arXiv:1711.08625v1 [math.RT] 23 Nov 2017

THE BRAUER INDECOMPOSABILITY OF SCOTT
MODULES FOR THE QUADRATIC GROUP Qd(p)
SHIGEO KOSHITANI AND İPEK TUVAY

Abstract. Let k be an algebraically closed field of prime characteristic p and P a finite p-group. We compute the Scott kG-module
with vertex P when F is a constrained fusion system on P and G
is Park’s group for F . In the case F is a fusion system of the quadratic group Qd(p) = (Z/p×Z/p)⋊SL(2, p) on a Sylow p-subgroup
P of Qd(p) and G is Park’s group for F , we prove that the Scott
kG-module with vertex P is Brauer indecomposable.

1. Introduction
Let p be a prime number and k an algebraically closed field of characteristic p. For a finite group G, a p-subgroup Q of G and a finite
dimensional kG-module M, the Brauer quotient M(Q) of M with respect to Q has a natural kNG (Q)-module structure. A kG-module
M is said to be Brauer indecomposable if M(Q) is indecomposable or
zero as a k(Q CG (Q))-module for any p-subgroup Q of G (see [11]).
Brauer indecomposability of p-permutation modules is important for
verifying categorical equivalences between p-blocks of finite groups (see
[8, 10, 11, 12]).
More precisely, if we want to have a stable equivalence of Morita
type between the two principal blocks B0 (kG) and B0 (kH) of finite
groups G and H, respectively, such that G and H have a common
Sylow p-subgroup P , then it is quite natural to expect that the Scott
k(G×H)-module M = Sc(G×H, ∆P ) with respect to ∆P = {(u, u) ∈
P × P |u ∈ P } would realize the stable equivalence of Morita type. If
this is the case, then for any non-trivial subgroup Q of P the Brauer
quotient M(∆Q) induces a Morita equivalence between the principal
blocks B0 (kCG (Q)) and B0 (kCH (Q)) (by so-called the gluing method
due to Broué, Rouquier and Linckelmann) , and it turns out that
M(∆Q) has to be indecomposable as (kCG (Q), kCH (Q))-bimodules
because blocks are indecomposable as bimodules.
Date: November 27, 2017.
Key words and phrases. Scott module, constrained fusion system, the quadratic
group.
1

A relationship between Brauer indecomposability of p-permutation
modules and saturated fusion system is given by R. Kessar, N. Kunugi
and N. Mitsuhashi. Namely, in [11, Theorem 1.1], they prove that if
there exists a Brauer indecomposable kG-module with vertex P then
FP (G) is a saturated fusion system. They also show that the converse
of this statement is not necessarily true. However, when P is abelian
and M is the Scott kG-module with vertex P they prove the converse
is also true. Later, this result is extended for saturated fusion systems
on P which are not necessarily abelian under the special case that P is
normal in F and G is Park’s group for F ([17, Theorems 1.3 and 1.4];
for the construction and details of Park’s group we refer the reader to
[15]). Moreover, in [17] the following question is posed:
Question 1.1. Let P be a finite group, F a saturated fusion system
on P and X a characteristic P − P -biset for F . For G = Park(F , X)
and ι Park’s embedding of P into G, is the Scott kG-module Sc(G, ιP )
Brauer indecomposable?
We answer this question positively in some cases. One of these is the
following result which is the main result of this paper.
Theorem 1.2. Let P be a Sylow p-subgroup of Qd(p) = (Z/p × Z/p) ⋊
SL(2, p) and F = FP (Qd(p)). Then for G = Park(F , Qd(p)) and ι
Park’s embedding of P into G, the Scott kG-module with vertex ιP is
Brauer indecomposable.
As an aside, we prove the following theorem.
Theorem 1.3. Let M be a finite group of order 3·2n for an integer n ≥
3. Assume that M is of 2-length 2 and O2′ (M) = 1. Let further P be a
Sylow 2-subgroup of M, and F = FP (M). Then for G = Park(F , M)
and ι Park’s embedding of P into G, the Scott module kG-module with
vertex ιP is Brauer indecomposable.
We shall use the following notation and terminology. In this paper G
is always a finite group, and k is an algebraically closed field of a prime
characteristic p. For the group Qd(p), see Section 4. For a subset S of
G and g ∈ G, we define g S = {gsg −1|s ∈ S} and S g = {g −1sg|s ∈ S}.
We write H ≤ G if H is a subgroup of G and H < G if H is a
proper subgroup of G. Modules in this paper are finitely generated
and left modules. For a subgroup H and for kH- and kG-modules N
and M respectively, we write IndG
H (N) = kG ⊗kH N for the induced
module of N from H to G (and hence this becomes a kG-module),
and ResG
H (M) for the restriction of M to H (and hence this becomes
a kH-module). We denote by kG = k the trivial kG-module. For a
2

