN (Ri )) for some Ri ∈ E cr (in which case gi ∈ Ker(ρ)), or
gi ∈ NN (T ). It follows that NG is a p-group if µ = λ, and a p′ -group if µ = λ′ .
We provide the remaining details for the case µ = λ. As T ρ is a maximal p-subgroup
p
of NG we obtain NΓ = M0 T . This shows that OL
(NΓ ) ≤ M0 . On the other hand,
0
p
p
the image of NΓ in L0 /OL0 (NΓ ) is a p-group, so OL0 (NΓ ) contains each of the groups
p
O p (NN (U )) for U ∈ Γ. Thus OL
(NΓ ) = M0 . We now employ Theorem A in order to
0
view L as an expansion (L0 )+ of L, and to provide a correspondence between the partial
normal subgroups of L0 and the partial normal subgroups of L. Then lemma II.7.3 yields
p
p
OL
(N ) = OL
(NΓ )+ = M0 (λ)+ .
0

Here M0 (λ)+ = M(λ) since M0 (λ) = M(λ) ∩ L0 , and this completes the proof in the
case that µ = λ. The proof for µ = λ′ is essentially the same. 
Notation 5.3. For elements x, y ∈ L write [x, y] = 1 if S(x,y) ∩ S(y,x) ∈ ∆ and xy = yx.
For non-empty subsets X and Y of L, write [X, Y ] = 1 if [x, y] = 1 for all x ∈ X and all
y ∈Y.
Lemma 5.4. Assume ΣT (F ) ⊆ ∆. Then the following hold.
(a) Op (LT )CF (T )cr ⊆ ∆. In particular, the hypothesis of 5.2 is fulfilled, with LT in
the role of L and with CT in the role of N .
p
(b) OL
(CT ) = hO p (NCT (V ) | V ∈ CF (T )cr i.
T
(c) Let X ∈ ΣT (F ) and let V ∈ CF (T )cr . Then [NN (X), O p (NCT (V ))] = 1.
p
(CT )CS (N ) E LT .
(d) OL
T
Proof. Point (a) follows from the observation that Op (L)T CF (T )cr ⊆ ΣT (F ) and that
T ≤ Op (LT ). As CF (T ) = FCS (T ) (CT ) by 1.5, we may apply 5.2 with LT and CT in the
roles of L and N , and thereby obtain (b). Point (c) is given by 3.10. As CS (N ) E LT
p
(CT ) E LT by definition II.7.1, we have also point (d). 
by 4.8, and OL
T
29

Definition 5.5. For any proper locality (L, ∆, S) on F , fix an expansion (Ls , F s , S) of
L to a proper locality on F whose set of objects is the set F s of F -subcentric subgroups
of S. We shall refer to such a locality Ls as a subcentric closure of L. For any N E L,
let N s be the partial normal subgroup of Ls whose intersection with L is N . Set
p
(CT )CS (N s ),
N ⊥ = OL
T

where T = S ∩ N , LT = NL (T ), and CT = CL (T ). When no confusion is likely we shall
write
T ⊥ := S ∩ N ⊥ ,
even though T ⊥ depends on N rather than on T . The uniqueness of Ls (up to a unique
isomorphism which restricts to the identity map on L), as given by Theorem II.A1, shows
that CS (N s ) does not depend on the choice of subcentric closure of L, and thus N ⊥ and
T ⊥ are well-defined.
Lemma 5.6. ∆ may be chosen so that ΣT (F ) ⊆ ∆, and so that:
(*) For each X ∈ ΣT (F ) such that CS (T ) ≤ X, each f ∈ NN (X), and each w ∈
W(CT ) ∩ D, we have S(f )◦w ∩ CS (T )T ∈ ∆.
Proof. Assume to begin with that ∆ = F s . Set M = hCT N i, Te = CS (T )T , D = FTe (M),
and Γ = D c . Then M E L and S ∩ M = Te by I.5.5. As CS (M) ≤ CS (T ) ≤ M we may
appeal to 4.5 with M in the role of N , and conclude that Γ ⊆ ∆, (M, Γ) is a localizable
pair, (MΓ , Γ, Te) is a proper locality on D, and that there is a natural conjugation action
of H on D. Then 4.1 applies to MΓ in the role of L, and yields ΣT (F ) ⊆ Γ. Let ∆0 be
the overgroup closure of Γ in S. As L = MH by the Frattini Lemma, the action of H
on D implies that ∆0 is F -closed.
Note that, since MΓ is a proper locality on D, D is Γ-generated. Then by 2.8(a), with
M in the role of N , we have F cr ⊆ ∆0 , and the restriction L0 of L to L0 is a proper
locality on F , with MΓ = M ∩ L0 . Set N0 = N ∩ L0 . Thus N0 E MΓ , and it now
suffices to show that (*) holds with L0 , N0 and CL0 (T ) in place of L, N and CT .
Let X ∈ ΣT (F ) with CS (T ) ≤ X, let each f ∈ NN0 (X), and let w ∈ W(CL0 (T )) ∩
D(L0 ). Set U = X ∩ T and V = Sw ∩ CS (T ). We now apply 4.1 to the locality MΓ in
the role of L (and with Te in the role of S). Since Sw ∩ Te = T V ∈ D c , and Z(T ) ≤ V ,
it follows from 4.1(b) that V ∈ CF (T )c . Since U ∈ E c by 2.6(b), and since E = FT (N0 )
by 3.14, it follows from 4.1(a) that U V ∈ D c . Thus U V ∈ ∆0 . Since f −1 ∈ NN (X), and
−1
since U V ≤ U CS (T ) ≤ X, we have (U V )f ≤ S. One then observes (f ) ◦ w ∈ D(L0 )
−1
via (U V )f . Thus (*) holds, and the proof is complete. 
Theorem 5.7. Let (L, ∆, S) be a proper locality and let N E L be a partial normal
subgroup.
p
(CT ) and N ⊥ are partial normal subgroups of L.
(a) Both OL
T
(b) S ∩ N ⊥ = T ⊥ ≤ CS (N ), and if ΣT (F ) ⊆ ∆ then T ⊥ = CS (N ).
30

(c) Let f ∈ N and g ∈ N ⊥ , and suppose that either (f, g) ∈ D or (g, f ) ∈ D. Then
[f, g] = 1.
(d) Let (L+ , ∆+ , S) be an expansion of L, and for any K E L let K+ be the unique
partial normal subgroup of L+ (given by Theorem II.A2) whose intersection with
L is equal to K. Then (N + )⊥ = (N ⊥ )+ .
Proof. By 5.6 there is a choice of ∆ such that:
(*) ΣT (F ) ⊆ ∆ and,
(**) For each X ∈ ΣT (F ) such that CS (T ) ≤ X, each f ∈ NN (X), and each w ∈
W(CT ) ∩ D, we have S(f )◦w ∈ ∆.
p
(CT ), and assume (*). Then, by 5.4(b), K is generated (as a partial subgroup
Set K = OL
T
of L) by the union Y0 of the groups O p (NCT (P )) taken over all P ∈ CF (T )cr . Recursively
define Yn for n > 0 to be the set of all Π(w) with w ∈ W(Yn−1 ) ∩D. Then K is the union
of the sets Yn , by I.1.9. Now let X ∈ ΣT (F ) with CS (T ) ≤ X. Then [NN (X), Y0 ] = 1
by 5.4(c). Let n be any index such that [NN (X), Yn] = 1, let f ∈ NN (X), and let
w = (g1 , · · · , gk ) ∈ W(Yn ) ∩ D. Assume now that (**) holds, so that (f ) ◦ w ∈ D via
some Q ≤ CS (T )T . Set

w′ = (f, g1 , f −1 , f, g2, · · · , f −1 , f, gk , f −1 , f ).
Then w′ ∈ D via Q. Since gif

−1

= gi for all i we obtain

f g = Π(f ◦ w) = Π(w′ ) = Π(w ◦ f ) = gf,
by D-associativity (I.1.4(b)). Thus [NN (X), Yn+1 ] = 1, and induction yields the following result.
(1) If (*) and (**) hold then [NN (X), K] = 1 for all X ∈ ΣT (F ) such that CS (T ) ≤
X.
Continue to assume (*) and (**), let f be an arbitrary element of N , and let u =
(a, f1 , · · · , fm ) be an N -decomposition of f . Then a ∈ NN (Te) by 3.12, and so a = tf0
where t ∈ T and where f0 ∈ O p (NN (Te)). Set u′ = (t, f0 , · · · , fm ), let s ∈ S ∩ K, and set
u′′ = (s, t, s−1 , s, f0 , s−1 , · · · , s, fm , s−1 ).
−1

Then u′′ ◦ (s) ∈ D via (Sf )s . Since s ∈ K ≤ CT , and since (fi )s = fi for all i with
0 ≤ i ≤ m by (1), we obtain
f s = Π(u ◦ (s)) = Π(u′′ ◦ (s)) = Π((s) ◦ u) = sf.
Thus S ∩ K ≤ CS (N ). By definition 5.5 we then have S ∩ N ⊥ = T ⊥ = (S ∩ K)CS (N s ).
Here CS (N ) = CS (N s ) by (*) and 4.4, so we have shown:
(2) If (*) and (**) hold then T ⊥ = CS (N ).
31

Let (L+ , ∆+ , S) be an expansion of L and let N + be the unique partial normal subgroup
of L+ whose intersection with L is equal to N . Then CS (N ) = CS (N + ) by 4.4. Thus
+
+
T ⊥ = CS (N + ). Set L+
T = NL+ (T ). Then (LT , ∆ , S) is a proper locality on NF (T )
+
+
by 1.5(a), so (LT , ∆ , S) is an expansion of the locality (LT , ∆, S), and (N ⊥ )+ E L+
T.
Observe that
(N ⊥ )+ LT = (N ⊥ )+ L = N ⊥ .
⊥
Thus (N ⊥ )+ is the partial normal subgroup of the locality L+
T which corresponds to N
p
via Theorem II.A2. Recall that we have N ⊥ = KT ⊥ , where K = OLT (CT )T ⊥ . Let K+
p
be the partial normal subgroup of LT corresponding to K. Then K+ = OL
+ (CL+ (T ))
T

