Groups of local characteristic p

Barbara Baumeister Andy Chermak Andreas Hirn Mario Mainardis

Ulrich Meierfrankenfeld Gemma Parmeggiani

Chris Parker Peter Rowley

Bernd Stellmacher Gernot Stroth An alternative title: An important step in a third generation proof for the classification of finite simple groups.

CFSG: Each finite simple group is isomorphic to one of the following:

a group of prime order.

an alternating group.

a group of Lie-type.

one of the 26 sporadics.

One of our goals: Uniform approach to the groups Lie-type.

Given a finite (simple) group G. How can one show that G is isomorphic to a group of Lietype G^{∗}.

G^{∗} is (essentially) the automorphism group of a building B^{∗}.

Buildings are fairly easy to identify up to isomorphism:

Theorem Let B and B^{∗} spherical buildings with the same set of types I and rank at least four. Let C and C^{∗} be chambers of B and B^{∗} respectively. Suppose that for all J ⊂ I, Res _{J}(C) ≅ Res _{J}(C^{∗}). Then B ≅ B^{∗}.

This theorem follows from Tits’ classification of the spherical bulidings, but also has fairly elementary proof (assuming (4.1.2) from Tits’ notes) .

But how can we show that G acts on a building? That is how can we describe the building in group theoretical terms?

Building ⟷ BN-pair ⟷ parabolic subgroups.

parabolis subgroups = overgroups of the Borel subgroup,

Borel subgroup = (Normalizer of a) maximal unipotent subgroups

Maximal unipotent subgroups = Sylow psubgroup, where p is the characteristic of the underlying field.

This raises two more questions: How to determine the correct prime p? How to find the parabolic subgroups, that is the overgroups of the Sylow p-subgroup?

An example: Let G = GL_{n}(q). Then G acts on the building consisting of the (flags of) proper subspaces of 𝔽_{q}^{n}.

The maximal parabolics are the normalizers of the subspaces:

M = (\begin{matrix}∗ & 0 \\ ∗ & ∗ \end{matrix})

Put Q = (\begin{matrix}I_{k} & 0 \\ ∗ & I_{n−k} \end{matrix})

Then Q is a p-group, M = N_{G}(Q) and C_{G}(Q) ≤ Q, ( mod Z(G))

G is a finite K_{p}-group, and p a fixed prime.

G has characteristic p if C_{G}(O_{p}(G)) ≤ O_{p}(G).

p-local subgroup: Normalizer of a non-trivial p-subgroup.

G has local characteristic p if all p-local subgroups of G have characteristic p.

G has parabolic characteristic p if all the plocal, parabolic subgroups of G have characteristic p.

Goal: Understand and classify the finite groups of local characteristic p with O_{p}(G) = 1.

Disclaimer: For p odd we do not expect to be able to achieve a complete classification. Some groups with a relatively small p-local structure will remain unclassified. In particular, we currently have no idea how to treat the case where G has a strongly p-embedded subgroup.

Characteristics of the known simple groups

1. Groups of Lie-Type

Let G be a finite simple group of Lie type defined over a field of characteristic r. If p = r, then G is of local characteristic p. If p ≠ r and a Sylow p-subgroup of G is not cyclic, then G is usually not of parabolic characteristic p. Some exceptions: U_{3}(3) ≅ G_{2}(2)′ , Sp_{4}(2)′ ≅ L_{2}(9), P Ω_{5}(3) ≅ Ω_{6}^{−}(2), L_{3}(4) and U_{4}(3) all have local characteristics 2 and 3. L_{4}(3) has parabolic characteristics 2 and 3.

2. Alternating groups

The alternating groups usually have no local characteristic. But Alt(p^{n}+ϵ),ϵ ≤ 2 has parabolic characteristic p.

3. Characteristics of the sporadics

Group local char.

parabolic char.

M_{11} 3 3
M_{12}  2, 3

J_{1}

M_{22} 2 2
J_{2}  2
M_{23} 2 2
HS  2
J_{3}  2  2
M_{24}  2  2
McL  3  3
He  2
Ru  2, 5
Suz  2
ON 7 7
Co_{3}  3, 5
Co_{2}  2  3, 5
F i_{22}  2  2
HN  2, 3
Ly  5  5
T h  2, 5  2, 3, 5
F i_{23}  3
Co_{1}  2, 3, 5
J_{4} 2, 11 2, 11
F i′_{24}  2, 3, 7
B  2, 3, 5
M  2, 3, 5, 7, 13

Notation

G is a group of local characteristic p with O_{p}(G) = 1. M is the set of maximal p-local subgroups of G. If T is a set of subgroups of G and A ≤ G, then

T (A) = {T ∈ T | A ≤ T }.

S is a fixed Sylow p-subgroup of G. M(S) is the set of maximal p-local subgroups containg S.

Back to the example G = GL_{n}(𝔽_{q}). The corresponding building is A_{n−1}. The residue corresponding to M has type A_{k−1} × A_{n−k−1}. So the residue can be read of from the structure of M/Q ≅ GL_{k}(q) × GL_{n−k−1}(q).

