Central intersections of element centralisers

arXiv:1702.00245v1 [] 1 Feb 2017

Julian Brough

FB Mathematik, TU Kaiserslautern, Postfach 3049
67653 Kaiserslautern, Germany
E-mail: brough@mathematik.uni-kl.de

MSC:
Primary: 20E34
Seconday: 20D99
Keywords:
Finite groups, element centralisers, CA-groups, F-groups
Abstract
In 1970 R. Schmidt gave a structural classification for CA-groups. In this paper
we consider a condition upon the intersection of element centralisers which turns out
to be equivalent to the definition of a CA-group. We then weaken which centralisers
we chose to intersect and structurally classify this new family of groups. Furthermore
we apply a similar weakening to the class of F-groups introduced by Itô in 1953 and
classified by Rebmann in 1971.

1

Introduction

A finite group is called a CA-group if the centraliser of every non-central element is abelian.
If G is a CA-group and x, y ∈ G \ Z(G), then CG (x) can never be properly contained in
CG (y). Furthermore we shall see in Lemma 2.1 that G being a CA-group is equivalent to
saying for all x and y non-central elements in G either CG (x) = CG (y) or CG (x) ∩ CG (y) =
Z(G). In addition to the class of CA-groups, Itô [Itô53] introduced the notion of an Fgroup. This is a group in which every non-central element centraliser contains no other
non-central element centraliser. That is, G is an F-group if for any x ∈ G \ Z(G), then
CG (x) ≤ CG (y) implies y ∈ Z(G). We shall also see in Lemma 2.6 that G being an Fgroup is equivalent to Z(CG (x)) ∩ Z(CG (y)) = Z(G) for all x, y ∈ G \ Z(G) such that
CG (x) 6= CG (y).
The aim of this paper is to consider these intersection conditions for a specific subset
of centralisers, in particular, the set of minimal centralisers in a group (those which do not
properly contain any other element centraliser). Thus we define a group to be a CAmin group if for two non-central elements x and y with minimal element centralisers in G either
CG (x) = CG (y) or CG (x)∩CG (y) = Z(G). Similarly we call G an Fmin -group if for two noncentral elements x and y with minimal element centralisers in G either CG (x) = CG (y) or
Z(CG (x)) ∩ Z(CG (y)) = Z(G). Note that the analogous condition by considering maximal
centralisers for CA-groups was studied by Schmidt [Sch70] (although Schmidt defined such
groups using subgroup centralisers).
The aim of this paper is to prove a structural classification of CAmin -groups and Fmin groups.
1

Theorem 1.1. G is a CAmin -group (respectively Fmin ) if and only if G has one of the
following forms:
1. G/Z(G) is a Frobenius group with kernel L/Z(G) and complement K/Z(G) such that
both K and L are abelian.
2. G/Z(G) is a Frobenius group with kernel L/Z(G) and complement K/Z(G) such that
K is abelian, L is a CAmin -group (respectively Fmin ), Z(G) = Z(L) and L/Z(L) is
a p-group.
3. G/Z(G) ∼
= Sym(4) and if V /Z(G) ∼
= V4 , then V is non-abelian.
4. G has an abelian normal subgroup of index p, G is not abelian.
5. G ∼
= A × P , where A is abelian and P is a non-abelian p-group for some prime p;
therefore P is a CAmin -group (respectively Fmin ).
6. G/Z(G) ∼
= P GL2 (pn ) or P SL2 (pn ) with pn > 3.
Note a similarity to Rebmann’s structural classification of F-groups [Reb71]. In fact the
groups of type (1), (3) and (4) are CA-groups [Sch70]. Moreover for the families (2) and
(5) replacing CAmin -group by F-group yields the corresponding family for F-groups. While
the non-solvable case (6) contains all the non-solvable cases of F-groups. In particular we
obtain the following corollary as given in Rebmann [Reb71].
Corollary 1.2. Let G be a non-solvable F-group. Then G is a CA-group.
By using the above theorem is it easy to see that the class of CA-groups is strictly
smaller than the class of CAmin -groups, as P SL2 (q) and P GL2 (q) have a non-abelian
centraliser. However in Rebmann’s paper no example of an F-group which is not a CAgroup was provided. We finish this paper by providing a family of p-groups which are
F-groups but not CA-groups.
Proposition 1.3. Let G be an extraspecial group of order p2n+1 with n > 1. Then G is an
F-group which is not a CA-group.
Note that this family will also be a family of Fmin -groups which are not CAmin -groups.
Finally, observe that if there exists a solvable CAmin -group which is not an CA-group,
then such a p-group exists for some prime p. However, running over the GAP libraries we
have been unable to find a p-group which is CAmin -group and not a CA-group. Note that
[Roc75] studied such groups however we were unable to use the results and methods in this
paper to produce such an example.