+ ⊥
⊥
by II.7.3, and then K+ T ⊥ E L+
T by 5.4(d). Then, since (K T ) ∩ L = N , we conclude
that (N ⊥ )+ = K+ T ⊥ . Then (N ⊥ )+ = (N )+ )⊥ by definition 5.5.
Now let (L′ , ∆′ , S) be the restriction of L+ to a locality (necessarily proper, by
II.2.11) on an F -closed subset ∆′ of F s containing F cr . For any partial subgroup H
of L+ set H′ = H ∩ L′ . By a straightforward exercise with definition II.7.1 one has
p
′
⊥ ′
) = (N ′ )⊥ . Since any proper locality on F can
K′ = O(L
+ ′ (CL+ (T ) ), and hence (N
)
T

be obtained (up to isomorphism) from L by a procedure of first expanding and then
restricting, by II.A1, we thereby obtain (d). By 4.4 and (2), we obtain also point (b). In
particular (and without recourse to (*) or (**)):
(3) T ⊥ ≤ CS (N ).
Assume that (*) holds. Then ΣCS (T ) (CT ) ⊆ ∆ by 3.9(c), and so every element of LT has a
CT -decomposition by 3.12. Let g ∈ N ⊥ , and let v = (b, g1 , · · · , gn ) be a CT -decomposition
of g. Write b = sg0 where s ∈ CS (T ) and g0 ∈ O p (NK (T ⊥ )). Set v ′ = (s, g0, · · · , gn ).
Then Sv = Sv ′ and g = Π(v ′ ). As g ∈ N ⊥ and g0 · · · gn ∈ K ≤ N ⊥ , it follows that
s ∈ S ∩ N ⊥ . Thus s ∈ CS (N ) by (3). Let f ∈ N , and let u = (f1 , · · · , fm ) by an
N -decomposition of f . Suppose that (f, g) ∈ D. Then u ◦ v ′ ∈ D, and f g = Π(u ◦ v ′ ).
Since s commutes with each fi we have
f g = sΠ(f1 , · · · , fm , g0 , · · · , gn ).
Since each [fi , gj ] = 1 for all i and j, by 5.4(c), we have S(fi ,gj ) = S(gj ,fi ) and P fi gj =
P gj fi for all P ∈ ∆ with P ≤ S(fi ,gj ) . From this fact, a straightforward argument by
induction on m + n yields:
f g = sΠ(g1 , · · · , gn , f0 , · · · , fm ),
and then f g = gf . Evidently the assumption (f, g) ∈ D can be replaced by the symmetric
assumption (g, f ) ∈ D, so:
(4) If ΣT (F ) ⊆ ∆ then (c) holds.
Since L = N LT , and since N ⊥ E LT , one may employ (4) and the splitting lemma
(I.3.12) to obtain N ⊥ E L in the case that ΣT (F ) ⊆ ∆. But then N ⊥ E L in general,
by (d). Similarly, since K ≤ N ⊥ and K E LT , we obtain K E L. Thus (a) holds, and
the proof is complete. 
32

Section 6: ∂(F ), δ(F ), and F ∗ (L)
As always, (L, ∆, S) is a proper locality on F . We are now in position to consider
all of the partial normal subgroups N of L simultaneously. And as always, there is the
difficulty in the back-ground, that the fusion system E of N need not be stable with
respect to the processes of expansion (given by Theorem II.A) and restriction (given by
II.2.11). Since S ∩ N is stable with respect to these two processes, one step towards
addressing the difficulty is with a simple notational device, as follows.
For N E L, and T = S ∩ N , define E s (N ) to be the fusion system on T of the form
FT (N s ); where N s is the partial normal subgroup of an expansion (Ls , F s , S) of L (as
in Theorem II.A1) whose intersection with L is N (as in Theorem II.A2).By 3.14, E s (N )
is stable with respect to restriction from F s to F -closed sets ∆ containing ΣT (F ).
Definition 6.1. A partial normal subgroup M E L will be said to be large in L if
S ∩ M⊥ ≤ M. Let M := M(L) be the set of all
normal subgroups of L.
S large partial
s
Define ∂(F ) to be the overgroup closure in S of {Op (L)E (M)cr | M ∈ M}.
Lemma 6.2.
(a) ∂(F ) depends only on F (and not on the choice of a proper locality on F ).
(b) ∂(F ) is F -closed, and F cr ⊆ ∂(F ) ⊆ F s .
(c) Let M ∈ M and set D = E s (M). Then D is (cr)-generated, and D cr ⊆ F q .
Proof. By Theorem II.A1, any proper locality L′ on F is isomorphic to a “version” of
L obtained by first expanding ∆ to F s , and then restricting to an F -closed subset ∆′
of F s containing F cr . By Theorem II.A2 there is a canonical isomorphism N 7→ N ′
from the poset of partial normal subgroups of L to that of L′ , and 5.5(c) implies that
(N ⊥ )′ = (N ′ )⊥ . Then S ∩ N ⊥ = S ∩ (N ′ )⊥ (by II.A2). As Op (L) = Op (L′ ) by II.2.3,
we obtain (a).
Let M ∈ M, set R = S ∩ M, and set D = E s (M). Let L+ be the expansion of L to
∆+ := F s , and let M+ be the partial normal subgroup of L+ whose intersection with L
is M. Then D = FR (M+ ) by definition. We have
CS (M+ ) = S ∩ (M+ )⊥ = S ∩ (M⊥ )+ = S ∩ M
by 5.5, so CS (M+ ) ≤ M+ .
Let Γ be the overgroup closure of D cr in R, and let ∆0 be the overgroup closure of
Op (L)Γ in S. Then Γ ⊆ F q by 4.3(c), so ∆0 ⊆ ∆+ , and thus ∂(F ) ⊆ F s . We have
F cr ⊆ ∂(F ) as a consequence of 2.6(b). As R ∈ Γ ⊆ F q , D is (cr)-generated by 4.3(d),
and thus (c) holds. Then 2.8 applies with M in the role of N , and 2.8(c) shows that
Γ is invariant under the action of NL (R). By the splitting lemma (I.3.12) Γ is then
F -invariant, so ∆0 is F -invariant. As ∂(F ) is the union of the various sets ∆0 taken over
all M ∈ M, the proof of (b) is complete. 
Lemma 6.3. Let N E L be a partial normal subgroup of L, set T = S ∩ N , and
E = FT (N ). Further, set T ⊥ = S ∩ N ⊥ and E ⊥ = FT ⊥ (N ⊥ ). Then ΣT (F ) ⊆ ∂(F ),
and if ∂(F ) ⊆ ∆ then:
(a) Op (L)E cr (E ⊥ )cr ⊆ ∂(F ).
33

(b) E is (cr)-generated.
Proof. We begin by proving (a) and (b) under the assumption that ΣS∩K (F ) ⊆ ∆ for
all K E L. Thus, set M = N N ⊥ and R = T T ⊥ . As ΣT (F ) ⊆ ∆ by assumption, 5.7(b)
yields T ⊥ = CS (N ), and then I.5.1 yields M E L and R = S ∩ M. Set D = FR (M).
As ΣR (M) ⊆ ∆ we have D = E s (M) by 3.14. Note that
S ∩ M⊥ = CS (M) ≤ CS (N ) = T ⊥ ≤ M,
so that M ∈ M. Then D is (cr)-generated and D cr ⊆ ∆ by 6.2(c). We may now appeal
to 2.10, with N1 = N and N2 = N ⊥ , to obtain
D cr = E cr (E ⊥ )cr
via 2.10(d). This yields (a). By 2.10(c) E is the fusion system of a proper locality, and
then (b) follows from II.2.10.
Assume now that ∆ = F s . Then ΣS∩K (F ) ⊆ ∆ for all K E L, and we may may make
further use of 2.10. Namely, 2.10(c) yields:
(*) Both E cr and (E ⊥ )cr are NL (R)-invariant.
Let V ∈ CF (T )cr . By II.1.15 there exists an NF (T )-conjugate V ′ of V such that X ′ :=
V ′ ∩ T ⊥ is fully normalized in NF (T ). Then X ′ is fully normalized in CF (T ) by II.1.17.
Set LT = NL (T ) and CT = CL (T ).
Note that CF (T )cr is NL (R)-invariant, by an application of 2.10(c) to the partial normal subgroup T CT of the locality LT . Thus V ′ ∈ CF (T )cr . As CF (T ) = FCS (T ) (CL (T ))
by 1.4, and since CF (T ) is inductive by II.6.1, the hypothesis of 2.6(c) is fulfilled with
(LT , CT , N ⊥ ) in the place of (L, M, N ). We therefore conclude that X ′ ∈ (E ⊥ )cr . Set
X = V ∩ T ⊥ . Then X is an NF (T )-conjugate of X ′ . As NF (T ) = FS (LT ), and
LT = E ⊥ NLT (R) by the Frattini Lemma, it follows from (*) that X ∈ (E ⊥ )cr . Thus:
(**) V ∩ T ⊥ ∈ (E ⊥ )cr for all V ∈ CF (T )cr .
Now let Y ∈ ΣT (F ), and write Y = Op (L)U V as in 3.7. Notice that since T ⊥ = S ∩ E ⊥
is weakly closed in F we have NL (CS (T )T ) ≤ NL (R). As E cr is NL (R)-invariant by (*),
we obtain U ∈ E cr from 2.6(b). Then U (V ∩ T ⊥ ) ∈ D cr by 2.10(d), and thus Y ∈ ∂(F ).
Having thus shown that ΣT (F ) ⊆ ∂(F ), the proof of the lemma is complete. 
Our next aim is to show that the intersection of the set of large partial normal subgroups of L containing Op (L) is itself large. Lemmas 6.4 through 6.7 will achieve that
result.
Recall that for any partial subgroup H ≤ L, Z(H) is defined to be the set of all z ∈ H
such that, for all h ∈ H, hz is defined and is equal to h.
Lemma 6.4. Assume ∂(F ) ⊆ ∆, let N be a partial normal subgroup of L, and set
T = S ∩ N . Then T ⊥ = CS (N ), and Z(N ) = CT (N ). Moreover, the following are
equivalent.
(1) T ⊥ = Z(N ).
(2) T ⊥ ≤ N .
(3) N ⊥ = Z(N ).
34

