The ¬E-uniqueness Theorem

Ulrich Meierfrankenfeld Bernd Stellmacher Gernot Stroth

February 23, 2003

1 Introduction

Let G be a finite group and p a prime dividing the order of G. We say that G has characteristic p if C_{G}(Ω_{p}(G)) ≤ Ω_{p}(G) and we say that G has local characteristic p if all p-local subgroups of G have characteristic p. This paper is part of the project to classify all finite groups of local characteristic p. The classification is divided into two main part: The Euniqueness case (E!) and the non E-uniqueness case (¬E!). To explain these cases we need to introduce some notation:

Let G be a finite group of local characteristic p, S ∈ Syl_{p}(G). Z = Ω_{1}Z(S), L = {L ≤ G | C_{G}(Ω_{p}(L)) ≤ Ω_{p}(L)}, M is set of maximal elements of L; If T is a set of subgroups of G and H ≤ G, then T (H) = {T ∈ T | H ≤ T } and

T_{H} = {T ∈ T | T ≤ H}.

We say that T ∈ L is a uniqueness subgroup of G if T is contained in a unique maximal p-local of G, that is if |M(T)| = 1.

For L ∈ L let Y_{L} be the the largest p-reduced normal subgroup of G ( see 4.1). For H a finite group, 𝔽_{p}^{∗}(H) is defined by 𝔽_{p}^{∗}(H)/Ω_{p}(H) = 𝔽^{∗}(H/Ω_{p}(H)). C̃ is a maximal p-local containing ℕ_{G}(Z) (in symbols: C̃ ∈ M(N_{G}(Z))). E = Ω^{p}(𝔽_{p}^{∗}(CC̃(YC̃))). E! now means that E is a uniqueness subgroup and ¬E! means that E is contained in

at least two different maximal p-locals of G.

2 Some unnecessary comments on groups of parabolic characteristic p

Let G be a finite group and p a prime dividing the order of p. A subgroup P of G is called a parabolic if it contains a Sylow p- subgroup of G. A parabolic P is called a local parabolic if Ω_{p}(P) ≠ 1. A parabolic is called regular, if it contains the normalizer of Sylow p-subgroup. G is of (regular) parabolic characteristic p if all (regular) local parabolics are of characteristic p. We eventually hope to extend the classification of groups of local characteristic p to the groups of regular parabolic characteristic p.

The Monster and the Baby Monster are example of groups which are of parabolic characteristic 2, but not of local characteritic 2. J_{1} is a group which is of regular parabolic characteristic 2, but not of parabolic characteristic 2.

3 An unnecessary section on bricks

Definition 3.1 [def-brick] Let G be a finite group. A brick of G is a perfect subnormal subgroup B of G such that B has a unique maximal normal subgroup M_{B}. Br(G) denotes the sets of all bricks of G.

Lemma 3.2 [minimal subnormal supplement] Let G be a finite group and D a normal subgroup of G with G/D perfect.

(a) [a] There exists a the unique minimal subnormal supplement B = B(G, D) to D in G.

(b) [b] B is normal in G

(c) [c] If G/D is simple, then B is the unique brick of G with B ≰ D. Moreover [B, D] ≤ M_{B} = B ∩ D.

(d) [d] If G is perfect, then G = BD^{∞}.

Proof: (a): Let B_{1} and B_{2} be minimal subnormal supplements to D in G. We need to show that B_{1} = B_{2}. If G = B_{i} for some i this is obvious. So we may assume that B_{i} ≤ M_{i} for a proper normal subgroup M_{i} of G. Then G = M_{i}D. Put M = [M_{1}, M_{2}]. Since G/D is perfect, G = [G, G]D = [M_{1}D, M_{2}D]D = MD. By induction theee exists a unique minimal supplement B to M ∩ D in M. Since G = MD and M ≤ M_{i}, M_{i} = M(D ∩ M_{i} and so M_{i} = B(D ∩ M_{i}. By induction B = B(M_{i}, D ∩ M_{i}) = B_{i} and thus B_{1} = B_{2}.