2
2.1

Preliminaries
Conditions on element centraliser intersections

Let G be a non-abelian CA-group with x and y non-central elements in G. Consider
CG (x) ∩ CG (y). If z ∈ CG (x) ∩ CG (y), then hCG (x), CG (y)i ≤ CG (z). As these centralisers
are abelian either CG (x) = CG (y) or z ∈ Z(G). Moreover we shall show in the next lemma
that this condition is equivalent to a CA-group.
Lemma 2.1. Let G be a finite non-abelian group. Then G is a CA-group if and only if for
any pair of non-central elements x and y such that CG (x) 6= CG (y) then CG (x) ∩ CG (y) =
Z(G).
2

Proof. We commented above that any CA-group satisfies this condition. Hence it remains
to show to converse. Let z be a non-central element in G and x, y ∈ CG (z) \ Z(G). Then
hzi ≤ CG (y) ∩ CG (x) and so CG (x) ∩ CG (y) 6= Z(G). Therefore CG (x) = CG (y), and so x
and y commute. In particular, any two elements in CG (z) commute. Thus CG (z) is abelian
and G is a CA-group.
Furthermore, with this observation we have the following corollary.
Corollary 2.2. Let G be a CA-group such that Z2 (G) > Z(G). Then G is meta-abelian.
Proof. We want to show that G′ is abelian. However, by [Hup67, Theorem III.2.11], G′ ≤
CG (Z2 (G)). Thus it is enough to show CG (Z2 (G)) is abelian. Let x, y ∈ CG (Z2 (G)) \ Z(G).
Then Z(G) < Z2 (G) ≤ CG (x)∩CG (y). Thus x and y commute and so CG (Z2 (G)) is abelian.
We now make clear the definition of a minimal element centraliser for use in the definitions of CAmin -groups and Fmin -groups.
Definition 2.3. An element centraliser CG (x) for x a non-central element is called a
minimal centraliser if CG (y) ≤ CG (x) implies CG (y) = CG (x).
Thus to relax the notion of a CA-group, we want to consider the intersection property
for minimal element centralisers. Note that any non-abelian group must have at least two
minimal non-central element centralisers. Otherwise if C = CG (x) is the unique minimal
centraliser in G, then for all y ∈ G, we have that C ≤ CG (y). Therefore x ∈ ∩y∈G CG (y) =
Z(G). Thus C = G and x ∈ Z(G).
Note that we could also consider maximal centralisers (CG (x) called a maximal centraliser if CG (x) < CG (y) implies y ∈ Z(G)). In fact Schmidt considered the set of groups
in which any two distinct maximal non-central element centralisers have intersection equal
to the center of the group [Sch70]. (These were referred to as D-groups) However he only
classified the soluble D-groups, although he did discuss in depth the non-solvable case too.
Definition 2.4. Let CAmin denote the set of finite groups G such that CG (x) ∩ CG (y) =
Z(G) for any two distinct minimal centralisers CG (x) and CG (y).
By definition, a group is an F-group if and only if for any non-central element x in G
we have CG (x) is both a maximal and minimal centraliser in G. Therefore, for an F-group,
the definitions of D and CAmin are equivalent. Furthermore, the following corollary follows
from Lemma 2.1.
Corollary 2.5. Let G be a finite group. Then G is a CA-group if and only if G is an
F-group and a CAmin -group.
As with the notion of CA-groups we shall weaken the notion of an F-group. However,
first we need an analogous lemma for Lemma 2.1.
Lemma 2.6. Let G be a finite non-abelian group. Then G is an F-group if and only if
for any pair of non-central elements x and y such that CG (x) 6= CG (y) then Z(CG (x)) ∩
Z(CG (y)) = Z(G).
Proof. Assume G is an F-group and let CG (x) 6= CG (y) for x and y non-central elements.
If z ∈ Z(CG (x)) ∩ Z(CG (y)), then hCG (x), CG (y)i ≤ CG (z). Hence as in an F-group every
centraliser is both maximal and minimal, it follows that CG (z) = G. Or in other words
z ∈ Z(G). Therefore Z(CG (x)) ∩ Z(CG (y)) = Z(G).
3