subgroup H of G we can define the Scott kG-module Sc(G, H) with
respect to H, which is an indecomposable direct summand of IndG
H (kH )
and has kG in its socle (see [14, Chapter 4 §8]). For a kG-module M
and a p-subgroup Q of G we denote by M(Q) the Brauer quotient of
P
R
M with respect to Q. Namely, M(Q) = M Q / R<Q TrQ
R (M ) where
M Q = {m ∈ M| um = m for all u ∈ Q} and TrQ
R is the trace map
from M R to M Q (see [16, §27]). For a subgroup H and a normal
subgroup N of G we write G = N ⋊ H if G is a semi-direct product
of N by H. For a positive integer n we denote by Sn the symmetric
group of degree n. For a finite set Ω we denote by Sym(Ω) the set
of all bijections of Ω, namely this is isomorphic to S|Ω| where |Ω| is
the number of the elements of Ω. For a subgroup A of the symmetric
group Sn and a group H, we write H ≀ A for the wreath product of H
by A. For subgroups Q, R of G, let HomG (Q, R) denote the set of all
group homomorphisms from Q to R which are induced by conjugation
by some element of G. For a p-subgroup P of G, the fusion system
FP (G) of G on P is the category whose objects are the subgroups of
P and whose morphism set from Q to R is HomG (Q, R). When FP (G)
is the fusion sytem of G on P , cg for g ∈ G is defined in [1, p.3], and
for the notation Op (F ) when F is a fusion system see [1, p.18]. For
the other notations in finite group theory such as Op (G), Op′ (G) and
Z(G), see [7]; in representation theory of finite groups see [14] or [16];
and in fusion systems, see [1] or [13].
The paper is divided into five sections. Section 2 contains results
needed for the proofs of the theorems in later sections. Section 3 analyzes Park’s group and Park’s embedding corresponding to a constrained fusion system. The proof of Theorem 1.2 is presented in Section 4 and the proof of Theorem 1.3 is in Section 5.
2. Lemmas
The following lemma is a generalization of Green’s Indecomposability
Theorem.
Lemma 2.1 ([8, Lemma 4.2]). Let G be a finite group and let k an
algebraically closed field of characteristic p. If H is a subgroup of G
such that |G : H| is a power of p, then IndG
H (k) is an indecomposable
kG-module.
When proving Brauer indecomposability results in the next sections,
we are dealing with Brauer quotients of induced modules several times.
The following lemma makes these computations in the particular case
which we are interested in.
3

Lemma 2.2. Let H be a subgroup of a finite group G and P a Sylow psubgroup of H. Suppose that FP (H) = FP (G), then for M = IndG
H (k)
and for any Q ≤ P we have
N (Q)

N (Q)

Q C (Q)

M(Q) = IndNHG (Q) (k) and ResQGCG (Q) (M(Q)) = IndQCHG(Q) (k).
N (Q)

