The Structure Theorem and Its Spinoffs

work in progress joint with Bernd Stellmacher and Gernot Stroth

Groups of local characteristic p

Let G be finite group and p a prime. Definition 1 G has characteristic p if

C_{G}(O_{p}(G)) ≤ O_{p}(G).

H is a p-local subgroup of G if H = N_{G}(P) for some non-trivial p-subgroup of G. G has local characteristic p if all p-local subgroups of G have characteristic p. G is a K_{p}-group if the composition factors of the plocals subgroups of G are known finite simple groups. Goals:

1. Understand the finite groups of local characteristic p.

2. Classify K_{p}-groups of local characteristic p whose p-local structure is not too small.

The Structure Theorem

Definition 2 Let H be group, 𝔽 a field and V an 𝔽H-module.

(a) H acts nilpotently on V if there exists an ascending series

0 = V_{0} ≤ V_{1} ≤ V_{2} …, V_{n−1} ≤ V_{n}

of 𝔽H-submodules of V such that H centralizes each of the factor V_{i+1} /V_{i}.

(b) V is H-reduced if [V, N] = 0 whenever N ⊴ H and N acts nilpotently on V.

(c) If H is finite, then the largest elementary abelian normal H-reduced p-subgroup of H is denoted by Y_{H}.

Definition 3 Let A and B be subgroups of G. The relation ≪ on the subgroups of G is defined by

A ≪ B : ⟺ A ⊆ C_{G}(Y_{A})B and Y_{A} ≤ Y_{B}.

Furthermore, we define

A^{†} := C_{G}(Y_{A})A

S^{†} = {L ≤ G | L = L^{†}} = {L ≤ G | C_{G}(Y_{L}) ≤ L} Lemma 4 (a) For all L ≤ G, A ≪ A^{†} and A^{†} ∈ S^{†}.

(b) ≪ is reflexive and transitive.

(c) Restricted to S^{†}, ≪ is a partial ordering.

Definition 5 S^{†}(S) = {L ∈ S^{†} | S ≤ L} and F(S) is the set of maximal elements of ≪ in S^{†}(S).

Definition 6 Let Q be a p-subgroup of a finite group G. We say that Q is large subgroup of G provided that C_{G}(Q) ≤ Q and

Q ⊴ N_{G}(A)

for all 1 ≠ A ≤ Z(Q).

Theorem 7 (Structure Theorem)

Let p be a prime, G be a finite K_{p}-group of local characteristic p. Suppose that Q is a large psubgroup of G and Q ≤ S ∈ Syl_{p}(G). Let M ∈ F(S) with Q ⋬ M. Put M^{∘} = ⟨Q^{M}⟩, \overline{M} = M/C_{M}(Y_{M}) and I = [Y_{M}, M^{∘}].

Suppose that Y_{M} ≤ O_{p}(N_{G}(Q)). Then one the following holds.

1. \overline{M^{∘}} ≅ SL_{n}(q), Sp_{2n}(q) or Sp_{4}(2)′ and I is the corresponding natural module.

2. There exists a normal subgroup K of \overline{M} such that

(a) K = K_{1} × ⋯ × K_{r}, K_{i} ≅ Sl_{2}(q) and

Y_{M} = V_{1} × ⋯ × V_{r}

where V_{i} := [Y_{M}, K_{i}] is a natural K_{i}-module.

(b) Q permutes the K_{i}’s transitively.

Suppose that Y_{M} ≰ O_{p}(N_{G}(Q)). Then one of the following holds:

(a) There exists a normal subgroup K of \overline{M} such that K = K_{1} ∘ K_{2} with K_{i} ≅ SL_{m_{i}}(q), Y_{M} ≅ V_{1} ⊗ V_{2} where V_{i} is a natural module for K_{i} and \overline{M^{∘}} is one of K_{1}, K_{2} or K_{1} ∘ K_{2}.