For the converse direction assume that CG (x) < CG (y) for both x and y non-central
elements in G. Then Z(CG (y)) ≤ Z(CG (x)) and therefore Z(CG (y)) = Z(G), which implies
y ∈ Z(G).
Thus we now make the following definition.
Definition 2.7. Let Fmin denote the set of finite groups G such that Z(CG (x))∩Z(CG (y)) =
Z(G) for any two distinct minimal centralisers CG (x) and CG (y).
Therefore we have the following inclusions: (Theorem 1.1 and Proposition 1.3 show
that these are strict inclusions)
Fmin -groups
(

)

CAmin -groups
)

(

F-groups

CA-groups
We finally observe that the intersection of CAmin -groups with F-groups equals the set
of CA-groups.

2.2

Exhibiting a partition

In the works of Rebmann and Schmidt [Reb71], [Sch70] providing an abelian normal partition of the central quotient G/Z(G) yielded a powerful tool to structurally classify families
of groups; in particular they could apply the following classifications by Baer and Suzuki.
Neither theorem appears as one statement but as several across the papers, therefore we
combine the results into one statement.
Theorem 2.8. [Bae61a][Bae61b] Let G be a solvable group with a normal non-trivial partition β, then G is one of the following:
1. A component of β is self normalising in G and G is a Frobenius group.
2. G ∼
= Sym(4) and β is the set of maximal cyclic subgroups of G.
3. G has a nilpotent normal subgroup N which lies in β with |G : N | = p and every
element in G \ N has order p.
4. G is a p-group, for p a prime.
Theorem 2.9. [Suz61] Let G be a non-solvable group with a normal non-trivial partition β.
Then G ∼
= P GL2 (pn ), P SL2 (pn ) for p prime and pn > 3, Sz(2n ) for n ≥ 3 or a component
of β is self normalising and G is a Frobenius group.
We aim to show that for G a CAmin -group or an Fmin -group, as for F-groups, the central
quotient G/Z(G) exhibits a normal abelian partition. For the case of CAmin -groups we
require the following preliminary result.
Lemma 2.10. Let G be a CAmin -group, then each minimal centraliser is abelian.