Proof. From [4, 1.4], we have M(Q) = ⊕g IndNGG (Q)∩ gH (k) where g runs
over representatives of double cosets NG (Q)\TG (Q, H)/H for
TG (Q, H) = {g ∈ G | Qg ≤ H}. We claim that TG (Q, H) = NG (Q)H.
Indeed, if g ∈ TG (Q, H) the condition Qg ≤ H implies that there
exists x ∈ H such that Qgx ≤ P since P is a Sylow p-subgroup of
H. Thus cgx : Q → Qgx is in FP (G) = FP (H) which implies gx to
be in HCG (Q). Hence g ∈ HCG (Q) which shows the containment
TG (Q, H) ≤ HNG (Q) and the other direction is trivial. Therefore,
there exists only one coset above and the first statement of the lemma
is established.
For the second statement, observe that
M
N (Q)
Q C (Q)
ResQGCG (Q) (M(Q)) =
IndQ CGG (Q)∩ gNH (Q) (k)
g

where g runs over a set of representatives of double cosets
Q CG (Q)\NG (Q)/NH (Q) in NG (Q). By the equality of the fusion systems, we have CG (Q)NH (Q) = NG (Q), so there is only one coset above.
N (Q)
Q C (Q)
Therefore, ResQGCG (Q) (M(Q)) = IndQ CGH (Q) (k).


The following lemma will be used in the proofs of Theorems 1.2
and 1.3. Note that this lemma can also be seen as a generalization of
Green’s Indecomposability Theorem.
Lemma 2.3. Let H be a subgroup of a finite group G and P a Sylow
p-subgroup of H where FP (H) = FP (G). Assume that there exists a
normal p-subgroup N of G such that CG (N ∩ H) is a p-group, then we
have
Sc(G, P ) = IndG
H (k).
Proof. Since P is a Sylow p-subgroup of H, the kG-Scott module
Sc(G, P ) is a direct summand of IndG
H (k) (see [14, Chapter 4, Corollary
8.5]). Thus, there is a kG-module X such that
IndG
H (k) = Sc(G, P ) ⊕ X.
Set R = N ∩H, then R is a normal subgroup of P . If R is trivial, the
condition on the centralizer of R in G implies that G is a p-group, so
IndG
H (k) is indecomposable by Green’s Indecomposability Theorem and
hence IndG
H (k) = Sc(G, P ) in this case. Assume R is non-trivial and set
4

N (R)

G
I = IndG
H (k) and S = Sc(G, P ). By Lemma 2.2 I(R) = IndNH (R) (k).
Moreover, by the equality of fusion systems, we have

|NG (R)|/|CG (R)| = |NH (R)|/|CH (R)|.
So the quotient |NG (R) : NH (R)| is a power of p since CG (R) is a
N (R)
p-group. Thus Lemma 2.1 implies that I(R) = IndNGH (R) (k) is indecomposable. Therefore, the identity
I(R) = S(R) ⊕ X(R)
implies that either X(R) = 0 or S(R) = 0. If the latter case occurs,
since R is normal in P , as NG (P ) ∩ NG (R)-modules
0 6= S(P ) ∼
= S(R)(P ) = 0
(see [5, Proposition 1.5(3)]) where the left hand side is non-zero since S
has vertex P . Hence we get a contradiction. Thus, we have X(R) = 0.
On the other hand, since X | IndG
H (k) by the Mackey formula
M
IndN
ResG
N ∩ gH (k)
N (X)
g

where g runs through representatives of double cosets in N\G/H. By
Green’s Indecomposability Theorem each summand above is indecomposable since N is a p-group. On the other hand, N being normal in G
implies N ∩ gH = g (N ∩ H) = gR. Hence there is a direct summand
X ′ of X such that X ′ (gR) 6= 0. This contradicts with X(R) being zero.
Therefore X must be zero.