Proof. We have ΣT (F ) ⊆ ∆ by 6.3. Then T ⊥ = CS (N ) by definition, and Z(N ) =
CT (N ) by 4.8. The implication (3) =⇒ (1) is then immediate. Since the implication
(1) =⇒ (2) is trivial, it remains only to show that (2) =⇒ (3).
Assume (2). Then T ⊥ = Z(N ), and so E ⊥ is the trivial fusion system on an abelian
p
(CT )T ⊥ , it follows that
p-group. As E ⊥ is the fusion system of N ⊥ , where N ⊥ = OL
T
N ⊥ = T ⊥ , and thus (3) holds. 
p
Next, recall from II.7.2 that for any N E L, OL
(N ) is the smallest partial normal
subgroup K E L such that N = (S ∩ N )K.

Lemma 6.5. Assume ∂(F ) ⊆ ∆, and let M and N be partial normal subgroups of L.
p
Suppose that M ∩ N ≤ Z(N ). Then OL
(N ) ≤ M⊥ .
Proof. Set R = S ∩ M, T = S ∩ N , and E = FT (N ). Then [R, T ] ≤ M ∩ N , so
[R, T ] ≤ Z(N ) by hypothesis. Let U ∈ E cr . Then R normalizes U , and then R normalizes
U T ⊥ since T ⊥ := S ∩ N ⊥ E S. Moreover, we have U T ⊥ ∈ ∂(F ) by 6.3(a), so U T ⊥ ∈ ∆,
and thus both R and NN (U ) are subgroups of the group NL (U T ⊥ ). As Z(N ) ≤ Op (N )
by 2.7, we obtain
[R, NN (U )] ≤ Z(N ) ≤ U,
and hence [R, O p (NN (U ))] = 1 by II.2.7(c).
p
(CR )CS (R). The canonical
Set LR = NL (R) and CR = CL (R). Then CR = OL
R
p
p
projection LR → LR /OLR (CR ) maps O (NN (U )) to a p-group, and so O p (NN (U )) ≤
p
(CR ). Thus O p (NN (U )) ≤ M⊥ for all U ∈ F cr , and the lemma now follows from
OL
R
5.2. 
Lemma 6.6. Assume ∂(F ) ⊆ ∆, and let M and N be partial normal subgroups of L.
Assume that CS (M) ≤ M and that M ∩ N ⊥ ≤ Z(N ). Then N ⊥ = CS (N ).
Proof. Since Z(N ) ≤ Z(N ⊥ ), the hypothesis that M∩N ⊥ be contained in Z(N ) enables
p
(N ⊥ ) ≤ M⊥ .
an application of the preceding lemma with N ⊥ in the role of N . Thus OL
p
Since CS (M) ≤ M, 6.4 yields M⊥ = Z(M) = CS (M), and thus OL (N ⊥ ) is a p-group.
Since S ∩ N ⊥ = CS (N ) by 6.6, the lemma follows. 
Lemma 6.7. Assume ∂(F ) ⊆ ∆, and let M and N be partial normal subgroups of L.
Assume that Op (L)CS (M) ≤ M and that Op (L)CS (N ) ≤ N . Then Op (L)CS (M∩N ) ≤
M ∩ N.
Proof. Set K = M ∩ N . Obviously Op (L) ≤ K, so the problem is to show that CS (K) ≤
K. Observe:
M ∩ (N ∩ K⊥ ) = K ∩ K⊥ = Z(K) ≤ Z(K⊥ ).
Since Z(K) ≤ N we then have Z(K) ≤ Z(N ∩ K⊥ ), and we may apply 6.5 with N ∩ K⊥
p
(N ∩ K⊥ ) ≤ M⊥ . As CS (M) ≤ M, M⊥ = CS (M) by 6.4.
in the role of N . Thus OL
Thus N ∩ K⊥ is a p-group, and indeed a normal p-subgroup of L. As Op (L) ≤ K, we
obtain N ∩ K⊥ ≤ K, and so N ∩ K⊥ ≤ Z(K). Now the hypothesis of 6.6 is satisfied, with
N in the role of M, and with K in the role of N . We conclude that K⊥ = CS (K). As
K⊥ E L, and Op (L) ≤ K, we obtain CS (K) ≤ K, as required. 
35

Definition 6.8. If ∂(F ) ⊆ ∆, define F ∗ (L) to be the intersection of the large partial
normal subgroups of L containing Op (L):
F ∗ (L) =

\

{N | Op (L)CS (N ) ≤ N E L}.

More generally, define F ∗ (L) to be F ∗ (L+ ) ∩ L, where (L+ , ∆+ , S) is the expansion of
L to a proper locality on F with ∆+ = ∆ ∪ ∂(F ).
Corollary 6.9. If ∂(F ) ⊆ ∆ then F ∗ (L) is a large partial normal subgroup of L containing Op (L), and is the unique smallest such. Moreover, and in general (i.e. whether
or not ∂(F ) ⊆ ∆), we have F ∗ (L+ ) = F ∗ (L)+ for any expansion (L+ , ∆+ , S) of L to a
proper locality on F . 
Proof. The first assertion is immediate from 6.7. Now drop the assumption that ∂(F ) is
contained in ∆, and let (L+ , ∆+ , S) be an expansion of L to some ∆+ with D ∪ ∂(F ) ⊆
∆+ ⊆ F s . Let (Ls , F s , Σ) be the expansion of L to F s . Let X+ be the set of large
partial normal subgroups N + of L+ containing Op (F ), and similarly define Xs relative
to Ls .
By 6.3 we have ΣT (F ) ⊆ ∂(F ) for all T = S ∩ N with N E L. Then by 3.14, the
correspondence (Theorem II.A2) between the set of partial normal subgroups of L+ and
the set of partial normal subgroups of Ls restricts to a bijection X+ → Xs . Then
F ∗ (L+ ) =

\

X+ = L+ ∩ (

\

Xs ) = L+ ∩ F ∗ (Ls ).

This shows that F ∗ (L+ ) is determined by F ∗ (Ls ). Then
F ∗ (L) = L ∩ F ∗ (Ls ) = L ∩ F∗ (L+ ).
The correspondence given by Theorem II.A2 then yields F ∗ (L+ ) = F ∗ (L+ ). 
Set F ∗ (F ) = FS∩F ∗ (L) (F ∗ (L)), and define δ(F ) to be the overgroup-closure of F ∗ (F )s
in S.
Lemma 6.10. δ(F ) depends only on F (and not on the choice of proper locality on F ).
Moreover, δ(F ) is F -closed, and
F cr ⊆ ∂(F ) ⊆ δ(F ) ⊆ F s .
Proof. As ∂(F ) depends only on F by 6.2(a), and since Ls depends only on F by Theorem
II.A1, it follows that δ(F ) depends only on F . Now let M be an arbitrary large partial
normal subgroup of L containing Op (L), set R = S ∩ M, and set D = FR (M). As
ΣR (F ) ⊆ ∂(F ) by 6.3, it follows from 3.14 that D is independent of the choice of ∆,
provided only that ∂(F ) ⊆ ∆ ⊆ F s . Taking ∆ = F s , we obtain D c ⊆ F q by 4.5(a), so
R ∈ F q , and then D s ⊆ F s by 4.3(c). In particular, by taking M = F ∗ (L), we obtain
F ∗ (F )s ⊆ F s .
36

As D cr ⊆ ∆, it follows from 2.8 that D s is F -invariant, and hence the overgroup-closure
of D s in S is F -closed. By specializing again to M = F ∗ (L), we conclude that δ(F ) is
F -closed. As F cr ⊆ ∂(F ) by 6.2(b), it now only remains to show that ∂(F ) ⊆ δ(F ).
Let P ∈ D cr . Then there exists Q ∈ P F such that P ∩ F ∗ (L) is fully normalized
in F , by II.1.15. We note that D cr is F -invariant (again by 2.8), so Q ∈ D cr , and
then Q ∩ F ∗ (L) ∈ F ∗ (F )cr by 2.6(c). Thus Q ∈ δ(F ), and then P ∈ δ(F ) as δ(F ) is
F -invariant. Thus ∂(F ) ⊆ δ(F ), and the proof is complete. 
Section 7: Regular localities
In the preceding section we produced a mapping δ which, to each saturated fusion
system F on a p-group S, assigns the overgroup closure in S of F ∗ (F )s . By 6.8(a) and
Theorem II.A1 there is a unique (up to unique isomorphism) proper locality L on F
whose set of of objects is δ(F ).
Definition 7.1. A locality (L, ∆, S) on F is regular provided that L is proper, and
∆ = δ(F ).
Proposition 7.2. Let (L, ∆, S) be a regular locality, let Z ≤ Z(L) be a subgroup of
Z(L), and let ρ : L → L/Z be the canonical projection. Then L/Z is regular, and
F ∗ (L)ρ = F ∗ (L/Z).
Proof. Set L = L/Z, and write X for the image under ρ of any subset of L, or any set
of subgroups of L. Set F = FS (L) and set F = FS (L). The restriction σ : S → S of ρ
to S is fusion-preserving, and we shall denote also by σ the associated homomorphism
F → F of fusion systems. Then σ maps the set Hom(F ) of F -homomorphisms onto
Hom(F ), so II.1.19 applies and yields the following information concerning a subgroup
X of S and its preimage X in S.
(1) X is fully normalized in F if and only if X is fully normalized in F .
(2) Op (NF (X)) ≤ Op (NF (X)).
c
cr
(3) If X ∈ F then X ∈ F c , and if X ∈ F then X ∈ F cr .
cr