(b) Let g ∈ G. The also B^{g} is a minimal subnormal supplement to D in G and so B = B^{g} by the uniqueness of B.

(c) Let M be a normal subgroup of B. Suppose that M ≰ D. The MD/D ⊴ BD/D = G/D. Since G/D is simple, G = MD and so the minimality of B implies M = B. Thus B ∩ D is the unique maximal normal subgroup of G and B is a brick. Let B̃ be any brick of G with B̃ ≰ D. Then B̃D/D is a non-trivial subnormal subgroup of the simple G/D and so B̃D = G. Thus B ≤ B̃. Moreover, B̃/B̃ ∩ D is simple and so B̃ ∩ D = MB̃. In particular B ≰ MB̃ and so B = B̃.

(d) Since G/B is perfect and G/B = DB/B we get G/D = D^{∞}B/B. □

Proposition 3.3 [bricks and subnormal subgroups] Let B be a brick of the finite group G and N ⊴ ⊴G. Then either B ≤ N or N normalizes B and [B, N] ≤ M_{B}.

Proof: If N = G, B ≤ N. So we may assume that N is contained in a maximal normal subgroup D of G. If B ≤ D we are done by induction. So suppose that B ≰ D. Then by 3.2 D = B(G, D) and [B, N] ≤ [B, D] ≤ M_{B}. □

Lemma 3.4 [products of bricks] Let B_{1} and B_{2} be bricks of the finite group G. Then ⟨B_{1}, B_{2}⟩ = B_{1}B_{2} and exactly one of the following holds.

1. [a] B_{1} = B_{2}

2. [b] B_{1} ≤ M_{B_{2}},

3. [c] B_{2} ≤ M_{B_{1}}.

4. [d] [B_{1}, B_{2}] ≤ M_{B_{1}} ∩ M_{B_{2}}.

Proof: If B_{1} ≰ B_{2} and B_{2} ≰ B_{1} then by 3.3 [B_{1}, B_{2}] ≤ M_{B_{1}} ∩ M_{B_{2}}. So we may assume B_{1} ≤ B_{2}. But then B_{1} = B_{2} or B_{1} ≤ M_{B_{2}}. So one of (1)-(4) holds. Since B_{i} is perfect its easy to see that at most one of (1)-(4) can hold. Moreover in all four cases, ⟨B_{1}, B_{2}⟩ = B_{1}B_{2}. □

Lemma 3.5 [Ginfty] Br(G) = Br(G^{∞}) and G^{∞} = ∏_{B∈Br(G)} B.

Proof: Note that a brick of G^{∞} is a brick of G and all bricks of G are contained in G^{∞}. Thus Br(G) = Br(G^{∞}). Let D be a maximal normal subgroup of G^{∞} Then by 3.2 there exists a brick B with G^{∞} = BD^{∞}. By induction D^{∞} is the products of its bricks. So also G^{∞} is the products of its bricks. □

4 The Largest p-reduced normal subgroup

Let L be a finite group of characteritic p. An elementary abelian normal subgroup V of L is called p-reduced if any normal subgroup of G which acts unipotently on V has to act trivially. Note that this is equivalent to Ω_{p}(L/ C_{L}(V)) = 1.

Lemma 4.1 [YL] Let L be a finite group of characteritic p and S ∈ Syl_{p}(L)

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

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

(c) [c] Let S_{L} = C_{S}(Y_{L}) and L^{f} = N_{G}(S_{L}). Then L = L_{f} C_{L}(Y_{L}), S_{L} = O_{p}(L^{f}) and Y_{L} = Ω_{1}Z(S_{L}).

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

Proof: (a) Let Y_{L} be the subgroup generated by the p-reduced normal subgroups of L. Let N be a normal subgroup acting unipotently on Y_{L}. Then N also acts unipotently on all the generators of Y_{L}. Hence N centralizes all the generators of Y_{L} and so Y_{L}. Thus Y_{L} is p-reduced.