3. Park’s construction in the case of constrained fusion
systems

Recall that a saturated fusion system F on P is called constrained
if CP (Op (F )) ≤ Op (F ). This definition is analogous with the definition of p-constrained groups. Recall that a finite group G is called
p-constrained if Op (G/Op′ (G)) contains its own centralizer in G/Op′ (G)
and called strictly p-constrained if G is p-constrained and Op′ (G) = 1.
It is shown in [3, Proposition 4.3] that every constrained fusion system
F on P is realized by a unique strictly p-constrained group containing
P as a Sylow p-subgroup, which is called the model for F .
Let F be a constrained fusion system on P and M be the model
for F . Since P is a Sylow subgroup of M and F = FP (M), the
group M can be regarded as a characteristic P − P -biset for F (for
definition and existence result of a characteristic biset for a saturated
fusion system, we refer the reader to [2, Proposition 5.5]). Thus M
can be used to construct Park’s group for F . Recall that Park(F , M)
5

is the group of bijections of the P − P -biset M preserving the right
P -action. Letting G = Park(F , M), Park’s embedding ι : P ֒→ G is
defined as ι : u 7→ (x 7→ ux) where F ∼
= FιP (G) ( [15, Theorem 3]). As
mentioned in [15] G is isomorphic to the wreath product P ≀ Sn where n
is the index |M : P |. Throughout the rest of the paper, let us identify
G with P ≀ Sn whenever G = Park(F , M) where M is the model for the
constrained fusion system F and n is the index of P in M.
Lemma 3.1. Let F be a constrained fusion system on P and M the
model for F . Let also G = Park(F , M) then
−1
−1
−1
ιP = {(m−1
1 umσu
(1) , . . . , mn umσu
(n) ; σu ) | u ∈ P }

where {m1 , . . . , mn } is a set of left coset representatives of P in M.
F
Proof. Let us write M = ni=1 mi P as a disjoint union of left cosets
of P . Since any automorphism in G is determined by its value on the
coset representatives mi , we can compute the subgroup ιP as follows.
Let u be an arbitrary element of P , then the image of ι(u) in G is
(x1 , . . . , xn ; σu ) where
ι(u)(mi ) = umi = mσu (i) xσu (i) .

Let B denote the base group of the wreath product. When Op (F ) 6=
P , the subgroup ιP does not lie in B. In fact, ιOp (F ) is the largest
subgroup of ιP which lies inside B as the following lemma shows.
Lemma 3.2. Let F be a constrained fusion system on P and M the
model for F . Set G = Park(F , M). Then ιOp (F ) = B ∩ ιM and as a
subgroup of G
ιOp (F ) = {(um1 , . . . , umn ; id) | u ∈ Op (F )}
where {m1 , . . . , mn } is a set of left coset representatives of P in M.
Proof. Since Op (F ) is normal in M, any u ∈ Op (F ) satisfies m−1
i umi ∈
Op (F ) so umi P = mi P . Hence, for all u ∈ Op (F ) we have σu = id and
umi = mi (m−1
i umi ) for all i = 1, . . . , n. These imply
−1
ι(u) = (m−1
1 um1 , . . . , mn umn ; id)

which shows why the second statement is true.
M acts on itself by left multiplication, thus ι extends to M so that
ιM ≤ G. Set R = B ∩ ιM, then R is a normal p-subgroup of ιM, so
R is a normal subgroup of ιM and ιP . Hence R is fully F -normalized
and by [13, Proposition 3.16]
NF (R) ∼
= FN (R) (NιM (R))
ιP

6

which yields NF (R) = F . So, R is normal in F and this implies
R ≤ ιOp (F ). By the first paragraph, we have ιOp (F ) ≤ B, hence
ιOp (F ) ≤ R.

