Norddeutsches Gruppentheorie-Kolloquium

11-12. Oktober 2002

Christian-Albrecht-Universität zu Kiel

A Pushing Up Theorem

Ulrich Meierfrankenfeld Andy Chermak work in progress

History

Aschbacher’s local CGT Theorem Baumann/Niles Pushing Up SL_{2}(q) Gaubermann-Niles: A pair of characteristic subgroups for pushing up in finite groups. Timmesfeld: A pushing up result and some consequences for the embedding of 2-constrained subgroups. Timmesfeld: Simultaneous pushing up.

p-reduced normal subgroups

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

An elementary abelian normal p-subgroups V of H is called p-reduced if O_{p}(H/C_{H}(V)) = 1.

Lemma: Let H be a finite group of characteristic p and T ∈ Syl_{p}(H). Then

(a) There exists a unique maximal p-reduced normal subgroup Y_{L} of L.

(b) Let T ≤ R ≤ H and X a p-reduced normal subgroup of R. Then ⟨X^{H}⟩ is a p-reduced normal subgroup of H. In particular, Y_{R} ≤ Y_{H}.

(c) Let T_{H} = C_{T} (Y_{H}) and H_{T} = N_{H}(T_{H}). Then H = H_{T} C_{H}(Y_{H}), T_{H} = O_{p}(H_{T}) and Y_{H} = Ω_{1}Z(T_{H}).

(d) Y_{T} = Ω_{1}Z (T), Z_{H} := ⟨Ω_{1}Z (T)^{H}⟩ is p-reduced for H and Ω_{1}Z (T) ≤ Z_{H} ≤ Y_{H}.

(e) Let V be p-reduced normal subgroup of H and K a subnormal subgroup of H. Then [V, O^{p}(K)] is a p-reduced normal subgroup of K.

Point-Stabilizers

Let T ∈ Syl_{p}(H). Then

P_{H}(T) := O^{p′}(C_{H}(Ω_{1}Z (T)))

is the point-stabilizer of H with respect to T. Lemma: Let H be a finite group of characteristic p, T ∈ Syl_{p}(H) and L a subnormal subgroup of H. Then

(a) C_{L}(Ω_{1}Z (T)) = C_{L}(Ω_{1}Z (T ∩ L))

(b) P_{L}(T ∩ L) = O^{p′}(P_{H}(T) ∩ L)

(c) C_{L}(Y_{L}) = C_{L}(Y_{H})

(d) Suppose L = ⟨L_{1}, L_{2}⟩ for some subnormal subgroups L_{1}, L_{2} of H. Then (da) P_{L}(T ∩ L) = ⟨P_{L_{1}}(T ∩ L_{1}), P_{L_{2}}(T ∩ L_{2})⟩. (db) For i = 1, 2 let P_{i} be a point stabilizer of L_{i}. Then ⟨P_{1}, P_{2}⟩ contains a point stabilizer of L.

Notation

For a p-group R we let PU_{1}(R) be the class of all finite CK- groups L containing R such i) L is of characteristic p, ii) R = O_{p}(N_{L}(R)) iii) N_{L}(R) contains a point stabilizer of L. Let R be a group and Σ a set of groups containing R. Then

O_{R}(Σ) = ⟨N ≤ R | N ⊴ L ∀L ∈ Σ⟩ So O_{R}(Σ) is the largest subgroup of R which is normal in all the L ∈ Σ. For example if R ∈ Syl_{p}(G) and R ≤ L ≤ G for all L ∈ Σ then

O_{R}(Σ) = O_{p}(⟨Σ⟩)

Goal

Given a finite p-group R and a subset Σ of PU_{1}(R). Suppose that O_{R}(Σ) = 1. For each L ∈ Σ, determine the structure of ⟨B(R)^{L}⟩.

Here B(R) is the Baumann subgroup of R defined as follows: A(R) is the set of elementary abelian subgroups of maximal order in R. J(S) = ⟨A(R)⟩ is the Thompson subgroup of R B(R) = C_{R}(Ω_{1}Z (J(R))).

Example

Let G be a finite group of local characteristic p, that is all p-local subgroups of G have characteristic p.

Let S ∈ Syl_{p}(G), 1 ≠ x ∈ Ω_{1}Z (S) and C = C_{G}(x). Then one of the following holds:

1. C is contained in a unique maximal p-local subgroup of G.

2. Put R = O_{p}(C) and let Σ be the set of maximal p-locals subgroups of G containing C. Then

Σ ⊆ PU_{1}(R) and O_{R}(Σ) = 1

Complication

Let I be a finite set and for i ∈ I let R_{i} be a finite p group and

Σ_{i} ⊆ PU_{1}(R_{i}) with O_{R_{i}}(Σ_{i}) = 1

Put

R = \ /_{i∈I} R_{i}

and

Σ = { \ /_{i∈I} L_{i} | L_{i} ∈ Σ_{i} ∀i ∈ I}

Then

Σ ⊆ PU_{1}(R) and O_{R}(Σ) = 1

FF-modules

Theorem: Let H be a finite group, V a faithful, irreducible 𝔽_{p}H-module, L a point stabilizer for H on V and 1 ≠ A ≤ O_{p}(L). Suppose that

i) |V/C_{V} (A)| ≤ |A/C_{A}(V)|.

ii) F^{∗}(H) is quasi-simple and H = ⟨A^{H}⟩.

