Finite
p -groups
with only three characteristic
subgroups
Csaba Schneider
Centro de Álgebra
Universidade de Lisboa
[email protected]
∼schneider
www.sztaki.hu/
Encuentro en Teoría de Grupos
Bilbao, 18 June 2010
Joint work with Péter P. Pálfy and Stephen Glasby
The problem
Theorem
If G is a nite group that does not have a proper non-trivial
characteristic subgroup (characteristically simple) then G ∼
= Tk
where T is a nite simple group.
Philip Hall: What if G has precisely one non-trivial proper
characteristic subgroup?
Taunt (MPCPS, 1955): Finite groups having a unique proper
characteristic subgroup I.
UCS csoport: Unique proper, non-trivial Characteristic Subroup.
Examples: C , S , extraspecial group with order 27 and exponent
3, S .
4
5
3
The problem
Theorem
If G is a nite group that does not have a proper non-trivial
characteristic subgroup (characteristically simple) then G ∼
= Tk
where T is a nite simple group.
Philip Hall: What if G has precisely one non-trivial proper
characteristic subgroup?
Taunt (MPCPS, 1955): Finite groups having a unique proper
characteristic subgroup I.
UCS csoport: Unique proper, non-trivial Characteristic Subroup.
Examples: C , S , extraspecial group with order 27 and exponent
3, S .
4
5
3
The problem
Theorem
If G is a nite group that does not have a proper non-trivial
characteristic subgroup (characteristically simple) then G ∼
= Tk
where T is a nite simple group.
Philip Hall: What if G has precisely one non-trivial proper
characteristic subgroup?
Taunt (MPCPS, 1955): Finite groups having a unique proper
characteristic subgroup I.
UCS csoport: Unique proper, non-trivial Characteristic Subroup.
Examples: C , S , extraspecial group with order 27 and exponent
3, S .
4
5
3
Taunt (1955)
Taunt investigated direct products of UCS groups and solvable UCS
groups that are not p-groups.
Taunt: The prime-power UCS groups can be further subdivided
into two types, comprising groups all of whose elements other than
the identity are of prime order, and groups having some elements of
order the square of a prime. We hope to return to the consideration
of such groups later.
UCS
p -groups
UCS p-group: G > N > 1, where N is the characteristic subgroup.
Two cases for G 0 :
(i) G 0 = N: G non-abelian.
(ii) G 0 = 1: G abelian and G ∼
= Cp2 × · · · × Cp2 .
Two cases for G p :
(i) G p = N: G has exponent p .
(ii) G p = 1: G has exponenent p.
2
UCS
p -groups
UCS p-group: G > N > 1, where N is the characteristic subgroup.
Two cases for G 0 :
(i) G 0 = N: G non-abelian.
(ii) G 0 = 1: G abelian and G ∼
= Cp2 × · · · × Cp2 .
Two cases for G p :
(i) G p = N: G has exponent p .
(ii) G p = 1: G has exponenent p.
2
UCS
p -groups
UCS p-group: G > N > 1, where N is the characteristic subgroup.
Two cases for G 0 :
(i) G 0 = N: G non-abelian.
(ii) G 0 = 1: G abelian and G ∼
= Cp2 × · · · × Cp2 .
Two cases for G p :
(i) G p = N: G has exponent p .
(ii) G p = 1: G has exponenent p.
2
Basic properties of UCS
p -groups
Let p be odd, and let G be a nite p-group with p-class 2.
Set G = G /Φ(G ).
Facts:
(i) The groups G and Φ(G ) are vector spaces over Fp .
(ii) The group is UCS i Aut(G ) acts irreducibly on both G and
Φ(G ).
(iii) The image of this action is G 6 GL(d , p ) where G = Fdp .
(iv) If G is non-abelian UCS, then the module Φ(G ) is isomorphic,
as an Aut(G )-module, to a quotient module of G ∧ G .
(v) If G has exponent p , then G and Φ(G ) 6 G ∧ G are
isomorphic Aut(G )-modules.
2
2- and 3-generator UCS
p -groups
with exponent
Let p be odd.
2-generator UCS p-groups with exponent p:
Precisely one group with dim G = 2 and dim Φ(G ) = 1.
3-generator UCS p-groups with exponent p:
Precisely one group: with dim G = dim Φ(G ) = 3.
p
4-generator UCS
p -groups
with exponent
p
Let p be odd. Let G be a 4-generator p-group of exponent p and
nilpotency class 2.
Let V = G (dim V = 4) and let W = Φ(G ). Then, as
Aut(G )-modules, W = (V ∧ V )/U with some U 6 V ∧ V .
The isomorphism types of such groups correspond to GL(V )-orbits
on the subspaces of V ∧ V .
Theorem (Brahana (1940))
There are 18 orbits of GL(V ) on V ∧ V . That is there are 18 isom
types of 4-generator groups with exponent p and nilpotency class 2.
4-generator UCS
p -groups
with exponent
p
Let p be odd. Let G be a 4-generator p-group of exponent p and
nilpotency class 2.
Let V = G (dim V = 4) and let W = Φ(G ). Then, as
Aut(G )-modules, W = (V ∧ V )/U with some U 6 V ∧ V .
The isomorphism types of such groups correspond to GL(V )-orbits
on the subspaces of V ∧ V .