This suggest that our main task is to indentify L/O_{p}(L) for various members L of L.

Our favorite method for this is to study the action of L on p-reduced normal subgroups, i.e. elementary abelian normal p-subgroups Y of L with

O_{p}(L/C_{L}(Y)) = 1.

Y_{L} is the largest p-reduced normal subgroup of L.

Unfortunately the action of L on Y_{L} does not yield any information about C_{L}(Y_{L}). An elementary argument shows that Ω_{1}Z (S) ≤ Y_{L} and so C_{L}(Y_{L}) ≤ C_{G}(Ω_{1}Z (S)). So to make up for this misfortune we also study the group N_{G}(Z). For this we pick

C̃ ∈ M with N_{G}(Z) ≤ C̃.

For a group H, define F_{p}^{∗}(H) by F_{p}^{∗}(H)/O_{p}(H) = F^{∗}(H/O_{p}(H)). Put E := O^{p}(F_{p}^{∗}(CC̃(YC̃))). We now distinguish two cases: E-uniqueness (E!): M(E) = {C̃} and non E-uniqueness (¬E!): |M(E)| ≥ 2. For the rest of this talk we assume that E! holds.

The Structure Theorem

Let M ∈ M(S). Suppose E! and that M ≠ C̃. Then there exists K ⊴ M/C_{M}(Y_{M}) and V ≤ Y_{M} with V ⊴M such K and V are as in the following table:

K  p  V  example
SL_{n}(q)  p  natural  SL_{n+1} (q)
SL_{n}(q)  p  ext. square  Ω_{2n}(q)
SL_{n}(q)  p  sym. square  Sp_{2n}(q)
SL_{n}(q^{2})  p  nat. ⊗ nat.^{σ}  SU_{2n}(q)
SL_{n}(q) ∘ SL_{m}(q)  p  nat. ⊗ nat. SL_{n+m} (q)
Sp_{2n}(q)  p  natural  B
A_{6}  2  natural  Suz
3A_{6}  2  6-dim  M_{24}
Ω_{n}^{±}(q)  p  natural  Ω_{n+2}^{±} (q)
Ω_{10}^{±} (q)  p  half spin  E_{6}(q)
E_{6}(q)  p  V (λ_{1})  E_{7}(q)
M_{11}  3  5-dim  Co_{3}
2M_{12}  3  6-dim  Co_{2}
M_{22}  2  10-dim  M(22)
M_{24}  2  11-dim  M(24)
SL_{2}(q)^{m} p = 2  natural^{m}  -

A finite group L is p-minimal if a Sylow psubgroup of L is contained in a unique maximal subgroup of L but is not normal in L.

P ≤ G is a minimal parabolic subgroup if O_{p}(P) ≠ 1 and P is parabolic and p-minimal.

P denotes the set of minimal parabolics of G.

Define the rank of G to be the minimal n such that there exist P_{i} ∈ P(S),1 ≤ i ≤ n, with

O_{p}(⟨P_{1}, P_{2}, …, P_{n}⟩) = 1.

If no such n we define the rank to be 1.

Fix P ∈ P(S) with P ≰ C̃. We say that gp(P) = 1 if there exists M ∈ M(P) with Y_{M} ≰ O_{p}(C̃). We say gb(P) = 2 if gb(P) ≠ 1 and ⟨(Y_{P})C̃⟩ is not abelian. The Small World Theorem: Suppose E!. Then one of the following holds:

1. G has rank 1 or 2.

2. gb(P) = 1 or gb(P) = 2.

Rank 2 Theorem, I

Suppose E!, gp(P) ≠ 1 and that G has rank 2. Then there exists P̃ ∈ P(S) such that one of the following holds:

1. (P, P̃) is a weak BN-pair. (slightly simplified statement)

2. The structure of P and P̃ is as in one of the following groups.

1. For p = 2: U_{4}(3).2^{e}, G_{2}(3).2^{e}, D_{4}(3).2^{e}, HS.2^{e}, F_{3}, F_{5}.2^{e} or Ru.

2. For p = 3: D_{4}(3^{n}).3^{e}, F i_{23} , F_{2}.

3. For p = 5: F_{2}.

4. For p = 7: F_{1}.

gb(P)=1: The H-Structure Theorem

Supppose E!, and gp(P) = 1. Let M ∈ M(P) with M^{∘} := ⟨O_{p}(C̃)^{M}⟩ maximal and Y_{M} ≰ O_{p}(C̃). Then either M^{∘}S is p-minimal or there exists M^{∘}S ≤ H ≤ G with O_{p}(H) = 1 such that H has same residuell type has one of the following groups:

1. A group of Lie-type in characteristic p.

2. For p = 2: M_{24} , He, Co_{2}, F i_{22} , Co_{1}, J_{4}, F i_{24} , Suz, B, M, U_{4}(3) or G_{2}(3).

3. For p = 3: F i_{24} , Co_{3}, Co_{1} or M.