As a consequence of (3) we have F ≤ ∆. For any P ∈ ∆ the group NL (P )/Z is of
characteristic p, by II.2.6(c), so L is a proper locality. We next show:
s

(4) F s = F .
s

In one direction: let Z ≤ P ≤ S with P fully normalized in F and with P ∈ F . Let Q
be the pre-image in S of Op (NL (P )). Then Q = Op (NL (P )), and Q ∈ F c by (3). Thus
s
F s ⊇ F . In the other direction: let P ∈ F s . Then ZP ∈ F s , as is any F -conjugate of
s
P . In order to show that P ∈ F we may therefore assume that Z ≤ P , that P is fully
normalized in F , and (by II.1.18) that Q := Op (NF (P )) is fully normalized in F . Let
D be the pre-image of CS (Q) in S. Thus [Q, D] ≤ Z. Let (Ls , F s , S) be the expansion
of L to a proper locality on F whose set of objects is F s , and set M = NLs (P ). Thus
M is a subgroup of Ls of characteristic p. Here Ls is generated by L as a partial group,
by point (c) in Theorem II.A1, so Z ≤ Z(Ls ). Thus Z ≤ Z(M ), and D centralizes the
37

chain Q ≥ Z ≥ 1 of normal subgroups of M . Now II.2.6(c) shows that D = Q, and thus
Q is centric in F . Thus (4) holds. We next show:
(5) Let M E L be a partial normal subgroup of L containing Op (L). Then M is
large in L if and only in M/Z is large in L/Z.
Evidently CS (M)/Z ≤ CS (M), so if M/Z is large in L/Z then M is large in L. Conversely, assume that M is large in L, set R = S ∩ M, set M = M/Z, and let N be
⊥
the pre-image in L of M . Then N E L, and [N , R] ≤ Z. As S ∩ F ∗ (L) ∈ ∆ and
F ∗ (L) ≤ M we have R ∈ ∆, and thus N is a subgroup of the group NL (R). Then
[O p (N ), R] = 1. By 6.3 and 6.10 we have ΣR (F ) ⊆ ∆, so O p (N ) ≤ M⊥ by 6.5. Thus
S ∩ O p (N ) ≤ Z(N ), and O p (N ) is a p′ -group, normal in NL (R). As NL (R) is of characteristic p we conclude that N is a p-group, so N ≤ Op (L), and thus N ≤ M. This
shows that M is large in L, and completes the proof of (5).
It is immediate from (5) that F ∗ (L)/Z = F ∗ (L/Z). Then (4) yields ∆ = δ(F) and
completes the proof. 
Lemma 7.3. Let P ≤ S be a subgroup of S, and suppose that Op (L)P ∈ δ(F ). Then
P ∈ δ(F ).
Proof. Set M = F ∗ (L), R = S ∩ M, and D = FR (M). As Op (L)P ∈ δ(F ) we have
R ∩ Op (L)P ∈ D s . As Op (L) ≤ R we have R ∩ Op (L)P = Op (L)(R ∩ P ), and then
R ∩ P ∈ D s by II.6.2(c). That is, P ∈ δ(F ). 
For the remainder of this section (L, ∆, S) will be a regular locality on F . The main
goal is to show that every partial normal subgroup of L is itself a regular locality.
Lemma 7.4. Let (L, ∆, S) be a regular locality on F , and let N E L be a partial
normal subgroup of L such that Op (L)CS (N ) ≤ N . Adopt the notation of 2.1, and set
Γ = {U ≤ T | U ∈ ∆}. Then (N , Γ, T ) is a regular locality on E, and F ∗ (N ) = F ∗ (L).
Proof. The hypothesis that Op (L)CS (N ) be contained in N is equivalent to F ∗ (L) ≤ N .
Then ∆ is the overgroup closure in S of Γ, and so Sw ∩ T ∈ Γ for each w ∈ W(N ) ∩ D.
Then (N , Γ, T ) is a locality by 1.2, and indeed a locality on E by the definition E as
FT (N ). Here E cr ⊆ ∂(F ), so E cr ⊆ ∆ by 6.10; and N is then a proper locality by
II.2.6(a). We note that NL (T ) acts on L (and hence on N , and also on E) by conjugation,
since P ∩ T ∈ ∆ for all P ∈ ∆. Then Op (N ) = Op (L) by the Frattini Lemma (I.3.12).
Set Γ+ = E s , and let ∆+ be the overgroup closure of Γ+ in S. As T ∈ F q by 4.5(a),
we have Γ+ ⊆ F s by 4.3(c), and then ∆+ ⊆ F s by II.6.2(a). As NL (T ) acts on E, it
follows that ∆+ is F -closed. Let (L+ , ∆+ , S) be the expansion of L to a proper locality
on F , via Theorem II.A1. For any K E L let K+ E L+ be the partial normal subgroup of
L+ , given by Theorem II.A2, whose intersection with L is K. Notice that (N + , Γ+ , T ) is
a proper locality on E, and so N + is an expansion of N . As NL (T ) = NL+ (T ) permutes
the set M(N + ) of large partial normal subgroups K of N + containing Op (N ), it follows
from the Frattini Lemma that F ∗ (N + ) E L+ .
Set K+ = F ∗ (N + ). Applying 6.4 with N + and K+ in the roles of L and N , we obtain
38

T ∩ (K+ )⊥ ≤ Z(K+ ). Then 6.5 yields
p
+ ⊥
+ ⊥
OL
) .
+ ((K ) ) ≤ (N

But (N + )⊥ = (N ⊥ )+ by 5.7(d), and N ⊥ = Z(N ) by 6.4. Thus (K+ )⊥ is a normal
p-subgroup of L+ , and so (K+ )⊥ ≤ Op (L+ ). As Op (L+ ) = Op (N + ) = Op (K+ ), we
conclude that K+ is a large partial normal subgroup of L+ , and hence F ∗ (L+ ) ≤ K+ .
The reverse inclusion holds since F ∗ (L)+ is a large partial normal subgroup of N +
containing Op (N + ), and since K+ is by definition the intersection of all such. Moreover,
Theorem II.A2 then yields:
K+ ∩ L = F ∗ (L)+ ∩ L = F ∗ (L),
which in turn yields F ∗ (E) = F ∗ (F ). Then Γ = δ(E), and the proof is complete.



Lemma 7.5. Let (L, ∆, S) be a regular locality on F , and let N E L be a partial normal
subgroup of L such that L = Op (L)N . Adopt the notation of 2.1, and set Γ = {U ≤ T |
U ∈ ∆}. Then (N , Γ, T ) is a regular locality on E, and F ∗ (N ) = F ∗ (L) ∩ N .
Proof. Set K = F ∗ (L) ∩ N . Then
F ∗ (L) = F ∗ (L) ∩ L = F ∗ (L) ∩ Op (L)N = Op (L)K,
by the Dedekind Lemma. Then 7.3 implies that P ∩ T ∈ Γ for each P ∈ ∆ such that
Op (L) ≤ P . Then Sw ∩ T ∈ Γ for each w ∈ W(N ) ∩ D, and so (N , Γ, T ) is a locality on
E.
As in the proof of 7.4, it will be necessary to consider “versions” of L and of N with
sets of objects other than ∆ and Γ. Let ∆0 be the overgroup closure of Γ in S. Thus
P ∈ ∆0 if and only if P ∩ T ∈ Γ, so Op (L)∆ ⊆ ∆0 ⊆ ∆, and the restriction of L
to ∆0 is then equal (as a partial group) to L. Set Γ+ = E s . As CL (S) is a p-group,
we have O p (CL (T )) = 1, and so T ∈ F q by II.2.8(c). Then Γ+ ⊆ F s by 4.3(c). Let
+
s
+
in S. Then ∆+
∆+
0 is F -closed, and ∆0 ⊆ ∆0 ⊆ F .
0 be the overgroup closure of Γ
+
Let (L+ , ∆+
0 , S) be the ∆0 -expansion of L to a proper locality on F , and for each
partial normal subgroup H of L let H+ be the partial normal subgroup of L+ such that
H = H+ ∩ L. As Op (L)N + ∩ L = L we then have L+ = Op (L)N + .
Let P ∈ E cr and set Q = Op (L)P . Then Q ∈ ∆+
0 , and
NL+ (Q) = Op (L)NN + (Q) = Op (L)NN + (P ).
Since P ∈ E cr we have P = Op (NN + (P )), and then Q = Op (NL+ (Q)). Then Q ∈ F cr
by II.2.8(a), so Q ∈ ∆0 . Then P = Q ∩ T ∈ Γ, and thus E cr ⊆ Γ. We have thus shown
that (N , Γ, T ) is proper.
Set H = K⊥ ∩ N . As Op (N ) ≤ K, 5.7(c) yields [Op (N ), H] = 1. As [Op (L), H] ≤
p
p
(H) ≤ CS (F ∗ (L)),
(H)] = 1. Thus T ∩ON
Op (N ) it follows from II.2.6(c) that [Op (L), ON
and 6.4 then yields
p
p
(H)).
(H) ≤ Z(ON
T ∩ ON
39

p
(H) is a p-group by 3.17, and so H ≤ T . As H E L we get H ≤ Op (L). Then
Then ON
H ≤ K, and since CT (K) ≤ H we conclude that Op (N )CT (K) ≤ K. Thus F ∗ (N ) ≤ K,
and Op (L)F ∗ (N ) ≤ F ∗ (L).
In order to obtain the reverse inclusion, observe first of all that

CS (Op (L)F ∗ (N + )) ≤ Op (L)CS (F ∗ (N )+ ) = Op (L)CT (F ∗ (N + )) ≤ Op (L).
This shows:
F ∗ (L+ ) ≤ Op (L)F ∗ (N + ).