(b) (\overline{M^{∘}}, p, I) is as given in the following table:

\overline{M^{∘}}  p  I
SL_{n}(q)  p  natural
SL_{n}(q)  p  ⋀^{2}(natural )
SL_{n}(q)  p  S^{2}(natural )
SL_{n}(q^{2})  p natural ⊗ natural ^{q}
3 Alt(6), 3 Sym (6), 2  2^{6}
ΓSL_{2}(4), Γ GL_{2}(4)  2  natural
Sp_{2n}(q)  2  natural
Ω_{n}^{±}(q)  p  natural
O _{4}^{+} (2)  2  natural
Ω_{10}^{±} (q)  2  half-spin
E_{6}(q)  p  q^{27}
Mat_{11}  3  3^{5}
2 Mat_{12}  3  3^{6}
Mat_{22}  2  2^{10}
Mat_{24}  2  2^{11}

2F -stability

Definition 8 Let A be an elementary abelian pgroup and V a finite dimensional GF (p)A-module. Then A is

(a) quadratic on V if [V, A, A] = 0,

(b) cubic on V if [V, A, A, A] = 0,

(c) nearly quadratic on V if A is cubic and

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

for every v ∈ V \ [V, A] + C_{V} (A),

(d) an offender on V if |V/C_{V} (A)| ≤ |A/C_{A}(V)|,

(e) a 2F -offender if |V/C_{V} (A)| ≤ |A/C_{A}(V)|^{2},

(f) non-trivial on V if [V, A] ≠ 0.

Let A be an elementary abelian p-subgroup A of G. Then A is F -stable in G if none of the elementary abelian p-subgroups of N_{G}(A)/C_{G}(A) are non-trivial offenders on A.

Similarly, A is 2F -stable in G if none of the elementary abelian p-subgroups of N_{G}(A)/C_{G}(A) are non-trivial nearly quadratic 2F -offenders on A.

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

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

A is a nearly quadratic 2F -offender on V Definition 9 Let S be Sylow p-subgroup of G.

B(S) := C_{S}(Ω_{1}Z (J(S)))

C^{∗}(G, S) := ⟨C_{G}(Ω_{1}Z (S))), N_{G}(C) | 1 ≠ C char B(S)⟩

Definition 10 Let G be a finite group and H ≤ G.

(a) H is called a parabolic subgroup of G if H contains a Sylow p-subgroup of G.

(b) G has parabolic characteristic p if all p-local, parabolic subgroups of G have characteristic p.

Theorem 11 Let G be a finite group of parabolic characteristic p and S ∈ Syl_{p}(G). Suppose M ∈ F(S) such that Y_{M} is 2F -stable. Then

(a) C^{∗}(G, S) ≤ M.

(b) C^{∗}(H, T) ≤ H∩M < H for all H ≤ G with B(S) ≤ H and H ≰ N, where B(S) ≤ T ∈ Syl_{p}(H).

(c) If N ∈ F(S) with N ≠ M, then Y_{N} is not F - stable.

Corollary 12 Let G be a finite group of parabolic characteristic p and S ∈ Syl_{p}(G). If S is contained in at least two maximal p-local subgroups of G, then there exists M ∈ F(S) such that Y_{M} is not 2F -stable.

The Fitting Submodule

Let 𝔽 be a field, H a finite group and V a finite dimensional 𝔽H-module.

Definition 13

(a) rad_{V} (H) is the intersection of the maximal 𝔽Hsubmodules of V

(b) Let W be an 𝔽H submodule of V and N ⊴ H. Then W is N-quasisimple if W is H-reduced, W/ rad_{W} (H) is simple for 𝔽H, W = [W, N] and N acts nilpotently on rad_{W} (H).

(c) S_{V} (H) is the sum of all simple 𝔽H-submodules of V.

(d) E_{H}(V) := C_{F^{∗}(H)}(S_{V} (H)).

(e) W is a component of V if either W is a simple 𝔽H-submodule with [W, F^{∗}(H)] ≠ 0 or W is an E_{H}(V)-quasisimple 𝔽H-submodule.