Lemma 3.3. Let F be a constrained fusion system on P and M the
model for F . Set G = Park(F , M). Then CG (ιOp (F )) is a p-group.
Proof. Let (x1 , . . . , xn ; β) ∈ CG (ιOp (F )), then by Lemma 3.2 we have
(x1 , . . . , xn ; β)(um1 , . . . , umn ; id)(xβ(1) , . . . , xβ(n) ; β −1) = (um1 , . . . , umn ; id)
for all u ∈ Op (F ). This implies that for all i = 1, . . . , n and for all
u ∈ Op (F )
−1
−1
−1
xi m−1
β −1 (i) umβ (i) xi = mi umi
which is equivalent to mi xi m−1
β −1 (i) to be in CM (Op (F )) for all i =
1, . . . , n. Since M is p-constrained and Op (F ) is F -centric [13, Proposition 4.12] implies mi xi m−1
β −1 (i) ∈ Z(Op (F )). So for all i, there exists
zi ∈ Z(Op (F )) such that mi xi m−1
β −1 (i) = zi or equivalently
mi xi = zi mβ −1 (i) .
Since Z(Op (F )) is a characteristic subgroup of Op (F ) and Op (F ) is
normal in M, Z(Op (F )) is normal in M. Thus for any i, there exists
zi′ ∈ Z(Op (F )) with zi mβ −1 (i) = mβ −1 (i) zi′ . This implies that
mi xi = mβ −1 (i) zi′ ,
so mi P = mβ −1 (i) P for all i. Hence β = id and CG (ιOp (F )) ≤ B.



Theorem 3.4. Let F be a constrained fusion system on P , M the
model for F and G = Park(F , M) and ι Park’s embedding of M into
G. Then Sc(G, ιP ) = IndG
ιM (k).
Proof. By Lemma 3.3, CG (ιOp (F )) is a p-group. Also B E G such that
B ∩ ιM = ιOp (F ) by Lemma 3.2. The result follows from Lemma 2.3.

4. The quadratic group case
Consider the semidirect product of an elementary abelian group V of
order p2 by SL(2, p) where SL(2, p) acts naturally on V . Glauberman
called this group as the quadratic group and denoted by Qd(p) (see
[6, p.1104]). Actually this group plays a very important role in finite
group theory and representation theory of finite groups (see [13, pp.
81 and 95]).
7

Set M = Qd(p) and a Sylow p-subgroup P of M as P = V ⋊ < α >
where


1 1
α=
.
0 1

Note that P is isomorphic to the extraspecial group of order p3 with
exponent p. We have


 r l
| r ∈ (Z/p)∗ , l ∈ Z/p .
NM (P ) = V ⋊
0 r −1
Set M1 = NM (P ), then there are p − 1 left cosets of P in M1 , and they
explicitly are
{νr P | r = 1, . . . , p − 1},


r 0
where νr =
and the left cosets of M1 in M are
0 r −1

{M1 , βM1 , . . . , β p−1M1 , γM1 }




1 0
0 −1
where β =
and γ =
. Let us gather these infor1 1
1 0
mation and choose representatives of left cosets of P in M as follows:
(
β s νr+1 if 1 ≤ j ≤ p2 − p
mj =
γ νr+1
if p2 − p < j ≤ p2 − 1
where s is the quotient and r is the remainder when j − 1 is divided
by p − 1. Thus we get a set of left cosets {mj P | j = 1, . . . , p2 − 1} of
P in M.
Set F = FP (M) and G = Park(F , M). Then G = P ≀ Sn for n =
p2 − 1 and Op (F ) = V . The proof of the following lemma is obvious
since P E M1 and M1 is the union of the first p − 1 cosets. We keep
the same notation as in Section 3.
Lemma 4.1. For u ∈ P , the permutation σu fixes all elements in
{1, . . . , p − 1}.
Lemma 4.2. For u ∈ P − V , the permutation σu is a product of p − 1
disjoint p-cycles in Sym({p, . . . , n}).
Proof. For u ∈ P − V , we have u = vαi for some v ∈ V and i ∈
{1, . . . , p − 1}. So σu = σαi by Lemma 3.2. Hence, without loss of
generality we can choose u = α. From Lemma 4.1, σu (i) = i for
i = 1, . . . , p − 1. Moreover, we have
−1

αβ s = β s(s+1)