(c) Let Y = ⟨X^{L}⟩ and C = C_{L}(Y). Let N/C = Ω_{p}(L/C). Then N = (N ∩ S)C and in particular, N = (N ∩ L)C. As X is p reduced, N ∩ L centralizes X. The same is true for C and so also for N. Since N is normal in L and Y = ⟨X^{L}⟩, N centralizes Y. Thus N = C and Y is p-reduced.

(b) Put C = C_{L}(Y_{L}). By Frattini, L = L^{f}C. Since O_{p}(L/C) = 1 we conclude O_{p}(L_{f}) ≤ C Hence O_{p}(L_{f}) ≤ C ∩ S = S_{L} and so O_{p}(L_{f}) = S_{L}). Let X = Ω_{1}(Z(S_{L})). Then clearly Y_{L} ≤ X and L_{f} normalizes Y. Put Y = ⟨Y^{L}⟩ = ⟨Y^{C}⟩. Clearly X is p-reduced for S_{L} and so by (b) applied to C, Y is p-reduced for C. Let N be a normal subgroup of L acting unipotently on Y. Since Y_{L} ≤ Y and Y_{L} is p-reduced for L, N ≤ C. As Y is p-reduced for C, N centralizes C and so Y is p-reduced for L. By maximality of Y_{L} we get Y ≤ Y_{L}. But Y_{L} ≤ X ≤ Y and so Y_{L} = X = Y.

(d) Clearly S centralizes Y_{S} and so Y_{S} ≤ Ω_{1}Z(S). Also Ω_{1}Z(S) is p-reduced for S and so Ω_{1}Z(S) ≤ Y_{S}. Thus Ω_{1}Z(S) = Y_{L}. The remaining parts now follow from (b). □

Lemma 4.2 [YL and subnormal subgroups] Let L be of characteristic p and K a subnormal subgroup of L.

(a) [a] Y_{L} ∩ K and [Y_{L}, Ω^{p}(K)] are p-reduced for K

(b) [b] [Y_{L}, Ω^{p}(K)] ≤ Y_{L} ∩ K = Y_{L} ∩ Y_{K} ≤ Y_{K}.

(c) [c] C_{K}(Y_{K}) = C_{K}([Y_{L}, Ω^{p}(K)]) = C_{K}(Y_{L} ∩ Y_{K}) = C_{K}(Y_{L}).

Proof: Note that Ω^{p}(K) = Ω^{p}(KY_{L}) and so [Y_{L}, Ω^{p}(K)] ≤ Y_{L} ∩ Ω^{p}(K) ≤ Y_{L} ∩ K. Let D be the largest normal subgroup of K acting unipotently on [Y_{L}, Ω^{p}(K)]. Since K acts acts unipotently on Y_{L}/[Y_{L}, Ω^{p}(K) we get that D is unipotent on Y_{L}. Since D is subnormal in L and Y_{L} is p-reduced, D centralizes Y_{L}, Y_{L} ∩ K and [Y_{L}, Ω^{p}(K)]. Thus Y_{L} ∩ K and [Y_{L}, Ω^{p}(K)] are p-reduced, D = C_{K}([Y_{L}, Ω^{p}(K). Thus (a)nd (b) hold and for (c) it remains to show that C_{K}(Y_{L}) ≤ C_{K}(Y_{K}).

For this we may assume by induction on the subnormal length that K is normal in L. Then also Y_{K} is normal in L. Let V be a normal subgroup of L contained in Y_{K} which is minimal with respect to C_{K}(V) = C_{K}(Y_{L}). Then Ω_{p}(K/C_{K}(V) = 1 and V is p-reduced for K. Let D be the largest normal subgroup of L acting unipotently on V. Then [K, D] ≤ K ∩ D ≤ C_{L}(V). Put W = C_{V} (D). Since D/Ω_{p}(D) is a p-group, the P × Q-Lemma implies C_{K}(W)/C_{K}(V) is a p-group. Hence C_{K}(W) = C_{K}(V) = C_{K}(Y_{K}). The minimality of V yields V = W. So D centralizes V and V is p-reduced for L. Thus V ≤ Y_{L} and

C_{K}(Y_{L}) ≤ C_{K}(V) = V_{K}(Y_{K}).
□

5 The Kieler Lemma and Point Stabilizers

Lemma 5.1 [Kieler Lemma for modules] Let G be a finite group L a subnormal subgroup of G, p a prime and S ∈ Syl_{p}(G). Let V be a GF (p)G module. Then

C_{L}(C_{V} (S)) = C_{L}(C_{V} (S ∩ L)).

