Nearly Quadratic Modules

Work in progress

joint with Bernd Stellmacher

Definitions

Let A be an abelian group, 𝔽 a field and V an 𝔽Amodule. The module V is a quadratic 𝔽A-module if [V, A, A] = 0, cubic 𝔽A-module if [V, A, A, A] = 0, nearly quadratic 𝔽A-module if V is a cubic 𝔽Amodule and

[V, A] + C_{V} (A) = [𝔽v, A] + C_{V} (A)

for every v ∈ V \ [V, A] + C_{V} (A). In the corresponding cases we also say that A is quadratic, cubic and nearly quadratic on V. Q_{V} (A) is the largest quadratic 𝔽A-submodule of V.

Motivation

Let G be a finite group, p a prime and V an elementary abelian p-subgroup of G. Suppose that

(i) C_{G}(O_{p}(G)) ≤ O_{p}(G) (ii) V ≰ O_{p}(G). (iii) V is weakly closed in G. Choose V ≤ L ≤ G minimal with V ≰ O_{p}(L). Put A := ⟨(V ∩ O_{p}(L))^{L}⟩. Then [V, A] ≠ 1 and

A is nearly quadratic on V Lemma 1 Let V be a nearly quadratic 𝔽A-module and W be an 𝔽A-submodule of V. Then

(a) Either

W ≤ [V, A] + C_{V} (A)

or

[V, A] ≤ [W, A] + C_{V} (A).

(b) One of the following holds:

1. A acts quadratically on V, Q_{V} (A) = V and

[V, A] + C_{V} (A) = C_{V} (A).

2. A does not act quadratically and

Q_{V} (A) = [V, A] + C_{V} (A).

(c) Both W and V/W are nearly quadratic 𝔽Amodules.

(d) A acts quadratically on W or on V/W.

Lemma 2 Let V be a nearly quadratic, but not quadratic 𝔽A-module. Let X and Y be 𝔽A-submodules of V such that

V = X ⊕ Y

Then A centralizes X or Y.

Lemma 3 Let V be a faithful, nearly quadratic, but not quadratic 𝔽A-module, and let Δ be a system of imprimitivity for A on V. Suppose that A acts nontrivially on Δ.

Then exists a unique A-orbit W^{A} ⊆ Δ with [W, A] ≠ 0. Moreover C_{A}(W)=1, B := N_{A}(W) acts quadratically on V, and one of the following holds:

1. char 𝔽 = 2, |W^{A}| = 4, dim_{𝔽} W = 1, B = 1 and A ≅ C_{2} × C_{2}.

2. char 𝔽 = 3, |W^{A}| = 3, dim_{𝔽} W = 1, B = 1, and A ≅ C_{3}.

3. char 𝔽 = 2, |W^{A}| = 2, and

[W, B] = C_{W} (B)

is an 𝔽-hyperplane of W.

Lemma 4 Let 𝕂 be a field, A a group and V_{1} and V_{2} be nilpotent 𝕂A-modules and

V = V_{1} ⊗_{𝕂} V_{2}.

Suppose that there exists a subfield 𝔽 ≤ 𝕂 such that V is a nearly quadratic, but not quadratic 𝔽Amodule. Then for j = 1, 2

[V_{j}, A] = C_{V_{j}}(A)

is a 𝕂-hyperplane of V_{j}.

Lemma 5 Let V be faithful, nearly quadratic,not quadratic 𝔽A-module. Suppose that there exists a field 𝕂 with 𝔽 ≤ 𝕂 such that V is a semi-linear, but not linear, 𝕂A-module. Put A_{𝕂} = C_{A}(𝕂). Then p := char 𝕂 ∈ {2, 3}, A is a elementary abelian pgroup, dim_{𝔽} 𝕂 = |A/A_{𝕂}| and one of the following holds:

1. A ≅ C_{3}, A_{𝕂} = 1 and dim_{𝕂} V = 1.

2. A ≅ C_{2} × C_{2}, A_{𝕂} = 1 and dim_{𝕂} V = 1.

3. |A/A_{𝕂}| = 2 and [V, A_{𝕂}] = C_{V} (A_{𝕂}) is a 𝕂-hyperplane of V.

Theorem 6 Let 𝔽 be field, H a group and V be a faithful semisimple 𝔽H-module. Let Q be the set of nearly quadratic, but not quadratic subgroups of H. Suppose that H = ⟨Q⟩. Then there exists a partition (Q_{i})_{i∈I} of Q such that

(a) H = ⨁_{i∈I} H_{i}, where H_{i} = ⟨Q_{i}⟩.

(b) V = C_{V} (H) ⊕ ⨁_{i∈I}[V, H_{i}].

(c) For each i ∈ I, [V, H_{i}] is a simple 𝔽H_{i}-module.

Theorem 7 ^{∗} Let H be a finite group, and V a faithful simple 𝔽_{p}H-module. Suppose that H is generated by elementary abelian, nearly quadratic, but not quadratic subgroups of H.

Let W a simple 𝔽_{p} F^{∗}(H)-submodule of V and

𝕂 = End_{F^{∗}(H)}(W).

Then H, V, W, 𝕂 and H/C_{H}(𝕂) are as in the following table:

H  V  W 𝕂 H/C_{H}(𝕂)
(C_{2} ≀ Sym (n))′  𝔽_{3}^{n}  𝔽_{3} 𝔽_{3}  −
SL_{n}(𝔽_{2}) ≀ C_{2}  𝔽_{2}^{n} ⊕ 𝔽_{2}^{n} 𝔽_{2}^{n} 𝔽_{2}  −
SL_{2}(𝔽_{2}) × SL_{2}(𝔽_{2}) 𝔽_{2}^{2} ⊗ 𝔽_{2}^{2} 𝔽_{4} 𝔽_{4}  −
Frob (39)  𝔽_{27}  V 𝔽_{27}  C_{3}
ΓGL _{n}(𝔽_{4})  𝔽_{4}^{n}  V 𝔽_{4}  C_{2}
ΓSL _{n}(𝔽_{4})  𝔽_{4}^{n}  V 𝔽_{4}  C_{2}
3. Sym (6)  𝔽_{4}^{3}  V 𝔽_{4}  C_{2}
SL_{n}(𝕂) ∘ SL_{m}(𝕂) 𝕂^{n} ⊗ 𝕂^{m} V any  1
(C_{2} ≀ Sym (4))′  𝔽_{3}^{4}  V 𝔽_{3}  1
F^{∗}(H) quasisimple  ?  V ?  1

∗ In the cases where |H/C_{H}(𝕂)| = 2 we currently use the FF-module Theorem ( and so a K-group assumption) to identify H Some examples for the last case ( under stronger assumptions, including 2F and K-group, these are all the examples.)

H  V  𝕂
SL_{n}(𝕂)  𝕂^{n}  any
Sp_{2n}(𝕂)  𝕂^{2n}  any
Ω_{n}^{ϵ}(𝕂)  𝕂^{n}  any
SL_{n}(𝕂)  ⋀^{2} 𝕂^{n}  any
SL_{n}(𝕂)  S^{2} 𝕂^{n} char 𝕂 odd
Spin _{10}^{+} (𝕂) 𝕂^{16} , spin  any
E_{6}(𝕂)  𝕂^{27}  any
Mat_{22} ,  𝔽_{2}^{10}  𝔽_{2}
Mat^{24}  𝔽_{2}^{11}  𝔽_{2}
Mat^{11}  𝔽_{3}^{5}  𝔽_{3}
2. Mat_{12}  𝔽_{3}^{6}  𝔽_{3}