(*)

Since F ∗ (N + ) = F ∗ (N )+ by 6.9, and since
Op (L)F ∗ (N )+ ∩ L = Op (L)(F ∗ (N )+ ∩ L) = Op (L)F ∗ (N ),
we get
(Op (L)F ∗ (N ))+ = Op (L)F ∗ (N )+ = Op (L)F ∗ (N + ).
Then (*) yields F ∗ (L) ≤ Op (L)F ∗ (N ), and thus F ∗ (L) = Op (L)F ∗ (N ).
We have still to show that (N , Γ, T ) is a regular locality. Thus, it remains to show
that Γ = δ(E). Let X ∈ δ(E) and set U = X ∩ F ∗ (N ). As E = E + by 3.14, the definition
of δ(E) yields U ∈ F ∗ (E)s . Here T ∩ F ∗ (E) ∈ E q by 4.5(a), so U ∈ E s by 4.3(c). Then
U ∈ F s since we have already seen that E s ⊆ F s . As U ≤ F ∗ (N ) ≤ F ∗ (L) it follows
that U ∈ F ∗ (F )s . Then U ∈ δ(F ) = ∆ and X ∈ ∆. Thus X ∈ Γ, and we have
δ(E) ⊆ Γ. Now let Y ∈ Γ and set V = Y ∩ F ∗ (L). Then V ∈ Γ since ∆ = δ(F ). Then
NL (V ) is of characteristic p, so also NF ∗ (N ) (V ) is of characteristic p by II.2.6(a). As
V ≤ F ∗ (L) ∩ N = F ∗ (N ) we conclude that V ∈ F ∗ (E)s , so V ∈ δ(E). Then Y ∈ δ(E),
and thus Γ = δ(E), as required. 
Lemma 7.6. Let (L, ∆, S) be a regular locality on F , and assume the “product setup”
of 2.9, with L = N1 N2 . Then:
(a) Ni ≤ (N3−i )⊥ for i = 1 and 2.
(b) Each (Ni , δ(Ei), Ti ) is a regular locality on Ei , and δ(Ei ) = {U ≤ Ti | U CS (Ni ) ∈
∆}.
(c) F ∗ (L) = F ∗ (N1 )F ∗ (N2 ).
Proof. We have N2 ≤ CL (T1 ) by 2.9. Applying II.7.4 to the locality NL (T1 ) then yields
(1)

p
(CL (T1 )) ≤ (N1 )⊥ .
O p NL (T1 )(N2 ) ≤ ON
L (T1 )

p
By definition, ON
(CL (T1 )) is the intersection of the set K of partial normal subgroups
L (T1 )
K of NL (T1 ) such that:

(2) K ≤ N2 and KT2 = N2 .
40

Evidently K contains the set of partial normal subgroups K of L such that (2) holds,
p
p
(N2 ) ≤ (N1 )⊥ . Since ΣT1 (F ) ⊆ ∆ we
(N2 ) ≤ O p NL (T1 )(N2 ). Then (1) yields OL
so OL
have CS (N1 ) = (N1 )⊥ by 5.7(b), and so T2 ≤ (N1 )⊥ . Thus N2 ≤ (N1 )⊥ , and point (a)
follows.
Set M = F ∗ (L), set M1 = M ∩ N1 , and set K = (M1 )⊥ ∩ N1 . Then K E L by 5.7(a),
and
K ∩ M = K ∩ M1 ≤ Z(K),
p
p
(K) = 1. Thus K is
(K) ≤ M⊥ . Now 4.7 yields M⊥ = Z(M) and OL
so 6.5 yields OL
a normal p-subgroup of L, and so K ≤ Op (L). Notice that since [Ni , T3−i ] = 1 we have
Op (L) = Op (N1 )Op (N2 ). As Op (L) ≤ M we then have Op (L) = Op (M1 )Op (M2 ), and
K ≤ Op (M1 ). We now compute:

CT1 (M1 ) ≤ CS (M1 M2 ) = CS (M1 ) ∩ CS (M2 )
= CT1 (M1 )T2 ∩ CT2 (M2 )T1 = Z(M1 )T2 ∩ Z(M2 )T1
= Z(M1 )(T2 ∩ Z(M2 )T1 ) = Z(M1 )Z(M2 )(T1 ∩ T2 ) ≤ Op (L)
≤ M1 M2 .
This shows that CT1 (M1 ) ≤ M1 , and that M1 M2 is a large partial normal subgroup of
L containing Op (L). We have shown:
(3) F ∗ (L) = M1 M2 , and
(4) Op (Ni )CS (Mi ) ≤ Mi (i = 1, 2).
Next, set Γi = {P T3−i ∩ Ti | P ∈ ∆} and set ∆0 = Γ1 Γ2 . Then (Ni , Γi , Ti ) is a proper
locality on Ei , ∆0 is an F -closed subset of ∆ containing F cr , and the restriction of caL to
+
s
∆0 is equal to L as a partial group, by 2.11 and by 2.10(a). Set Γ+
i = Ei and let ∆0 be
+ +
+
the overgroup closure of Γ1 Γ2 . Then ∆0 is an F -closed subset of F s by 2.10(e). Form
the corresponding expansions (Ni )+ and L+ via Theorem II.A1. Let Ki+ = F ∗ (Ni )+
be the partial normal subgroup of (Ni )+ corresponding to Ki = F ∗ (Ni ) via Theorem
+ ⊥
) (a) and 5.7(d). Then 5.7(c) implies that Ki E L and
II.A2. Note that Ni+ ≤ (N3−i
+
+
+
Ki E L . Let (K1 K2 ) be the partial normal subgroup of L+ whose intersection with
L is K1 K2 . Then (K1 K2 )+ = K1+ K2+ by II.5.3. We compute:
CS (K1+ K2+ ) = CS (K1+ ) ∩ CS (K2+ ) = CT1 (K1+ )T2 ∩ CT2 (K2+ )T1
= Z(K1+ )T2 ∩ Z(K2+ )T1 = Z(K1+ )Z(K2+ )(T1 ∩ T2 ) ≤ K1+ K2+ ,
and so (K1 K2 )+ is a large partial normal subgroup of L+ . As also Op (F ) = Op (L) ≤
K1 K2 , we conclude that F ∗ (L+ ) ≤ (K1 K2 )+ . This yields M ≤ K1 K2 . Since also
Ki ≤ Mi by (4), we obtain Ki = Mi . That is, (c) holds.
Let Ui ∈ F ∗ (Ei )s . Then Ui ∈ δ(Ei ) ⊆ (Ei )s , and then U1 U2 ∈ F s by 2.10(e). As
U1 U2 ≤ F ∗ (L) by (c), we obtain δ(E1 )δ(E2 ) ⊆ δ(F ). As δ(F ) = ∆ we have U1 (T2 ∩M2 ) ∈
∆ in particular, and so U1 (T1 ∩ T2 ) ∈ Γ1 . This shows that Z(N1 )δ(E1 ) ⊆ Γ1 .
Let Q1 ∈ Γ1 . Then Q1 = P ∩ T1 for some P ∈ ∆ with T2 ≤ P . Then P ∩ M ∈ F ∗ (F )s
as ∆ = δ(F ). Here P ∩ M = (P ∩ M1 )(T2 ∩ M2 ), so P ∩ M1 ∈ F ∗ (E1 )s by 2.10(e). Thus
41

Q1 ∈ δ(E1 ), and this shows that Γ1 ⊆ δ(E1 ). Since Z(N1 ) ≤ Sw for all w ∈ W(N1 ), the
expansion of (N1 , Γ1 , T1 ) to a regular locality on δ(E1 ) has the same underlying partial
group N1 . This completes the proof of (b), and of the lemma. 
Theorem 7.7. Let (L, ∆, S) be a regular locality on F , and let N E L be a partial
normal subgroup. Set T = S ∩ N , E = FT (N ), and Γ = {U ≤ T | CS (N )U ∈ ∆}. Then:
(a) (N , Γ, T ) is a regular locality on E.
(b) F ∗ (N ) = N ∩ F ∗ (L).
(c) N ⊥ = CL (N ). That is, N ⊥ is the set of elements g ∈ L such that xg = x for all
x ∈ N.
(d) Op (E) = Op (N ) E L.
(e) (g −1 , x, g) ∈ D for all x ∈ N and all g ∈ NL (T ). Moreover, the mapping
cg : N → N given by x 7→ Π(g −1 , x, g) is an automorphism of N , and the
mapping η : NL (T ) → Aut(N ) given by g 7→ cg is a homomomorphism of partial
groups with kernel N ⊥ .
(f) The image of the natural homomorphism NL (T ) → Aut(T ) is contained in
Aut(E).
Proof. If N is large and Op (L) ≤ N then (a) and (b) are given by 7.3 and 7.4. Set
M = N N ⊥ . Thus (a) and (b) hold with Op (L)M in the role of N . In order to prove
(a) and (b) in general, we may then assume that L = Op (L)M. By 7.5 we then obtain
(a) and (b) for M in the role of N . In this way we reduce to the case where L = M,
where (a) and (b) are then given by 7.6.
In proving (c) we may now assume that L is the regular locality N N ⊥ . Let f ∈ N
and g ∈ N ⊥ , set U = Sf ∩ T , and set V = Sg ∩ T ⊥ . Then U V ∈ ∆ by 7.6(b), and so
(f, g) ∈ D. Then [f, g] = 1 by 5.7(c), and this yields point (c). That Op (E) = Op (N )
is now a consequence of (a) and II.2.3. Point (d) will then follow from (f) and the
factorization L = NL (T )N given by the Frattini Lemma. Thus, it remains to prove (e)
and (f).
Set H = NL (T T ⊥ ) and set LT = NL (T ). Then (LT , ∆, S) is a locality, and N ⊥ E LT ,
so the Frattini Lemma yields LT = N ⊥ H. Here T T ⊥ ∈ ∆ by an application of (a) to
N N ⊥ in the role of N , so H is a subgroup of L. Let x ∈ N and set P = Sx ∩ T . Then
P ∈ Γ. Now let g ∈ LT and employ the Splitting Lemma so as to write g = yh with
y ∈ N ⊥ and h ∈ H, and with Sg = S(y,h) . Set Q = Sy ∩ T ⊥ . Then P Q ∈ ∆ by 2.11, and
one may verify that the word (h−1 , y −1 , x, y, h) is in D via (P Q)h . Thus (g −1 , x, g) ∈ D,
and (f) follows from 2.3. A similar argument shows that if (x1 , · · · , xn ) ∈ D(N ) then
(g −1 , x1 , g, · · · , g −1 , xn , g) ∈ D,
and hence that the conjugation map cg : N → N is an endomorphism of N as a partial
group. One checks that cg ◦ cg −1 is the identity map on N , so we have a mapping
η : LT → Aut(N ) given by g 7→ cg . The restriction of η to H is easily seen to be a
homomorphism (of groups), and a further exercise with the Splitting Lemma will verify
that η itself is a homomorphism of partial groups, completing the proof of (e). 
42