Proof: Without loss L is normal in G and G = LS. Also C_{V} (S) ≤ C_{V} (S ∩ L) and replacing G by C_{G}(C_{V} (S) we may assume that C_{V} (S) ≤ C_{V} (G). For T ∈ L/L ∩ S and v ∈ C_{V} (L ∩ S) define v^{T} = v^{t} for any t ∈ T. Note that this is independ from the choice of t ∈ T ( Also we slightly are abusing notation as v^{T} usually is define as {v^{t} | t ∈ T }. Define

π : C_{V} (S ∩ L) → V, v → ∑_{T ∈L/S∩L} v^{T}

Let v ∈ C_{V} (S ∩ L) and l ∈ L. Then

π(v^{l}) = ∑_{T ∈L/S∩L} v^{T l}

Since T → T l is a bijection of S/S ∩ L we conclude π(v^{l}) = π(v) and so Im π ≤ C_{V} (L). Also if v ∈ C_{V} (L) then π(v) = mv where m = |L/L ∩ S|. L ∩ S is a Sylow p-subgroup of L. Thus p does not devide m and π |_{C_{V} (L)} is one to one. We conclude that

C_{V} (S ∩ L) = ker π ⊕ C_{V} (L).

Let s ∈ S the map T → T^{s} is a bijection of L/S ∩ L and thus

π(v)^{s} = ∑_{T ∈L/S∩L} v^{T s} = ∑_{T ∈L/S∩L} = (v^{s})^{T^{s}} = π(v^{s})

and we conclude that ker π is S-invariant. Suppose that ker π ≠ 0, then also C_{ker π}(S) ≠ 0, but thus contradicts C_{V} (S) ≤ C_{V} (L) and C_{V} (L) ∩ ker pi = 0. Hence ker π = 0 and so C_{V} (S ∩ L) = C_{V} (L). Thus

C_{L}(C_{V} (S ∩ L) = L = C_{L}(C_{V} (S))

and the lemma is proved. □

Proposition 5.2 (Kieler Lemma) [kieler lemma] Let G be a group of local characteritic p, L a subnormal subgroup of G and S ∈ Syl_{p}(G). Then

C_{L}(Ω_{1}Z(S)) = C_{L}(Ω_{1}Z(S ∩ L))

Proof: By induction G/L we may assume that L is normal subgroup of G and G = LS. Put Z = Ω_{1}Z(S) and Y = Ω_{1}Z(S ∩ L). Since S normalizes Ω(L), L ∩ Z ≠ 1. Note that L ∩ Z ≤ Ω_{1}Z(S ∩ L). So S, C_{L}(Z) and C_{L}(Y) are all contained in C_{G}(L ∩ Z) we may assume that G = C_{G}(L ∩ Z). Since G is of local characteristic p, we now get that G is of characteristic p. Let V = Ω_{1}ZΩ(L)Z. Since Ω_{p}(L) ≤ S, Z ≤ C_{G}(Ω_{p}(L)) and so [Z, L] ≤ C_{G}(Ω_{p}(L)) ∩ L. Thus [V, L] ≤ Ω_{1}ZΩ(L)) and V is an elementary abelian normal p-subgroup of G. Note that V = Ω_{1}ZΩ_{p}(L)) ⊕ X for some X ≤ Ω_{1}Z and so by a theorem of Gaschütz, V = Ω_{1}ZΩ_{p}(L) ⊕ A for some normal subgroup A of G. But then [A, G] = 1, A ≤ Ω_{1}Z(G) and so Z = (V ∩ Z)A = (Z ∩ L)A. By assumption Z ∩ L ≤ Z(G) and thus Z = Ω_{1}Z(G) = C_{V} (G). Also C_{V} (S ∩ L) = Y A and so C_{L}(Y) = C_{L}(C_{V} (S ∩ L)). The proof is now completed by 5.1. □

Definition 5.3 [def-point] Let G be a finite group, p a prime and S ∈ Syl_{p}(G). Then

P_{G}(S) := O^{p′}(C_{G}(Ω_{1}Z(S))).

and

Pst(G) = {P_{G}(S) | S ∈ Syl_{p}(G)}.