(f) The Fitting submodule F_{V} (H) of V is the sum of all components of V.

(g) R_{V} (H) := ∑ rad_{W} (H), where the sum runs over all components W of V Theorem 14 (a) The Fitting submodule F_{V} (H) is

H-reduced.

(b) R_{V} (H) is a semisimple 𝔽 F^{∗}(H)-module.

(c) R_{V} (H) = rad_{F_{V} (H)}(H).

(d) F_{V} (H)/ R_{V} (H) is a semisimple 𝔽H-module Theorem 15 Let V be faithful and H-reduced. Then also F_{V} (H) and F_{V} (H)/ R_{V} (H) are faithful and Hreduced.

Definition 16 Let A be a subgroup of G such that A/C_{A}(V) is an elementary abelian p-group. A is a best offender of G on V if |B|·|C_{V} (B)| ≤ |A|·|C_{V} (A)| for every B ≤ A.

Definition 17 The normal subgroup of G generated by the best offenders of G on V is denoted by J_{G}(V).

A J_{G}(V)-component is non-trivial subgroup K of J_{G}(V) minimal with respect to K = [K, J_{G}(V)].

Theorem 18 (The Other P(G, V )-Theorem.)

Suppose that V is a faithful finite dimensional, reduced 𝔽_{p}G-module. Then

[E, K] = 1 and [V, E, K] = 0

for any two distinct J_{G}(V)-components E and K.

Definition 19 A finite group is a CK-group if all its compositions factors are known finite simple groups.

Theorem 20 (FF-Module Theorem, Guralnick-Malle) Let M be a finite CK group with F^{∗}(M) be quasisimple and V a faithful simple 𝔽_{p}M-module. Suppose that M = J_{M}(V).

Then (M, p, V) is one of the following:

M  p  V
SL_{n}(q) p  natural
Sp_{2n}(q) p  natural
SU_{n}(q) p  natural
Ω_{n}^{ϵ}(q) p natural
O _{2n}^{ϵ}(q) 2  natural
G_{2}(q) 2 q^{6}
SL_{n}(q) p ⋀^{2}(natural )
Spin _{7}(q) p  Spin
Spin _{10}^{+} (q) p  Spin
3. Alt(6) 2  2^{6}
Alt(7) 2 2^{4}
Sym (n) 2  natural
Alt(n) 2  natural

Theorem 21 (J-Module Theorem) Let M be a finite CK-group, V a faithful, reduced 𝔽_{p}M-module. Let J = J_{V} (M). Let J = J_{V} (M) be the set of J_{V} -components of V. Put W = [V, J]C_{V} (J)/C_{V} (J) and let K ∈ J.

(a) K is either quasisimple or p = 2 or 3 and K ≅ SL_{2}(p)′.

(b) [V, K, L] = 0 for all K ≠ L ∈ J.

(c) W = ⨁_{K∈J} [W, K].

(d) J^{p}J′ = 0^{p}(J) = F^{∗}(J) = \ / J.

(e) W is a semisimple 𝔽_{p}J-module.

(f) Let J_{K} = J/C_{J}([W, K]). Then K ≅ O^{p}(J_{K}) and one of the following holds:

1. [W, K] is a simple K-module and (J_{K}, [W, K]) fullfills the assumptions and so also the conclusion of Theorem 20 .

2. J_{k}, and [W, K] are as follows (where N denotes a natural module and N^{∗} its dual):

J_{K}  [W, K]  conditions
SL_{n}(q) N^{r} ⊕ N^{∗s} √r + √s ≤ √n
Sp_{2n}(q)  N^{r}  r ≤ n
SU_{n}(q)  N^{r}  r ≤ \frac{n}{4}
Ω_{n}^{ϵ}(q)  N^{r}  r ≤ \frac{n−2}{4}
O _{2n}^{ϵ}(q)  N^{r}  p = 2, r ≤ \frac{2n−2}{4}

Nearly Quadratic Modules

Lemma 22 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.

Theorem 23 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 24 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) as follows:

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