1. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Subgroup structure of simple (algebraic) groups
Raffaele Rainone
University of Southampton
PGR seminar
November 23, 2011
Raffaele Rainone Subgroup structure of simple (algebraic) groups

2. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Table of contents
1 Motivations
2 Preliminaries
Classification of simple algebraic groups
Classification of finite simple groups
Almost simple groups
3 Finite simple groups
O’Nan - Scott Theorem
Aschbacher Theorem
4 Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
Raffaele Rainone Subgroup structure of simple (algebraic) groups

3. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Group actions
Let G be a group and Ω be a set. An action of G on Ω is
G × Ω → Ω
such that
1.ω = ω;
g.(h.ω) = (gh).ω.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

4. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Transitive action
The orbit of α ∈ Ω is G.α = {g.α | g ∈ G}. We write G/Ω = {G.α | α ∈ Ω}.
Definition
The action of G on Ω is transitive if |G/Ω| = 1.
Example
1 Let Ω = {1, . . . , n}, we consider the standard action Sn;
2 Let G be any group and H ≤ G, the standard action of G on G/H is
transitive.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

5. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Transitive action
The stabilizer of α ∈ Ω is Gα = {g ∈ G | g.α = α} ≤ G.
Proposition
Let G act transitively on Ω. Then the action is equivalent to the standard action
of G on G/Gα for any α in Ω.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

6. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Primitive action
G acts on Ω. A system of imprimitivity is a non-trivial partition
Ω = Ω1 ∪ . . . ∪ Ωn
preserved by G, i.e. if α, β ∈ Ωi then ∀g ∈ G, g.α, g.β ∈ Ωj . Whenever such
partition exists we say the action to be imprimitive.
Definition
An action is primitive if it is transitive and it is not imprimitive.
Proposition
The action of G on Ω is primitive if, and only if, Gα is maximal.
The primitive action of G on Ω is equivalent to the standard action of G on
G/Gα.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

7. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classification of simple algebraic groups
Classification of finite simple groups
Almost simple groups
1 Motivations
2 Preliminaries
Classification of simple algebraic groups
Classification of finite simple groups
Almost simple groups
3 Finite simple groups
O’Nan - Scott Theorem
Aschbacher Theorem
4 Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
Raffaele Rainone Subgroup structure of simple (algebraic) groups

8. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classification of simple algebraic groups
Classification of finite simple groups
Almost simple groups
An algebraic group is a closed subgroup of GLn(k), where n ≥ 1 and k is an
algebraically closed field of characteristic p ≥ 0.
Example
SLn(k) = {A ∈ GLn(k) | detA = 1}
Raffaele Rainone Subgroup structure of simple (algebraic) groups

9. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classification of simple algebraic groups
Classification of finite simple groups
Almost simple groups
Abstract group theory
subgroup
normal subgroup
simple group
Algebraic group theory
closed subgroup
closed normal subgroup
simple algebraic group
Raffaele Rainone Subgroup structure of simple (algebraic) groups

10. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classification of simple algebraic groups
Classification of finite simple groups
Almost simple groups
Simple algebraic groups are classified by root system (Dynking), as
Classical
An, Bn, Cn, Dn
Exceptional
E6, E7, E8, F4, G2
Remark
SLn+1(k) ∼= PSLn+1(k), both in An.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

11. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classification of simple algebraic groups
Classification of finite simple groups
Almost simple groups
CFSG (1832–2004)
The finite simple groups are
(i) Cyclic, Cp, p prime;
(ii) Alternating An, n ≥ 5;
(iii) Lie type:
Classical
PSLn(q), PSUn(q), PSpn(q),
PΩ2n+1(q), PΩ+
2n(q), PΩ−
2n(q)
Exceptional
G2(q), F4(q), . . . , 2
F4(2)
(iv) 26 sporadic groups.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

12. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classification of simple algebraic groups
Classification of finite simple groups
Almost simple groups
A group G is almost simple if there exists a simple group T such that
T G ≤ Aut(T)
Raffaele Rainone Subgroup structure of simple (algebraic) groups

13. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
O’Nan - Scott Theorem
Aschbacher Theorem
1 Motivations
2 Preliminaries
Classification of simple algebraic groups
Classification of finite simple groups
Almost simple groups
3 Finite simple groups
O’Nan - Scott Theorem
Aschbacher Theorem
4 Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
Raffaele Rainone Subgroup structure of simple (algebraic) groups

14. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
O’Nan - Scott Theorem
Aschbacher Theorem
The O’Nan - Scott Theorem
For G, Sn or An. Let H be a subgroup of Sn or An not containing An. Then
either H ∈ A(G) or H ∈ S.
A1 subset stabilisers (intransitive);
A2 stabilisers of certain partitions of Ω, (imprimitive);
A3 stabilisers of cartesian product structures of Ω, (primitive wreath
product);
A4 stabilisers of affine structure of Ω;
A5 stabilisers of Tk
, (diagonal type).
S Primitive almost-simple groups.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

15. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
O’Nan - Scott Theorem
Aschbacher Theorem
structure of the Ai families
Ai Structure in Sn comments
A1 Sk × Sn−k k = n/2
A2 Sm St n = mt
A3 Sk Sd n = kd
A4 AGLd (p) = pd
: GLd (p) n = pd
, p prime
A5 Tk
.(Sk × Out(T)) n = |T|k−1
, k ≥ 2
Raffaele Rainone Subgroup structure of simple (algebraic) groups

16. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
O’Nan - Scott Theorem
Aschbacher Theorem
Example: A3
The family A1, A2 complete classifies imprimitive maximal subgroups of An.
Let n = k2
, put the points of Ω = {1, . . . , k2
} into a matrix
0
B
B
B
@
1 2 . . . k
k + 1 k + 2 . . . 2k
...
...
...
(k − 1)k . . . . . . k2
1
C
C
C
A
And H ∼= Sk × Sk is imprimitive in Sn. Adjoining the permutation that reflects in
the main diagonal we get
Sk S2
that is primitive and maximal (for k ≥ 5).
S3 S2 ≤ S9 is not maximal.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

17. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
O’Nan - Scott Theorem
Aschbacher Theorem
The S family
Let G be an almost simple group, i.e.
T G ≤ Aut(T)
Let M be a maximal subgroup of G. The action of G on G/M is primitive.
Therefore
G → S|G:M|
Raffaele Rainone Subgroup structure of simple (algebraic) groups

18. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
O’Nan - Scott Theorem
Aschbacher Theorem
Classification of Maximal subgroups of Sn, An
Theorem (Liebeck - Praeger - Saxl, 1987)
All the maximal subgroup of Sn and An are classified.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

19. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
O’Nan - Scott Theorem
Aschbacher Theorem
Raffaele Rainone Subgroup structure of simple (algebraic) groups

20. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
O’Nan - Scott Theorem
Aschbacher Theorem
Aschbacher theorem
Let G be a finite simple group of Lie type (classical, Cl(V )). Let H be a
maximal subgroup of G. Then either H ∈ C(G) or H ∈ S.
C1 stabilizers of totally singular or non-singular subspaces;
C2 stabilizers of decomposition V =
Lt
i=1 Vi , dim Vi = n/t;
C3 stabilizers of extension/subfield field of Fq of prime index b;
C5 stabilizers of tensor product decompositions V =
Nt
i=1 Vi ,
dim Vi = a;
C6 dimV = rm
and H is “local”;
C7 stabilizers of bilinear or quadratic form.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

21. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
O’Nan - Scott Theorem
Aschbacher Theorem
Aschbacher theorem
Let G be a finite simple group of Lie type (classical, Cl(V )). Let H be a
maximal subgroup of G. Then either H ∈ C(G) or H ∈ S.
C1 stabilizers of totally singular or non-singular subspaces;
C2 stabilizers of decomposition V =
Lt
i=1 Vi , dim Vi = n/t;
C3 stabilizers of extension/subfield field of Fq of prime index b;
C5 stabilizers of tensor product decompositions V =
Nt
i=1 Vi ,
dim Vi = a;
C6 dimV = rm
and H is “local”;
C7 stabilizers of bilinear or quadratic form.
S almost simple groups acting irreducibly and
tensor-indecomposably.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

22. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
O’Nan - Scott Theorem
Aschbacher Theorem
Ci rough tructure in GLn(q) comments
C1 maximal parabolic
C2 GLm(q) St n = mt
C3 GLa(qb
).b n = ab, b prime
C4 GLn(q0) q = qb
0 , b prime
C5 (GLa(q) ◦ . . . ◦ GLa(q)
´
.St n = at
C6 (Zq−1 ◦ r1+2a
).Sp2a(r) n = ra
, r prime
C7 Spn(q), SOn(q), Un(q1/2
) < SLn(q)
Raffaele Rainone Subgroup structure of simple (algebraic) groups

23. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
1 Motivations
2 Preliminaries
Classification of simple algebraic groups
Classification of finite simple groups
Almost simple groups
3 Finite simple groups
O’Nan - Scott Theorem
Aschbacher Theorem
4 Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
Raffaele Rainone Subgroup structure of simple (algebraic) groups

24. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
Liebeck-Seitz Theorem
Let G = Cl(V ) (over k, algebraically closed) and H be a closed maximal
subgroup. Then either H ∈ C(G) or H ∈ S.
C1 stabilizers of totally singular or non-singular subspaces;
C2 stabilizers of decomposition V =
Lt
i=1 Vi , dim Vi = n/t;
C3 stabilizers of extension/subfield field of Fq of prime index b;
C5 stabilizers of tensor product decompositions V =
Nt
i=1 Vi ,
dim Vi = a;
C6 dimV = rm
and H is “local”;
C7 stabilizers of bilinear or quadratic form.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

25. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
Liebeck-Seitz Theorem
Let G = Cl(V ) (over k, algebraically closed) and H be a closed maximal
subgroup. Then either H ∈ C(G) or H ∈ S.
C1 stabilizers of totally singular or non-singular subspaces;
C2 stabilizers of decomposition V =
Lt
i=1 Vi , dim Vi = n/t;
C5 stabilizers of tensor product decompositions V =
Nt
i=1 Vi ,
dim Vi = a;
C6 dimV = rm
and H is “local”;
C7 stabilizers of bilinear or quadratic form.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

26. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
Liebeck-Seitz Theorem
Let G = Cl(V ) (over k, algebraically closed) and H be a closed maximal
subgroup. Then either H ∈ C(G) or H ∈ S.
C1 stabilizers of totally singular or non-singular subspaces;
C2 stabilizers of decomposition V =
Lt
i=1 Vi , dim Vi = n/t;
C5 stabilizers of tensor product decompositions V =
Nt
i=1 Vi ,
dim Vi = a;
C6 dimV = rm
and H is “local”;
C7 stabilizers of bilinear or quadratic form.
S almost simple groups acting irreducibly and
tensor-indecomposably.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

27. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
From algebraic to finite: Frobenius automorphism
Let k be an algebraically closed field of characteristic p > 0.
σ : GLn(k) → GLn(k)
`
aij
´
→
`
apm
ij
´
Then, say q = pm
,
GLn(q) =
“
GLn(k)
”σ
= {x ∈ GLn(k) | σ(x) = x}
If G is any algebraic group G0 = Gσ
is a finite simple group.
Remark
It is possible to deduce the Aschbacher theorem by the Leibeck-Seitz theorem.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

28. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
Lang’s Theorem
Theorem
Let G ≤ GLn(k) be a connected linear algebraic group, where k is an a.c. field
of characteristic p > 0. Then the map g → g−1
σ(g) is surjective.
Thanks to this if we know the conjugacy classes of subgroups in G we know the
conjugacy classes of the image subgroups in Gσ
, as well.
Raffaele Rainone Subgroup structure of simple (algebraic) groups

29. Motivations
Preliminaries
Finite simple groups
Simple algebraic groups
Classical groups
Intermezzo
Exceptional groups
Algebraic
E6, E7, E8, F4, G2
Finite
of Lie type
G2(q), F4(q), E6(q), 2
E6(q), 3
D4(q), E7(q), E8(q)
Suzuki-Ree groups
2
B2(22n+1
), 2
G2(32n+1
), 2
F4(22n+1
)
Tits group
2
F4(2)
Raffaele Rainone Subgroup structure of simple (algebraic) groups