8

−1

νs+1 α(s+1)

if s + 1 is invertible in Z/p and
αβ p−1 = γ νp−1 αp−1.
Also,
αγ = βαp−1.
−1
Therefore, u(β s P ) = β s(s+1) νs+1 P if s + 1 is invertible in Z/p and
u(β p−1P ) = γ νp−1 P . Also, u(γP ) = βP . As a result, since νr nor−1
malize P , we have u(β s νr P ) = β s(s+1) νs+1 νr P in the case s + 1 is
invertible in Z/p and u(β p−1νr P ) = γ νp−1 νr P and u(γνr P ) = βνr P
for all r = 1, . . . , p − 1.
Since u(mi P ) = mσu (i) P , it is easy to see from the identities above
that, the cosets βνr P = mp+r−1 P for r = 1, . . . , p−1 determine disjoint
p−1
p-cycles of σu . Hence, σu = πr=1
σr where σr = (p + r −1, . . . , p2 −p + r)
and the identity
m(σr )i (p+r−1) P = αi (βνr P )
gives the i+1st position of σr ’s presentation as above for i = 0, 1, . . . , p−
1 and r = 1, . . . , p − 1.

Lemma 4.3. If Q is a subgroup of P of order p2 , then CG (ιQ) is a
p-group.
Proof. There are two types of subgroups of order p2 in P . If Q = V ,
from Lemma 3.3 CG (ιQ) is a p-group. If Q 6= V then Q =< t > × <
y > where t = (1, 0) is a central element of P and y = (a, b)α for some
a, b ∈ Z/p. We have CG (ιQ) = CG (ιt) ∩ CG (ιy). Since t ∈ Op (F ), by
Lemma 3.2 ιt = (tm1 , . . . , tmn ; id). So we have
−1
CG (ιt) = {(x1 , . . . , xn ; τ ) | mτ −1 (i) x−1
i mi ∈ CM (t)}

and since CM (t) = P , this becomes
−1
CG (ιt) = {(x1 , . . . , xn ; τ ) | mτ −1 (i) x−1
i mi ∈ P }.

For i = 1, . . . , p − 1, the condition for (x1 , . . . , xn ; τ ) to be in CG (ιt)
−1
implies mτ −1 (i) x−1
∈ P , which implies mτ −1 (i) P = P mi which is
i mi
equal to mi P since P E M1 . Thus τ (i) = i for i = 1, . . . , p − 1. From
Lemmas 3.1 and 4.2,
−1
ιy = (m−1
1 ymσy−1 (1) , . . . , mn ymσy−1 (n) ; σy )

where σy is a product of p − 1 disjoint p-cycles. So if an element
(x1 , . . . , xn ; τ ) centralizes ιy, then τ centralizes the permutation σy . By
[9, 4.1.19], we have CSym({p,...,n}) (σy ) ∼
= (Z/p) ≀ Sp−1. Here the wreathing
part comes from the permutations which permute the p − 1 cycles
lying in σy . Since τ (i) = i for i = 1, . . . , p − 1, we deduce that if
(x1 , . . . , xn ; τ ) ∈ CG (ιQ) then τ ∈ CSym({p,...,n}) (σy ). Suppose to the
9

contrary that CG (ιQ) is not a p-group. Then by Cauchy’s Theorem
there exists a prime q different from p and an element (x1 , . . . , xn ; τ ) in
CG (ιQ) of order q. In this case, τ has order q. By the argument above
τ σr τ −1 = σr′ where σr and σr′ are the disjoint p-cycles appearing in
the cycle decomposition of σy as in the proof of Lemma 4.2. So, there
exists an integer j appearing in the cycle σr′ such that τ (p + r − 1) =
′
j. By the property of j, we have mj P = αj βνr′ P for some j ′ ∈
{0, 1, . . . , p − 1} and mp+r−1 = βνr . The condition for (x1 , . . . , xn ; τ )
−1
to be in CG (ιt) implies that mp+r−1 x−1
to be in P . Since xj ∈ P ,
j mj
i
xj = vα for some v ∈ V and i ∈ Z/p. Thus the condition is equivalent
to βνr α−1 ν(r′ )−1 β −1 to be in < α >. But the matrix


r(r ′)−1 + ir(r ′ )
−ir(r ′ )
r(r ′ )−1 + ir(r ′ ) − r −1 r ′ −irr ′ + r −1 r ′
lies in < α > if and only if r = r ′ and i = 0. This contradicts with σr
and σr′ to be disjoint. Therefore CG (ιQ is a p-group in this case, too.