Definition 7.8. A partial subgroup K of a locality L is subnormal in L (denoted K EE
L) if there exists a sequence (N0 , · · · , Nk ) of partial subgroups of L such that K = N0 E
N1 E · · · E Nk = L.
Corollary 7.9. Let L be a regular locality and let K EE L be a partial subnormal
subgroup of L. Then K is a regular locality.
Proof. By induction on the length of a subnormal chain from K to L it suffices to show
that K is regular in the case that K E L, in which case we are done by 7.7(a). 
Lemma 7.10. Let L be a regular locality, let N E L be a partial normal subgroup, and
let K be a non-empty set of partial normal subgroups of N . Set T = S ∩ N , and let
Λ = NAut(N ) (T ) be the set of automorphisms of N (as a partial
T group) which leave T
invariant. Suppose that K is invariant under Λ, and set K = K. Then K E L.
Proof. We have K E N N ⊥ by 7.7(c), and K is S-invariant since, by I.2.9, S acts on L by
conjugation. Thus K E M := Op (L)N N ⊥. Set H = NL (S ∩ M). Then H is a subgroup
of NL (S ∩ F ∗ (L)). As L is regular we have D = {w ∈ W(L) | Sw ∩ F ∗ (L) ∈ ∆}, so
H acts on L by conjugation. Then O p (N ) is H-invariant, and we may employ I.3.13 to
conclude that K E L. 
′

p
(L) (and similarly for O p (L)). Recall
Recall from II.7.1 that O p (L) is defined to be OL
also the definition of [L, L] from 3.16.

Corollary 7.11. Let L be a regular locality and let N E L be a partial normal subgroup.
′
Then O p (N ), O p (N ), and [N , N ] are partial normal subgroups of L. Moreover, we have
′
p
p′
OL
(N ) = O p (N ) and OL
(N ) = O p (N ).
Proof. Let K be the set of partial normal subgroups of N defined by any one of the
following three conditions: (1) H ∈ K ifTHT = N , (2) H ∈ K if T ≤ H, and (3) H ∈ K
if N /H is an abelian group. Set K = K. Then K E L by 7.10, and K is variously
′
the set of all partial normal subgroups H′ of
O p (N ), O p (N ), or [N , N ]. Now let K′ be
T
p
p
(N ) ≤ O p (N ) since
L such that H′ T = N . Then OL (N ) = K′ by definition, and OL
p
p
′
O (N ) ∈ K . The reverse inclusion holds since OL (N ) ∈ K (with K as in (1)). Thus
′
p
p′
OL
(N ) = O p (N ), and similarly OL
(N ) = O p (N ). 
Section 8: Components
Throughout this section, (L, ∆, S) is a regular locality on F . That is, L is a proper
locality on F , and ∆ = δ(F ). By 7.9, every partial subnormal subgroup K of L is then a
regular locality, and O p (K) EE L by 7.11. We may write δ(L) for δ(F ), and thus write
also (K, δ(K), S ∩ K) for the regular locality K. A locality L is simple if L has exactly
two partial normal subgroups.
Definition 8.1. A partial subnormal subgroup K EE L is a component of L if K =
O p (K) and K/Z(K) is simple.
43

Write Comp(L) for the set of components of L, and let
E(L) = hComp(L)i
be the partial subgroup of L generated by the union of the components of L. Recall that
for subsets X, Y of L, the notation [X, Y ] = 1 indicates that the commutators [x, y] are
defined and are equal to 1 for all x ∈ X and all y ∈ Y .
Lemma 8.2. Let K ∈ Comp(L) and let N E K. Then Z(K) = Op (K), and either
N = K or N ≤ Z(K).
Proof. As K is itself a regular locality, we may as well assume that K = L. We have
Z(L) ≤ Op (L) by 2.7. Now let N E L be given with N  Z(L). Here Z(L)N E L by
I.5.1. As L/Z(L) is simple, the Correspondence Theorem (I.4.8) yields Z(L)N = L.
Set T = S ∩ N . Then
O p (N )S ≥ Z(L)O p (N )T = Z(L)N = L.
As O p (N ) E L, by 7.11, we conclude that O p (L) ≤ O p (N ). Then N = L, since
L = O p (L). Thus, either N ≤ Z(L) or N = L. The special case where N = Op (L) then
yields Op (L) ≤ Z(L), and completes the proof. 
Lemma 8.3. Let K EE L.
(a) If K ≤ N E L then K EE N .
(b) Op (K) ≤ Op (L), and if K E L then Op (K) E L.
(c) Op (L)K EE L.
Proof. Point (a) is immediate from the observation that the intersection of partial normal
subgroups is again a partial normal subgroup.
In proving (b): suppose first that K E L. Then Op (K) E L by 7.11, with {Op (K)}
in the role of K. Now (b) follows by an obvious induction argument on the length of a
subnormal chain from K to L.
Set L = L/Op (L) and let K be the image of K in L under the canonical projection
ρ : L → L. The Correspondence Theorem (I.4.7) yields K EE L, and the preimage H of
K is a partial subnormal subgroup of L. This yields (c). 
Lemma 8.4. Let K ∈ Comp(L). Then:
(a) Z(K) ≤ Op (L).
(b) [Op (L), K] = 1.
(c) K = O p (Op (L)K).
Proof. We have Z(K) ≤ Op (K) by 4.8, and then point (a) follows from 6.3(a).
Let X be a subgroup of K of order prime to p. Then Op (L)X is a subgroup of L. Set
Z0 = Op (L) and recursively define Zn := [Zn−1 , X] for n ≥ 1. Then Zn ≤ Op (L) for all
n, and since K EE L we obtain Zn ≤ K ∩ Op (L) for n sufficiently large. Then Zn+1 = 1
since Op (K) ≤ Z(K); and then Z1 = 1 by coprime action. In particular, we now have
44

[Op (L), O p (NK (P )] = 1 for all P ∈ δ(K). One easily verifies (via I.2.9 for example) that
CL (Op (L)) is a partial subgroup of L, and thus
hO p (NK (P )) | P ∈ δ(K)i ≤ CL (Op (L)).
Then O p (L) ≤ CL (Op (L)) by an application of 4.3, with K in the role of L. As
(x−1 , g −1 , x, g) ∈ D for all x ∈ Op (L) and all g ∈ L, we obtain (b).
As Op (L)K is subnormal in L by 6.3(b), Op (L)K is a regular locality, and we may
therefore take L = Op (L)K in proving (c). Then (b) yields K E L. Now O p (L) ≤ K as
KS = L, and then O p (K) ≤ O p (L) since
K = L ∩ K = O p (L)S ∩ K = O p (L)T.
The reverse inclusion O p (L) ≤ O p (K) is given by
O p (K)S = (O p (K)T )S = KS = L.