4

Proof. Let C be a minimal centraliser in G and x ∈ C. If x 6∈ Z(C), then there exists a
minimal centraliser D ≤ CG (x). It is clear that Z(CG (x)) ≤ Z(D). Hence x ∈ C ∩ D which
equals Z(G) or C = D. Thus assume that C = D. However as x ∈ Z(D), it means that
x ∈ Z(C).
Lemma 2.11. Let G be a CAmin -group. Then
β = {C/Z(G) | C a minimal centraliser in G}
forms a non-trivial normal partition of G/Z(G) consisting of abelian subgroups.
Proof. It is clear that the set β is closed under conjugation and by Lemma 2.10 every
subgroup in β is abelian. Thus to show β is a partition we need to show that every element
in G/Z(G) lies in a unique subgroup in β.
Take C/Z(G) and D/Z(G) distinct in β. Then C/Z(G)∩D/Z(G) = (C ∩D)/Z(G) = 1.
Thus it is enough to show that any xZ(G) lies in some C/Z(G).
Consider CG (x) which contains some minimal centraliser C. Then as in Lemma 2.10,
Z(CG (x)) ≤ Z(C), hence x ∈ C. In particular xZ(G) ∈ C/Z(G).
Lemma 2.12. Let G be an Fmin -group. Then
β = {Z(C)/Z(G) | C a minimal centraliser in G}
forms a non-trivial normal partition of G/Z(G) consisting of abelian subgroups.
Proof. It is clear that the set β is closed under conjugation and every subgroup in β is
abelian. Thus to show β is a partition we need to show that every element in G/Z(G) lies
in a unique subgroup in β.
Take Z(C)/Z(G) and Z(D)/Z(G) distinct in β. Then Z(C)/Z(G) ∩ Z(D)/Z(G) =
(Z(C) ∩ Z(D))/Z(G) = 1. Thus it is enough to show that any xZ(G) lies in some
Z(C)/Z(G).
Consider CG (x) which contains some minimal centraliser C. Then as in Lemma 2.10,
Z(CG (x)) ≤ Z(C), hence x ∈ Z(C). In particular xZ(G) ∈ Z(C)/Z(G).

3

Proof of main theorem

We now aim to classify CAmin -groups and Fmin -groups. In fact a similar argument as for
F-groups occurs when we replace F-group by CAmin -group or Fmin -group.

3.1

Classifying CAmin -groups

Due to the classification of partitions by Baer and Suzuki, first we shall consider the solvable
CAmin -groups.
Theorem 3.1. Let G be a solvable CAmin group. Then G is one of the following:
1. G/Z(G) is a Frobenius group with kernel L/Z(G) and complement K/Z(G) such that
both K and L are abelian.
2. G/Z(G) is a Frobenius group with kernel L/Z(G) and complement K/Z(G) such that
K is abelian, L is a CAmin -group, Z(L) = Z(G) and L/Z(L) is a p-group.
3. G/Z(G) ∼
= Sym(4) and if V /Z(G) ∼
= V4 , then V is non-abelian.
5