Proof of Theorem 1.2. Set M = Qd(p). Since M is p-constrained, F =
FP (M) is a constrained fusion system, so Theorem 3.4 implies that
Sc(G, ιP ) = IndG
ιM (k). Moreover since FιP (ιM) = FιP (G), Lemma 2.2
implies that
N (ιQ)

ιQ C (ιQ)

G
ResιQGCG (ιQ) ((Sc(G, ιP ))(ιQ)) = IndιQ CιM
(ιQ) (k)

for all Q ≤ P . Thus, using [8, Theorem 1.3] it is enough to show that
ιQ CG (ιQ)
IndιQ CιM
(ιQ) (k) is indecomposable for all fully F -normalized Q ≤ P .
If Q ≤ P such that |Q| ≥ p2 then by Lemma 4.3 CG (ιQ) is a p-group.
ιQ C (ιQ)
Hence Green’s Indecomposability Theorem implies that IndιQ CGιM (ιQ) (k)
is indecomposable in this case.
Since Q is fully F -normalized, [13, Proposition 2.5] implies Q is fully
F -centralized. Hence by [13, Lemma 2.10] ιQ CιP (ιQ) is a Sylow psubgroup of ιQ CιM (ιQ). Moreover FιP (ιM) = FιP (G) implies
FιQ CιP (ιQ) (ιQ CM (ιQ)) = FιQ CιP (ιQ) (ιQ CG (ιQ)).
If Q ≤ P of order p, there
Q ≮ V . If Q < V , then Q is
fully F -normalized element of
Q < V , we can set Q = Z(P ).

are two possibilities: either Q < V or
F -conjugate to Z(P ). Since Z(P ) is a
the set of subgroups with the property
Then Q CV (Q) = V . So we have

CιQ CG (ιQ) (ι(Q CV (Q))) = CιQ CG (ιQ) (ιV ) ≤ CG (ιV ).
Lemma 3.3 implies that CG (ιV ) is a p-group so CιQ CG (ιQ) (ι(Q CV (Q)))
is also a p-group. Hence we can apply Lemma 2.3 by changing G
10

with ιQ CG (ιQ), H with ιQ CιM (ιQ), P with ιQ CιP (ιQ) and N with
ιQ CB (ιQ) and deduce that
ιQ C (ιQ)

G
IndιQ CιM
(ιQ) (k) = Sc(ιQ CG (ιQ), ιQ CιP (ιQ)).

If Q ≮ V which is fully F -normalized, then Q CV (Q) = Z(P ) × Q
is a subgroup of P of order p2 . So CG (ιQ CιV (ιQ)) is a p-group by
Lemma 4.3 which implies that CιQ CG (ιQ) (ιQ CιV (ιQ)) is also a p-group.
Hence we can apply Lemma 2.3 by changing G with ιQ CG (ιQ), H with
ιQ CιM (ιQ), P with ιQ CιP (ιQ) and N with ιQ CB (ιQ) and deduce that
ιQ C (ιQ)

IndιQ CGιM (ιQ) (k) = Sc(ιQ CG (ιQ), ιQ CιP (ιQ))
ιQ C (ιQ)

in this case, too. Therefore, we deduce that IndιQ CGιM (ιQ) (k) is indecomposable for all possible fully F -normalized subgroups Q of P as
desired.