Theorem 8.5. Let (L, ∆, S) be a regular locality which is not a group of characteristic
p. Then Comp(L) 6= ∅. Let (K1 , · · · , Km ) be a non-redundant
S list of the components of
L, and let E(L) be the partial subgroup of L generated by Comp(L). Then
F ∗ (L) = Op (L)E(L) = Op (L)K1 · · · Km ,
and
[Ki , Kj ] = 1 (1 ≤ i 6= j ≤ m).
p

∗

Further, E(L) = O (F (L)), and [Op (L), E(L)] = 1.
Proof. If Op (L) is a large partial normal subgroup of L then Op (L) = F ∗ (L) ∈ ∆, and
then L is a group of characteristic p, contrary to hypothesis. Thus Op (L) is not large.
Set H = Op (L)⊥ , and let X be the set of partial subnormal subgroups K of H such
that K is not a p-group. Regard X as a poset via inclusion, and let K be minimal in
X. Then K = O p (K), and K/Op (K) is simple. As Op (K) ≤ Op (L) by 8.3(b), and since
K ≤ H = CL (Op (L)) by 7.7(c), we obtain K ∈ Comp(L). Here X 6= ∅ as H ∈ X, so we
have shown that Comp(L) 6= ∅.
Set M = F ∗ (L) and let K ∈ Comp(L). Let N E L be a partial normal subgroup of
L which is minimal with respect to the condition K EE N . If K = L then M = K and
{K} = Comp(L) by 8.2, and there is then nothing to prove. Thus we may assume that
K=
6 L. Then N =
6 L, and induction on |L| then yields K ≤ F ∗ (N ). Then K ≤ F ∗ (L)
by 7.7(b). Thus E(L) ≤ M, and we may therefore assume at this point that L = M.
With N as above we have L = N N ⊥ 6= N , and L =
6 N ⊥ . As L = F ∗ (L) =
F ∗ (N )F ∗ (N ⊥ ), induction on |S| yields K E L = E(L). Then L = KK⊥ is a product
of pairwise commuting components, and it suffices now to show that K⊥ contains each
component K′ of L other than K.
Let K′ ∈ Comp(L), and suppose that K =
6 K′ . Then K∩K′ ≤ Z(K) by 8.2. As L/Z(L)
is regular by 7.2, we may assume Z(L) = 1. Let K′ be a component with K =
6 K′ . Then
K′ E L and [S ∩ K, S ∩ K′ ] = 1. As K′ = O p (K′ ) we then have K′ ≤ O p (CL (S ∩ K) by
II.7.4, and then K′ ≤ K⊥ as desired, by definition 5.5. 
45

Corollary 8.6. Let (L, ∆, S) be a regular locality and let N be a partial normal subgroup
of E(L). Then N = E(N )Z, where Z ≤ Z(E(L)) and where E(N ) is a product of
components of L.
Proof. We may assume without loss of generality that L = E(L). Then Op (L) = Z(L)
by the final statement in 8.5. We have N = F ∗ (N ) by 7.7(b), and thus N = Op (N )E(N )
where E(N ) is a product of components of N . The components of N are by definition
components of L, and Op (N ) ≤ Op (E(L)) by 7.7(d). Set Z = Op (N ). Thus Z ≤ Z(L)
and N = E(N )Z. 
Section 9: Im-partial subgroups and E-balance
Let (L, ∆, S) be a regular locality on F , and let X ≤ S be fully normalized in F . We
shall see that there is then a regular locality LX on NF (X) which can be constructed in
a somewhat indirect way from the partial group NL (X). One of the goals of this section
is to show that if X ≤ F ∗ (L) then LX can be produced directly as a subset of L, and
that the inclusion map LX → L is a homomorphism of partial groups.
In the category of groups and homomorphisms one has the obvious equivalence between the notions of “subgroup of G” and “image of a homomorphism into G”. The
situation is different in the category of partial groups, where partial subgroups are indeed images of homomorphisms, but where images of homomomorphisms need not be
partial subgroups. As an example, take L to be the additive group of integers, and let H
be the subset {−1, 0, 1} of L. Then H is the image of a homomorphism of partial groups
(see I.1.2 and I.1.12), but H is not a partial subgroup of L (since partial subgroups of
groups are subgroups).
Definition 9.1. Let H and M be partial groups. Then H is an im-partial subgroup
of M if H is the image in M of a homomorphism of partial groups. Equivalently:
D(H) ⊆ D(L) and P iH is the restriction of ΠM to D(H). Write K 4 H to indicate that
K is an im-partial subgroup of H.
It is obvious that the relation 4 is transitive. Of the long list (analogous to I.1.8) of
properties of im-partial subgroups that one might compile, the following lemma provides
the few instances that will be needed here.
Lemma 9.2. Let M be a partial group, let H 4 M be an im-partial subgroup, and let
K ≤ M be a partial subgroup. Then:
(a) H ∩ K is an im-partial subgroup of K and of M.
(b) H ∩ K is a partial subgroup of H, and if K E M then H ∩ K is a partial normal
subgroup of H.
Proof. Let w ∈ W(H ∩ K) ∩ D(H). Then ΠM (w) = ΠH (w) ∈ H. As D(H) ⊆ D(M)
and K ≤ M we have also Π(w) ∈ K. Thus H ∩ K ≤ H and H ∩ K 4 K. Transitivity of
4 yields H ∩ K 4 M. Now suppose that K E M, and let x ∈ H ∩ K and h ∈ H such
that (h−1 , x, h) ∈ D(H). Then ΠH (h−1 , x, h) ∈ H. But also (h−1 , x, h) ∈ D(M), so that
xh ∈ K. Thus H ∩ K E H. 
46

Example 9.3. Let (H, Γ) be a localizable pair in a locality (L, ∆, S). Then HΓ is an
im-partial subgroup of L.
For the remainder of this section (L, ∆, S) will be a regular locality on F . Denote
by Ls the expansion of L to a proper locality whose set of objects is F s , and by Lc the
restriction of Ls to a proper locality on F whose set of objects is F c . Set
F ∗ (F ) = FS∩F ∗ (L) (F ∗ (L)) and E(F ) = FS∩E(L) (E(L)).
For any proper locality L′ on F that can be obtained from L by a process of expansions
and restrictions, define F ∗ (L′ ) and E(L′ ) by means of Theorem II.A2.
For any subgroup X ≤ S write FX for NF (X). If X is fully normalized in F then by
II.6.3 there exists a proper locality (LcX , (FX )c , NS (X)) on FX , and in fact the proof of
II.6.3 involves the construction of LcX as in im-partial subgroup of the locality Lc . Write
LsX for the expansion of LcX to a proper locality on FX whose set of objects is (FX )s ,
and LX for the restriction of LsX to a regular locality on FX .
Lemma 9.4. Let X be fully normalized in F , and let P be a subgroup of NS (X) containing X. Suppose that P ∈ (FX )s . Then P ∈ F s .
Proof. By II.6.3 FX is the fusion system of a proper locality. Subgroups of NS (P ) which
are fully normalized in FX are then fully centralized in FX , by II.1.13. As P ∈ (FX )s ,
there exists an FX -conjugate Q of P such that Q is fully centralized in FX , and then
Q is fully centralized in F by II.1.16. Set R = Op (CF (Q)). Then, since Q ∈ (FX )s ,
II.6.4 yields R ∈ CFX (Q)c . As CFX (Q) = CF (Q), II.6.4 yields also Q ∈ F s , and thus
P ∈ F s. 
Corollary 9.5. Let X be fully normalized in F , and set Γ = (FX )s . Then (NLs (X), Γ)
is a localizable pair, and the locality (NL (X)Γ , Γ, NS (X)) given by 1.2 is isomorphic to
LsX .
Proof. We note first of all that FX is the fusion system of a proper locality by 6.3,
hence Γ is FX -closed by 6.7. Secondly, NS (X) is a maximal p-subgroup of NL (S) as a
straightforward consequence of I.2.11(b) and of the hypothesis that X is fully normalized
in F . Thirdly, for any w ∈ W(NL (X)) with NSw (X) ∈ Γ we have NSw (X) ∈ F s by
9.4. This verifies the three conditions for a localizable pair in definition 1.1, and so 1.2
applies and yields a locality (NL (X)Γ , Γ, NS (X)). By 1.4(b) this locality is in fact a
proper locality on FX . Further, (NL (X)Γ is an expansion of the locality LcX constructed
in I.6.3, and is therefore isomorphic to LsX by Theorem II.A1. 
In what follows, we shall always identify LsX with the locality NLs (X)Γ in 9.5, and
LX with the restriction of LsX to δ(FX ).
Corollary 9.6. We have LX 4 LsX 4 Ls .
Proof. Immediate from 9.3. 
47

Lemma 9.7. Let X be fully normalized in F . Then X ∈ F s if and only if E(LX ) = 1.
Proof. Set P = Op (LX ), and note that P = Op (FX ) by II.2.3. Suppose first that
E(LX ) = 1. Then F ∗ (LX ) = P by an application of 8.5 to the regular locality LX .
Thus P is a large partial normal subgroup of LX , and so CLX (P ) ≤ P . Then P ∈ F c
by II.6.2, and thus X ∈ F s . Conversely, suppose that X ∈ F s , so that P ∈ F c . Then
S ∩ E(LX ) = 1, and then the regular locality E(LX ) is trivial. 
Lemma 9.8. Let H be a product of components of L, and let K be the product of those
components of L which are not contained in H. Assume that H E L, and let D be a
subgroup of S ∩ H⊥ such that D is fully normalized in F and such that S ∩ K ≤ D. Then
S ∩ H ≤ E(LD ).
Proof. Set Γ = {P ∈ ∆ | D E P }, set E = FS∩H (H), and set V = S ∩ K. Notice that
E(L) = HK and that [H, K] = 1, by 8.5. Then 2.10 applies to the regular locality E(L),
and 2.10(e) yields the following result.
(1) Let U ≤ S ∩ H. Then U V ∈ ∆ and U D ∈ Γ if and only if U ∈ E s .
In particular, Γ is non-empty, and then 1.2 shows that (NL (D), Γ) is a localizable pair.
Write L(D,Γ) for the locality (NL (D)Γ , Γ, NS (D)).
Let w ∈ W(H) and set P = NSw (D). Then D E P , and (1) shows that P ∈ Γ if and
only if Sw ∩ H ∈ E s . Thus P ∈ Γ if and only if w ∈ D(H), and this shows that H is a
partial subgroup of L(D,Γ) . As H E L we obtain:
(2) H E L(D,Γ) .
As H E L we have S ∩H strongly closed in F , and hence S ∩H is strongly closed in FD .
The hypothesis of II.6.10 is then fulfilled, with the subcentric locality (LsD , (FD )s , NS (D))
on FD in the role of L and with S ∩ H in the role of T . We may therefore conclude that
for each P ∈ (FD )cr there exists an FD -conjugate Q of P with Q ∩ H ∈ E c . Then Q ∈ Γ
by (1), and then (FD )cr ⊆ Γ as Γ is FD -closed. Then 1.4(b) yields:
(3) L(D,Γ) is a proper locality on FD .
We may now apply Theorem II.A to L(D,Γ) . Let M+ be a large partial normal
subgroup of LsD containing Op (FD ), and set M = M+ ∩ L(D,Γ) . Then S ∩ M = S ∩ M+ ,
and H ∩ M E L(D,Γ) . Set H0 = CH (H ∩ M). As H ∩ M E H where H is a regular
locality, 7.7(c) yields H0 E H. Then H0 = Z(H)H1 where H1 = O p (H1 ) is a product
of components of H, by 8.6. As L(D,Γ) is a subset of NL (S ∩ H), it follows from 7.7(e)
that L(D,Γ) acts on H by conjugation. This action plainly preserves H0 and H1 . Thus
H1 E L(D,Γ) . We now claim (see 3.7):
(4) ΣS∩H1 (FD ) ⊆ Γ.
For this it suffices to show that DXY ∈ Γ for X ∈ FS∩H1 (H1 )cr and Y ∈ CFD (S ∩H1 )cr .
For any such X and Y we have XY ∈ E cr and V XY ∈ E(F )cr by 2.10(d), so V XY ∈ ∆.
Thus DXY ∈ Γ, and (4) holds.
We have [H1 , S ∩ M] ≤ H1 ∩ M ≤ Z(H1 ), and then [H1 , S ∩ M] = 1 since H1 =
O p (H1 ). Let H1+ be the partial normal subgroup of LsD whose intersection with L(D,Γ) is
H1 . Then H1+ is the partial subgroup hH1 i of LsD generated by H1 , by (4) and 3.14(b),
48