4. G has an abelian normal subgroup of index p, G is not abelian.
5. G ∼
= A × P , where A is abelian and P is a non-abelian p-group for some prime p;
therefore P is a CAmin -group.
Proof. As G/Z(G) admits a non-trivial normal partition, we apply the classification of Baer
(Theorem 2.8) to determine G/Z(G).
Case (1)
Let L/Z(G) denote the Frobenius kernel of G/Z(G). Furthermore, let K/Z(G) denote an
element in the partition of G/Z(G) which is self-normalising. Then K = CG (x) for some
minimal centraliser CG (x) in G. We want to show that K/Z(G) is a Frobenius complement,
thus we need that K ∩ K g = Z(G) for all g ∈ G \ K.
As K is a minimal centraliser in G, it means K g is also a minimal centraliser in G and
therefore K ∩ K g = Z(G) or K. However, if K g = K, then gZ(G) ∈ NG/Z(G) (K/Z(G)) =
K/Z(G) as K/Z(G) was chosen to be self-normalising. Thus K/Z(G) is a Frobenius complement in G/Z(G). Furthermore as K is a minimal centraliser in G, then K is abelian
(Corollary 2.10).
Let x ∈ L \ Z(G). As G/Z(G) is Frobenius with kernel L/Z(G),
CG (x)/Z(G) ≤ CG/Z(G) (xZ(G)) ≤ L/Z(G),
so CG (x) ≤ L; or in other words CG (x) = CL (x).
If L has a unique minimal centraliser, then L is abelian by Corollary 2.10 and thus of
type (1). Hence assume L has two distinct minimal centralisers CL (x) < L and CL (y) < L
for x, y ∈ L \ Z(G). Then CL (x) = CG (x) and CL (y) = CG (x). If CG (x) is not a minimal
centraliser in G, then there exists CG (z) < CG (x). As CG (z) < CG (x) ≤ L, it follows
that z ∈ L \ Z(G). Thus CL (z) < CL (x) and so CL (x) is not minimal. Therefore CG (x)
and CG (y) are distinct minimal centralisers in G. As G is a CAmin -group, we have that
Z(L) ≤ CL (x) ∩ CL (y) = CG (x) ∩ CG (y) = Z(G). However Z(G) ≤ L and therefore
Z(G) ≤ Z(L) implying that Z(L) = Z(G). Furthermore, we have shown that L is a
CAmin -group. By Thompson, [Hup67, Theorem V.8.7], L/Z(G) is nilpotent (as it is a
Frobenius kernel). By [Bae61a, Remark 2.4], the only nilpotent groups with a partition are
p-groups for some prime p. This implies G is of type (2).
Case (2)
In this case G/Z(G) ∼
= Sym(4). Let V ≤ G such that V /Z(G) ∼
= V4 , the Klein-four subgroup. If V is abelian, then for all x ∈ V \ Z(G) we have V ≤ CG (x). As xZ(G) has order 2
it follows that CG/Z(G) (xZ(G)) = D8 or V4 (when xZ(G) is a double or single transposition
respectively), we also observe that V /Z(G) ≤ CG (x)/Z(G) ≤ CG/Z(G) (xZ(G)).
If there exists an x ∈ V such that CG (x)/Z(G) ∼
= V4 , then CG (x) = V and so is abelian
and thus a minimal centraliser. However V4 is not contained in the partition of G/Z(G)
which yields a contradiction. Thus for all x ∈ V \ Z(G) we have that CG (x)/Z(G) =
CG/Z(G) (xZ(G)) ∼
= D8 and xZ(G) ≤ Z(D8 ). In particular, it follows that V4 must be the
normal klein-four subgroup of Sym(4).
Inside CG (x)/Z(G) ∼
= D8 there exists a unique cyclic subgroup of order 4 and another
copy of V4 which is not normal in Sym(4). Let N, M ≤ CG (x) such that N/Z(G) ∼
= C4 and
M/Z(G) equals the non-normal copy of V4 . By Theorem 2.8, N/Z(G) lies in the partition
and therefore N is a minimal centraliser in G. The subgroup M contains x and therefore
hx, Z(G)i ≤ Z(M ). In particular, we see that M/Z(M ) must be cyclic and so M is abelian.
However x ∈ N ∩ M and so M cannot be a minimal centraliser in G. Thus CG (y) > M for
all y ∈ M . As M/Z(G) lies in a unique maximal subgroup isomorphic to D8 , it follows that
for each y ∈ M \ Z(G), then CG (y)/Z(G) equals the unique maximal subgroup containing
M/Z(G). Thus M/Z(G) ≤ Z(D8 ) which is a contradiction.
6