5. The 3·2n ordered group case
Proof of Theorem 1.3. Since G = P ≀ S3 , the index |G : ιM| is a power
of 2. Hence Lemma 2.1 together with [14, Chapter 4, Corollary 8.5]
implies that
Sc(G, ιP ) = IndG
ιM (k).
Since FιP (ιM) = FιP (G), Lemma 2.2 implies that
N (ιQ)

ιQ C (ιQ)

G
ResιQGCG (ιQ) ((IndG
ιM (k))(ιQ)) = IndιQ CιM (ιQ) (k)

for all Q ≤ P . So it is enough to show the indecomposability of
ιQ CG (ιQ)
IndιQ CιM
(ιQ) (k) for all Q ≤ P .
M is solvable and O2′ (M) = 1, so M is strictly 2-constrained and
F = FP (M) is a constrained fusion system. Since M has 2-length 2,
the non-trivial subgroup O2 (M) is strictly contained in P . Note also
that O2 (M) = O2 (F ).
If Q 6≤ O2 (F ), then there exists an element u ∈ Q such that σu is a
non-trivial 2-cycle by Lemma 3.2. Thus, if (x1 , x2 , x3 ; β) ∈ CG (ιu), then
β ∈ CS3 (σu ) =< σu >. Hence any element in CG (ιu) is an element of
ιQ C (ιQ)
order a power of 2 and CG (ιQ) is a 2-group. Therefore IndιQ CGιM (ιQ) (k)
is indecomposable by Green’s Indecomposability Theorem.
If Q ≤ O2 (F ), we have two cases: either CM (Q) is a group of composite order (that is |CM (Q)| = 3.2k for k ≤ n) or CM (Q) is a 2-group.
In the first case, since CιM (ιQ) ≤ CG (ιQ), 3 divides both of the orders
CG (ιQ) and CιM (ιQ) so that the index |CG (ιQ) : CιM (ιQ)| is a power
of 2, hence the corresponding induced module is indecomposable by
Lemma 2.1. If CM (Q) is a 2-group, then CM (Q) lies inside one of the
11

conjugates of P , since P is a Sylow p-subgroup of M. Suppose that
CM (Q) ≤ P . Then since Q ≤ O2 (F ), by Lemma 3.2
ιQ = {(um1 , um2 , um3 ; id) | u ∈ Q}
where {m1 , m2 , m3 } is a set of coset representatives of P in M. So
−1
(x1 , x2 , x3 ; β) ∈ CG (ιQ) if and only if mβ −1 (i) x−1
∈ CM (Q). Withi mi
out loss of generality m1 can be chosen as the representative of the
coset P . Putting i = 1 in the last condition, we get mβ −1 (1) P = P
since CM (Q) ≤ P . We get β(1) = 1 and so β is either identity or a
2-cycle. Therefore CG (ιQ) is a 2-group and the indecomposability of
ιQ CG (ιQ)
IndιQ CιM
(ιQ) (k) follows from Green’s Indecomposability Theorem. If
CM (Q) 6≤ P , there exists an element m ∈ M such that CM (Q) ≤ P m .
From this we get CM (m Q) ≤ P. Thus CG (ι(m Q)) is a 2-group by the
argument above. Since CG (ι(m Q)) ∼
= CG (ιQ), the centralizer CG (ιQ)
ιQ C (ιQ)
is a 2-group. Hence indecomposability of IndιQ CGιM (ιQ) (k) follows from
Green’s Indecomposability Theorem. Therefore, the Brauer indecomposability is established.

Acknowledgements
A part of this work was done while the second author was visiting Chiba
University in July 2017. She would like to thank Shigeo Koshitani and
the Center for Frontier Science, Chiba University for their hospitality. She
would like to thank also Naoko Kunugi for her hospitality.

Funding
The first author was supported in part by the Japan Society for Promotion
of Science (JSPS), Grant-in-Aid for Scientific Research (C)15K04776, 2015–
2018. The second author was partially supported by the Center for Frontier
Science, Chiba University and Mimar Sinan Fine Arts University Scientific
Research Project Unit with project number 2017/22.

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Center of Frontier Science, Chiba University, 1-33 Yayoi-cho, Inageku, Chiba 263-8522, Japan.
E-mail address: koshitan@math.s.chiba-u.ac.jp
Mimar Sinan Fine Arts University, Department of Mathematics, 34380,
Bomonti, Şişli, Istanbul, Turkey
E-mail address: ipek.tuvay@msgsu.edu.tr

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