Theorem (Brahana (1940))
There are 18 orbits of GL(V ) on V ∧ V . That is there are 18 isom
types of 4-generator groups with exponent p and nilpotency class 2.
The Klein correspondence
As dim V = 4, GL(V ) preserves a quadratic form Q modulo scalars
on the space V ∧ V :


Q
Theorem
X
i <j
αi ,j xi ∧ xj  = α1,2 α3,4 − α1,3 α2,4 + α1,4 α2,3 .
The group G is UCS i the form Q is non-degenerate on the
corresponding subspace U.
4-generator UCS
p -groups
with exponent
p
Theorem
For odd p, the 4-generator UCS p-groups with exponent p are as
follows:
(i)
G0
(ii)
G2
= Fp,4 /(Fp,4 )p ;
=
/ Fp,4 )p , [x1 , x2 ][x3 , x4 ]−1 , [x1 , x3 ], [x1 , x4 ], [x2 , x3 ], [x2 , x4 ] ;
Fp ,4
= Fp,4 / h(Fp,4 )p , [x1 , x3 ], [x1 , x4 ], [x2 , x3 ], [x2 , x4 ]i;
(iii)
G4
(iv)
=
/ (Fp,4 )p , [x1 , x2 ], [x3 , x4 ], [x2 , x3 ][x1 , x4 ]−1 , [x1 , x3 ]α [x2 , x4 ] ;
p
−1
G11 = Fp ,4 / (Fp ,4 ) , [x1 , x4 ], [x2 , x3 ], [x2 , x4 ][x1 , x3 ]
;
G6
Fp ,4
(v)
(vi)
G14
(vii)
G16
(viii)
G18
= Fp,4 / h(Fp,4 )p , [x1 , x2 ][x3 , x4 ]i;
= Fp,4 / h(Fp,4 )p , [x1 , x2 ], [x3 , x4 ]i;
= Fp,4 / (Fp,4 )p , [x2 , x3 ][x1 , x4 ], [x1 , x3 ]α [x2 , x4 ]−1 .
UCS
p -groups
with exponent
p2
A UCS group G with exponent p leads to an irreducible A-module
V and a submodule U 6 V ∧ V such that V ∼
= V ∧ V /U.
2
Such a module is called an ESQ-module. The group A is called an
ESQ-group.
Conversely, from an irreducible ESQ-module it is usually possible to
obtain a UCS-group with exponent p . The group A will occur as a
subgroup of Out(G ).
2
UCS
p -groups
with exponent
p2
A UCS group G with exponent p leads to an irreducible A-module
V and a submodule U 6 V ∧ V such that V ∼
= V ∧ V /U.
2
Such a module is called an ESQ-module. The group A is called an
ESQ-group.
Conversely, from an irreducible ESQ-module it is usually possible to
obtain a UCS-group with exponent p . The group A will occur as a
subgroup of Out(G ).
2
UCS
p -groups
with exponent
p2
A UCS group G with exponent p leads to an irreducible A-module
V and a submodule U 6 V ∧ V such that V ∼
= V ∧ V /U.
2
Such a module is called an ESQ-module. The group A is called an
ESQ-group.
Conversely, from an irreducible ESQ-module it is usually possible to
obtain a UCS-group with exponent p . The group A will occur as a
subgroup of Out(G ).
2
3-generator UCS
p -groups
with exponent
p2
If V = Fp , then g ∈ GL(V ) acts on V ∧ V as (det g )g −T .
If p > 3, then SO(V ) is ESQ group. Hence
3
Theorem
For odd prime, there is precisely one 3 generator UCS p-group with
exponent p .
2
4-generator UCS groups
Suppose that p 6= 5. Then the group C o C is an irreducible ESQ
subgroup of GL(4, p ).
5
4
Theorem
Suppose that V = FP and let G be an irreducible ESQ subgroup of
GL(V ). Then
I p 6= 5;
I Further, C 6 G 6 C o C .
I If 5 is a square in Fp then G = C o C .
4
5
5
4
5
4
Corollary
There exists a 4-generator UCS p-group with exponent p if and
only if p 6= 5. If 5 is a square in Fp then there is a unique
isomorphism type; otherwise there are 2 isom types.
2
5-generator UCS groups with exponent
p2
Theorem
Let G be a minimal irreducible ESQ subgroup of GL(d , q ) with d
prime. Then one of the following:
(i) G is not absolutely irreducible, r = |G | is a prime, and q d ≡ 1
(mod r ).
(ii) G is an absolutely irreducible non-abelian simple group.
(iii) G is absolutely irreducible, |G | = r s d where r 6= d is a prime,
q ≡ 1 (mod r ).
If d = 5 then (ii) does not occur (Di Martino and Wagner). In case
(i), |G | = 11; in case (iii), |G | = 55.
Corollary
There is a 5-generator UCS group with exponent p if and only if
p ≡ 1 (mod 11).
2
5
Final words
Theorem
If p is odd and k positive, then there exist UCS p-groups with
exponent p
I with 3k generators if and only if p and k;
I with 4 generators if and only if p 6= 5;
I with 5 generators if and only if p ≡ 1 (mod 11);
I with 7k generators for all p and k;
2
5