Case (3)
In this case N/Z is a component of β, which implies N is abelian.
Case (4)
In this case G/Z(G) is a p-group and therefore G is nilpotent. Therefore G = A × P for
A ≤ Z(G) and P a p-subgroup which is a CAmin -group.
We next show that each case occurring in Theorem 3.1 yields a CAmin -group.
Proposition 3.2. Any solvable group occurring in Theorem 3.1 is a CAmin -group.
Proof. The solvable groups in Theorem 1.1 are those of type (1) − (5). Any group of type
(5) is easily seen to be a CAmin -group. For the groups of type (1), (3) and (4), Schmidt
[Sch70] has shown that they are CA-groups and hence are CAmin -groups. Thus it only
leaves those of type (2). If x ∈ L \ Z(G), then it was shown in the proof of Theorem 3.1
that CG (x) = CL (x). If x ∈ G \ L, then as G/Z(G) is a Frobenius group xZ(G) lies in
some conjugate of K/Z(G) [Hup67, Page 496]. Thus assume x ∈ K. As K is abelian,
then K/Z(G) ≤ CG (x)/Z(G) ≤ CG/Z(G) (xZ(G)) ≤ K/Z(G). That is K = CG (x). It now
follows that any two distinct minimal non-central element centralisers have intersection
equal to Z(G).
Thus it only remains to study the non-solvable CAmin -groups.
Theorem 3.3. Let G be a non-solvable group. Then G is a CAmin group if and only if
G/Z(G) ∼
= P SL2 (pn ) or P GL2 (pn ) with pn > 3.
Proof. As G/Z(G) admits a non-trivial normal partition, we will use the classification of
Suzuki (Theorem 2.9) to determine G/Z(G).
Case (1)
If G/Z(G) is a Frobenius group, then as in the solvable case the kernel L/Z(G) is nilpotent
and the complement K/Z(G) is abelian. However, this implies G/Z(G) and therefore G is
solvable.
Case (2)
If G/Z(G) is isomorphic to Sz(2n ), then Schmidt[8] showed that the Sylow 2-subgroups of
Sz(2n ) are subgroups of some components for any non-trivial partition. However Sz(2n )
has non-abelian Sylow 2-subgroups and therefore cannot be a CAmin -group.
Case (3)
Assume G/Z(G) ∼
= P SL2 (pn ) or P GL2 (pn ) with pn > 3. In this case we want to show that
any group arising in this way is a CAmin -group.
It is well known that every element centraliser in P GL2 (q) and P SL2 (q) takes one of
the following forms:
1. A cyclic group of order q, q − 1 or q + 1.
2. A dihedral group of order 2(q + 1) or 2(q − 1).
Let x ∈ G \ Z(G) such that CG (x) is a minimal centraliser and set C to be the subgroup
of G such that CG/Z(G) (xZ(G)) = C/Z(G). If C/Z(G) is cyclic, then C is abelian and thus
CG (x) = C . If C/Z(G) is dihedral, then as xZ(G) ∈ Z(C/Z(G)), it follows that xZ(G) ∈
C ′ /Z(G) for C ′ /Z(G) the cyclic subgroup of index 2 in C/Z(G). Hence C ′ ≤ CG (x) ≤ C.
If CG (x) = C, then there exists a y ∈ G such that CG/Z(G) (yZ) = C ′ /Z(G) and so
CG (y) < CG (x) contradicting minimality. Thus every minimal centraliser in G is abelian
and its quotient is a centraliser in G/Z(G).

7

Let x, y ∈ G \ Z(G) such that CG (x) and CG (y) are distinct minimal centralisers in
G. Then CG (x)/Z(G) and CG (y)/Z(G) are centralisers in G/Z(G). It is enough to show
that (CG (x)/Z(G)) ∩ (CG (y)/Z(G)) is trivial in G/Z(G). Let kZ(G) ∈ (CG (x)/Z(G) ∩
CG (y)/Z(G)). Then CG (x)/Z(G) and CG (y)/Z(G) are distinct abelian centralisers in
G/Z(G) which are subgroups of CG/Z(G) (kZ). However, no centraliser in P SL2 (q) or
P GL2 (q) contains two distinct abelian centralisers in P SL2 (q) or P GL2 (q) respectively.
Therefore kZ = Z and hence the intersection is trivial. In particular, we have shown that
CG (x) ∩ CG (y) = Z(G) and G is a CAmin -group.
Corollary (Corollary 1.2). Any non-solvable F-group is a CA-group.
Proof. By Rebmann, any non-solvable F-group must be of the form G/Z(G) ∼
= P SL2 (q) or
P GL2 (q) with some extra condition on G′ . However, by the previous result any group such
that G/Z(G) has this structure is CAmin -group. Hence G is an F-group and CAmin -group,
and it follows it is a CA-group.