and it follows that [H1+ , S ∩ M] = 1. As H1+ = O p (H1+ ) by II.7.3, it follows from
definition 5.5 that H1+ ≤ (Ms )⊥ (where the “⊥” operation is taken in the locality LsD ).
As Ms is large in LsD we have S ∩ (Ms )⊥ = Z(Ms ) by 6.4, and thus S ∩ H1 ≤ M. As
H1 ∩ M ≤ Z(H1 ) we conclude that H1 = 1 and that H ≤ M. Thus S ∩ H ≤ F ∗ (LsD ).
As S ∩ F ∗ (LsD ) = S ∩ F ∗ (LD ) by Theorem II.A, the proof is complete. 
We are now in position to prove a version for regular localities of the “L-balance”
Theorem from finite group theory. For a discussion of this result for finite groups see
[GLS], and for a fusion-theoretic version see [Asch].
Theorem 9.9. Let L be a regular locality X ≤ S be fully normalized in F . Then
S ∩ E(LX ) ≤ E(L). Moreover, if X ≤ F ∗ (L) then LX 4 L and E(LX ) 4 E(L).
Proof. Assume false, and let X be a counterexample of maximal order. set R = S∩E(L)),
and let (K1 , · · · , Km ) be a listing (in arbitrary order) of the components of LX . As
[Ki , Kj ] = 1 for all i and j with i 6= j, I.5.1 shows that S ∩ E(LX ) is the product in
any order of the intersections S ∩ Ki . Let H be the product of all of the components Ki
such that S ∩ Ki  E(L). As LX 4 Ls we have H ∩ E(Ls ) E H by 9.2(b), and then
H ∩ E(Ls ) ≤ Z(H) by 8.6.
Set A = NR (X). Then [S ∩ H, A] ≤ Z(H), so A normalizes every P in FS∩H (H)cr .
For each such P , 9.2(b) yields
[NH (P ), A] ≤ NH (P ) ∩ E(L) ≤ Z(H).
Thus NH (P ) centralizes the chain Z(H)A ≥ Z(A) ≥ 1, and then [O p (NH (P )), A] = 1
by coprime action (II.2.5). By 5.2 O p (H) is generated by the set of all O p (NH (P )) with
P as above, and then [H, A] = 1 as H = O p (H). Thus [H, AX] = 1.
Let D ∈ (AX)F be fully normalized in F . As F is inductive there exists an F homomorphism φ : NS (AX) → NS (D) such that (AX)φ = D. Set Y = Xφ and
B = Aφ. Then B ≤ NR (Y ), and equality then holds since X is fully normalized in F ,
and since R is strongly closed in F . As NS (X) ≤ NS (AX), φ maps NS (X) into NS (Y ),
and so Y is fully normalized in F . We observe that the F -isomorphism NS (X) → NS (Y )
induced by φ induces also an isomorphism FX → FY of fusion systems, by II.1.5. Since
S ∩ E(LX ) and S ∩ E(LY ) depend only on FX and FY , we may now assume that X = Y
and that D is fully normalized in F . Moreover, from the preceding paragraph we have:
(1) H ≤ CE(LX (D).
Note that R ∈ F s by II.6.9. If R ≤ X then also X ∈ F s , and then 9.7 yields a
contradiction to the non-triviality of H. Thus R  X, and so X is a proper subgroup of
D. The maximality of |X| then implies that D is not a counterexample to the lemma,
and so:
(2) S ∩ E(LD ) ≤ E(L).
Set SD,X = NS (D) ∩ NS (X), and let FD,X be the fusion system NNF (D) (X) on
SD,X . It is a straightforward consequence of the definition (following II.1.4) of normalizers in fusion systems that FD,X = NNF (X) (D). Set ΓD,X = (FD,X )s , and write
49

(LsD,X , ΓD,X , SD,X ) for the proper locality on FD,X given by two applications of 9.5. By
9.4, a word w ∈ W(NL (D) ∩ NL (X)) is in D(LsD,X ) if and only if Sw ∩ SD,X ∈ F s .
Thus:
(3) LsD,X 4 LsD 4 Ls .
We may now appeal to lemma 9.8 with LX in the role of L, in order to obtain S ∩ H ≤
E(LsD,X ). We may write also:
(4) S ∩ H ≤ E(LD,X ),
where LD,X is the regular locality obtained by restriction from LsD,X .
Suppose that D ≤ Op (L). Then X ≤ Op (L), and so X ≤ E(L)⊥ . This yields A = R,
so that R ≤ D ≤ Op (L). Then E(L) = 1, and F ∗ (L) = Op (L). Thus Op (L) ∈ ∆, and
L is a group of characteristic p. Then LX = NL (X) is a group of characteristic p by
II.2.7(b), and E(LX ) = 1. As there is nothing to prove in this case, we may assume that
D  Op (L).
By induction on |L| we may then assume that the Theorem holds for LD in the role
of L. Thus:
(5) S ∩ E(LD,X ) ≤ S ∩ E(LD ).
Suppose next that X = D. Then R ≤ X, so Op (L)X ∈ ∆ ⊆ F s , and then X ∈ F s
by II.6.9. Then E(LX ) = 1 by 9.7, and there is again nothing to prove. Thus we
may assume that X is a proper subgroup of D, and the maximal choice of X then
yields S ∩ E(LD ) ≤ E(L). This result, in combination with (4) and (5), now yields
S ∩ H ≤ E(L). By definition, H is the product of those components K of L such that
S ∩ K  E(L), we conclude that H = 1, and that S ∩ E(LX ) ≤ E(L). This completes
the first part of the proof.
Suppose now that X ≤ F ∗ (L). Let w ∈ D(LX ), and set P = NSw (X) ∩ F ∗ (L). Then
s
X ≤ P , and Sw ∩ E(LX ) ≤ P by what has just been proved. Here Sw ∩ F ∗ (LX ) ∈ FX
s
s
by the definition of D(LX ), and then Sw ∩ E(LX ) ∈ FX by II.6.9. Thus X ≤ P ∈ FX ,
and then P ∈ F s by 9.4. Thus P ∈ δ(F ). Recall that LX 4 Ls by definition. In the
case that w = (g) for some g ∈ LX we now conclude that g ∈ L. Thus LX ⊆ L, and
D(LX ) ⊆ D(L). This yields LX 4 L.
We have already seen that S ∩ E(LX ) ≤ E(L). As E(LX ) 4 L we have E(LX ) ∩
E(L) E E(LX ) by 9.2(b). Each component K of LX has the property that K/Z(K) is
simple, so K has no proper partial normal subgroups containing S ∩ K. This shows that
E(LX ) ⊆ E(L), and then E(LX ) 4 E(L) by 9.2(a). 
Section 10: Theorems C, D, and E
Theorem C is given by Theorems 5.7 and 7.7 (which in fact contain a great deal more
information), and Theorem D is Theorem 8.5.
Let (L, ∆, S) be a regular locality on F . Point (a) of Theorem E is that if N E L
is a partial normal subgroup of L then N is a regular locality; which is already part
of Theorem C. Point (b) of Theorem E, concerning the regular locality LX on NF (X),
50

under the assumption that X is fully normalized in F and that X ≤ F ∗ (L), is given by
Theorem 9.9. Point (c) of Theorem E is Proposition 7.2.
References
[Asch]
[Ch1]
[Ch2]
[Ch3]
[GLS]

[Gold]
[Gor]
[He]

Michael Aschbacher, The generalized Fitting subsystem of a fusion system, Memoirs Amer.
Math. Soc. 209 (2011).
Andrew Chermak, Fusion systems and localities, Acta Math. 211 (2013), 47-139.
, Finite localities I, (preprint) (2016).
, Finite localities II, (preprint) (2016).
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The Classification of the Finite Simple
Groups, Number 2, Mathematical surveys and monographs, volume 40, number 2, American
Mathematical Society, 1991.
David Goldschmidt, A conjugation family for finite groups, Jour. of Alg. 16 (1970), 138-142.
Daniel Gorenstein, Finite groups, Second Edition, Chelsea, New York, 1980.
Ellen Henke, Products of partial normal subgroups, (arXiv:1506.01459) (2015).

Manhattan Kansas
E-mail address: chermak@math.ksu.edu

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