Then H ≅ SL_{n}(q), Sp_{2n}(q), G_{2}(q) or Sym (n), where p = 2 in the last two cases and n ≡ 2, 3 (mod 4) in the last case. Moreover, V is the corresponding natural module.

The Baumann Argument

Lemma: Let L be a finite group, R a psubgroup of L, V := Ω_{1}Z (O_{p}(L)), K := ⟨B(R)^{L}⟩, Ṽ = V/C_{V} (O^{p}(K)), and suppose that each of the following holds:

i) O_{p}(L) ≤ R and L = ⟨J(R)^{L}⟩N_{L}(J(R)). ii) C_{K}(Ṽ) is p-closed. iii) |Ṽ /CṼ (A)| ≥ |A/C_{A}(Ṽ)| for all elementary abelian subgroups A of R. iv) If U is an L/O_{p}(L) module with Ṽ ≤ U and U = C_{U}(B(R))Ṽ, then U = C_{U}(O^{p}(K))Ṽ. Then O_{p}(K) ≤ B(R).

The Reduction Theorem

Let PU_{3}(R) be the class of all finite CK-groups L such that

i) L is of characteristic p. ii) R ≤ L and L = ⟨R^{L}⟩ iii) L/O_{p}(L) ≅ SL_{n}(q), Sp_{2n}(q) or G_{2}(q), where q is a power of p and p = 2 in the last case. iv) Y_{L}/C_{Y_{L}}(L) is the corresponding natural module.

v) O_{p}(L) ≤ R and N_{L}(R) contains a point-stabilizer of L. vi) If L/C_{L}(Y_{L}) ≅ G̸_{2}(q) then R = O_{p}(N_{L}(R))

Let PU_{4}(R) be the class of all finite groups L containg R such that L is of characteristic p and

L = ⟨N_{L}(R), H | R ≤ H ≤ L, H ∈ PU_{3}(R)⟩. Theorem: Let R be a p-group. Then

PU_{1}(R) ⊆ PU_{4}(B(R)).

The First Pushing Up Theorem Theorem: Let T be a p-group with T = B(T) and Σ a subset of PU_{3}(T) with O_{R}(Σ) = 1. Then there exists H ∈ Σ such O^{p}(H) has one of the following structures (where q is a power of p) q^{n}SL_{n}(q)′; q^{2n}Sp_{2n}(q)′, p odd; q^{1+2n} Sp_{2n}(q)′, p = 2; 2^{6}G_{2}(2)′, p = 2; q^{1+6+8} Sp_{6}(q), p = 2; 2^{1+4+6} L_{4}(2), p = 2; or q^{1+2+2} SL_{2}(q)′, p = 3.

On the Proof

Lemma: Let L, H ∈ Σ.

(a) If L/O_{p}(L) ≅ S̸p_{2n}(q), n ≥ 2 and Y_{L} ≰ Y_{H}, then [Y_{L}, R] ≤ Z(H).

(b) If L/O_{p}(L) ≅ Sp_{2n}(q), n ≥ 2 and [Y_{L}, R] ≰ Y_{H}, then [Y_{L}, R, R] ≤ Z(H).

(c) If L/O_{p}(L) ≅ Sp_{2n}(q), n ≥ 2 and N is a normal p-subgroup of H with [Y_{L}, R, N] = 1, then [Y_{L}, N] ≤ Z(H)

Proposition Let L, H ∈ Σ. Then one of the following holds

1. O_{p}(H) ∩ O_{p}(L) is normal in H and L.

2. ⟨Y_{H} ^{L}⟩ is not abelian.

3. ⟨Y_{L}^{H}⟩ is not abelian. Proposition: There exists H, L in Σ such that ⟨Y_{H} ^{L}⟩ is not abelian. Proof: Otherwise ⋂_{H∈Σ} O_{p}(H) is normal in each L ∈ Σ. Also Ω_{1}Z (R) ≤ ⋂_{H∈Σ} O_{p}(H) and

_{H∈Σ}^{⋂} O_{p}(H) ≠ 1.

A guess

Let R be a finite p-group, Σ ⊆ PU_{1}(R) with O_{R}(Σ) = 1 and L ∈ Σ. Then we guess that there exist subgroups L_{i}, i ∈ I, of L such that

⟨B(R)^{L}⟩ = \ /_{i∈I} L_{i}

and for i ∈ I, O^{p}(L_{i}) has one of the following structures:(where q is a power of p) q^{n}SL_{n}(q)′; q^{2n}Sp_{2n}(q)′, p odd; q^{1+2n} Sp_{2n}(q)′, p = 2; 2^{6}G_{2}(2)′, p = 2; q^{1+6+8} Sp_{6}(q), p = 2; 2^{1+4+6} L_{4}(2), p = 2; q^{1+2+2} SL_{2}(q)′, p = 3. 2^{n−1} Alt(n), p = 2, n ≡ 2, 3 (mod 4) 2^{1+2+1·2m+2·2k} SL_{2}(2)′, p = 2. maybe a case similar to the last one with SL_{2}(2) replaced by Sp_{2n}(2).