3.2

Classifying Fmin -groups

We now repeat a similar argument as in the case of CAmin -groups. Most details are omitted,
however we include details for cases (1) and (2) in the solvable case to highlight that the
partition now consists of quotients of centres of centralisers.
Theorem 3.4. Let G be a solvable Fmin -group. Then G is one of the following:
1. G/Z(G) is a Frobenius group with kernel L/Z(G) and complement K/Z(G) such that
both K and L are abelian.
2. G/Z(G) is a Frobenius group with kernel L/Z(G) and complement K/Z(G) such that
K is abelian, L is an Fmin -group, Z(L) = Z(G) and L/Z(L) is a p-group.
3. G/Z(G) ∼
= Sym(4) and if V /Z(G) ∼
= V4 , then V is non-abelian.
4. G has an abelian normal subgroup of index p, G is not abelian.
5. G ∼
= A × P , where A is abelian and P is a non-abelian p-group for some prime p;
therefore P is an Fmin -group.
Proof. As G/Z(G) admits a non-trivial normal partition, we apply the classification of Baer
(Theorem 2.8) to determine G/Z(G) and then G. Note that cases (3) and (4) use the exact
same argument as in Theorem 3.1 and so we shall not repeat them.
Case (1)
Let L/Z(G) denote the Frobenius kernel of G/Z(G) and K/Z(G) an element in the partition
of G/Z(G) which is self-normalising. Then K = Z(CG (x)) for some minimal centraliser
CG (x) in G. Using the same argument as in Theorem 3.1 shows K/Z(G) is a Frobenius
complement and abelian. Furthermore, recall that for x ∈ L \ Z(G), then CG (x) = CL (x)
and CL (x) is minimal centraliser in L implies that CG (x) is minimal centraliser in G.
If L has a unique minimal centraliser, then L is abelian by Corollary 2.10 and thus of
type (1). Hence assume L has two distinct minimal centralisers CL (x) < L and CL (y) < L
for x, y ∈ L \ Z(G). Then CL (x) = CG (x) and CL (y) = CG (x) are distinct minimal
centralisers in G. Moreover Z(L) ≤ Z(CL (x)) ∩ Z(CL (y)) = Z(G) and so Z(L) = Z(G). In
particular, we have shown that L is an Fmin -group. Repeating the argument in Theorem 3.1
also implies G is of type (2).
Case (2)
In this case G/Z(G) ∼
= Sym(4). Let V ≤ G such that V /Z(G) ∼
= V4 . If V is abelian,
8

then for all x ∈ V \ Z(G) we have V ≤ CG (x). As xZ(G) has order 2 it follows that
CG/Z(G) (xZ(G)) = D8 or V4 (when xZ(G) is a double or single transposition respectively),
we also observe that V /Z(G) ≤ CG (x)/Z(G) ≤ CG/Z(G) (xZ(G)).
If there exists an x ∈ V such that CG (x)/Z(G) ∼
= V4 , then CG (x) = V and so is abelian
and thus a minimal centraliser. However V4 is not contained in the partition of G/Z(G)
which yields a contradiction. Thus for all x ∈ V \ Z(G) we have that CG (x)/Z(G) =
CG/Z(G) (xZ(G)) ∼
= D8 and xZ(G) ≤ Z(D8 ). In particular, it follows that V4 must be the
normal Klein-four subgroup of Sym(4).
Inside CG (x)/Z(G) ∼
= D8 there exists a unique cyclic subgroup of order 4 and another
copy of V4 which is not normal in Sym(4). Let N, M ≤ CG (x) such that N/Z(G) ∼
= C4 and
M/Z(G) equals the non-normal copy of V4 . By Theorem 2.8, N/Z(G) lies in the partition
and therefore N is the centre of a minimal centraliser in G. The subgroup M contains x
and therefore hx, Z(G)i ≤ Z(M ). In particular, we see that M/Z(M ) must be cyclic and so
M is abelian. However x ∈ Z(N ) ∩ Z(M ) and so M cannot be a minimal centraliser in G.
Thus CG (y) > M for all y ∈ M . As M/Z(G) lies in a unique maximal subgroup isomorphic
to D8 , it follows that for each y ∈ M \ Z(G), then CG (y)/Z(G) equals the unique maximal
subgroup containing M/Z(G). Thus M/Z(G) ≤ Z(D8 ) which is a contradiction.
Using the same arguments as in Proposition 3.2 gives the analogous result for Fmin groups.
Proposition 3.5. Any solvable group occurring in Theorem 1.1 is an Fmin -group.
We are therefore left to study the non-solvable Fmin -groups.
Theorem 3.6. Let G be a non-solvable group. Then G is an Fmin -group if and only if
G/Z ∼
= P SL2 (pn ) or P GL2 (pn ) with pn > 3.
Proof. As G/Z(G) admits a non-trivial normal partition, we will use the classification of
Suzuki (Theorem 2.9) to determine G/Z(G). Note that Cases (1) and (2) use exactly the
same argument as in Theorem 3.3 and so shall not be repeated.
Case (3)
Assume G/Z(G) ∼
= P SL2 (pn ) or P GL2 (pn ) with pn > 3. We saw that any such group is a
CAmin -group, which implies it is an Fmin -group.
Moreover, we now obtain the analogous corollary of Rebmann for non-solvable Fmin groups.
Corollary 3.7. Any non-solvable Fmin -group is a CAmin -group.
Finally, by combing the two theorems in this section we obtain Theorem 1.1.

4

A family of F-groups which are not CA-groups

As we saw in the introduction, given the classification of CAmin -groups, it is easy to see
that there is a non-solvable CAmin -group which is not a CA-group. However, as commented
in Rebmann, any non-solvable F-group is also a CA-group. Thus we need to consider the
solvable classification from Rebmann. In particular, using [Reb71, Corollary 5.1] if G is an
F-group that is not a CA-group, G must take one of the two forms:
1. G ∼
= A × P where A is abelian and P is a non-abelian p-group which is also an
F-group.
9

2. G/Z(G) is a Frobenius group with kernel L/Z(G) and complement K/Z(G), K is
abelian Z(L) = Z(G), L is an F -group and L/Z(L) is a p-group.
Note that if we have an F-group which is not a CA-group of the second type, then
the subgroup L cannot be a CA-group. In particular, if there exists an F-group which is
not a CA-group, then there exists such a p-group. Thus to find an F-group which is not
a CA-group, the first place to consider is in the set of p-groups. In particular we shall
consider the class of extraspecial groups.
First we state the following lemma which will be of use to us.
Lemma 4.1. Let G be a finite group in which the derived subgroup has order p for some
prime p. Then G is an F-group.
Proof. This result follows from the observation that the conjugacy class gG is contained
in the coset gG′ . Therefore |gG | equals p or 1. Hence |CG (g)| = |G|
p or |G| and so every
non-central element centraliser is both maximal and minimal. In particular G is an Fgroup.
for
Let G be an extraspecial group, usually denoted by one of the two groups p1+2n
±
some positive integer n. Then we have G′ = Φ(G) = Z(G) of order p. By the previous
lemma G is an F-group.
Assume n > 1, otherwise G is of order p3 and it is easy to see such groups are CA-groups.
Then G is isomorphic to the central product of H and P , where H is an extraspecial group
of order p1±2(n−1) and P is extraspecial of order p3 . Take x ∈ H \ Z(H). Then x 6∈ Z(G),
however P ≤ CG (x). Thus G is not a CA-group.
Proposition (Proposition 1.3). Let G be an extraspecial group of order p2n+1 with n > 1.
Then G is an F-group which is not a CA-group.
Note that not all F-groups which are not CA-groups occur from extraspecial groups.
In particular, using GAP we can find 5 groups of order 64 which are F-groups but not
CA-groups.

Acknowledgments
The author gratefully acknowledges financial support by the ERC Advanced Grant 291512.
In addition the author would like to thank Benjamin Sambale for reading and discussing a
preliminary version of this paper.

References
[Bae61a] R Baer. Partitionen endlicher Gruppen. Math. Z., 75:333–372, 1960/1961.
[Bae61b] R Baer. Einfache Partitionen endlicher Gruppen mit nicht-trivialer Fittingscher
Untergruppe. Arch. Math. (Basel), 12:81–89, 1961.
[Hup67] B. Huppert. Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer-Verlag, Berlin-New York, 1967.
[Itô53]

N. Itô. On finite groups with given conjugate types. I. Nagoya Math. J., 6:17–28,
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[Reb71] J. Rebmann. F -Gruppen. Arch. Math. (Basel), 22:225–230, 1971.
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[Roc75]

D. Rocke. p-groups with abelian centralizers. Proc. London Math. Soc. (3),
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