The group P_{G}(S) is called a point stabilizer of G.

Lemma 5.4 [alternative definition of PG(S)] Let G be a finite group, p a prime, S ∈ Syl_{p}(G) Then

P_{G}(S) = ⟨T ∈ Syl_{p}(G) | Ω_{1}Z(T) = Ω_{1}Z(S)⟩.

Proof: Let T ∈ Syl_{p}(G) with Ω_{1}Z(T) = Ω_{1}Z(S). Then clearly T ≤ P_{G}(S). Conversely if T ∈ Syl_{p}(C_{G}(Ω_{1}Z(S), then [Ω_{1}Z(S), T] = 1, Ω_{1}Z(S)T is a p-group and so Ω_{1}Z(S) ≤ Ω_{1}Z(T) and so Ω_{1}Z(T) = Ω_{1}Z(S). Since P_{G}(S) is just the group generated by the Sylow p-subgroups of C_{G}(Ω_{1}Z(S)), the lemma is proved. □

Lemma 5.5 [sylow subgroups and subnormal subgroups] Let G be a finite group, A_{1} and A_{2} subnormal subgroups of G and p a prime and S ∈ Syl_{p}(G).

(a) [a] A_{i} ∩ S is a Sylow p-subgroup of A_{i}.

(b) [b] For i = 1, 2 let S_{i} be a Sylow p-subgroup of A_{i}. Then ⟨S_{1}, S_{2}⟩ contains a Sylow p-subgroup of ⟨A_{1}, A_{2}⟩.

(c) [c] ⟨A_{1}, A_{2}⟩ ∩ S = ⟨A_{1} ∩ S, A_{2} ∩ S⟩.

Proof: (a) is well known [?].

(b) By induction on |G/A_{1}||G/A_{2}|. In particular we may assume that G = ⟨A_{1}, A_{2}⟩. If A_{i} = G for some i, (b) holds. So we may assume that A_{i} lies in a maximal normal subgroup M_{i} of G. Let {1, 2} = {i, j} and B_{i} = ⟨A_{i}, A_{j} ∩ M_{i}. Then B_{i} ≤ M_{i} and by induction ⟨S_{i}, S_{j} ∩ M_{j} contains a Sylow p-subgroup T_{i} of B_{i}. If A_{1} ≠ B_{1} or A_{2} ≠ B_{2} the by induction ⟨T_{1}, T_{2}⟩ contains a Sylow p-subgroup of G. But

⟨T_{1}, T_{2}⟩ ≤ ⟨⟨S_{1}, S_{2} ∩ M_{1}⟩, ⟨S_{2}, S_{1} ∩ M_{2}⟩⟩ ≤ ⟨S_{1}, S_{2}⟩

and (b) holds. So suppose that A_{1} = B_{1} and A_{2} = B_{2}. Then A_{2} ∩ M_{1} ≤ A_{1} and A_{1} ∩ M_{2} ≤ A_{1}. Thus

A_{1} ∩ M_{2} = A_{1} ∩ A_{2} = A_{2} ∩ M_{1}

.

Snce A_{i}∩M_{j} is normal in A_{i} we conclude that A_{1}∩A_{2} is normal in G⟨A_{1}∩A_{2}. Replacing G by G/A_{1} ∩ A_{2} we may assume that A_{1} ∩ A_{2} = 1.

Since G = ⟨A_{1}, A_{2}⟩ = A_{i}M_{j} we have A_{i} ≅ A_{i}/A_{1} ∩ A_{2} = A_{i}/A_{i} ∩ M_{j} ≅ G/M_{j} and so A_{i} is simple.

Suppose that A_{1} is perfect. Then A_{1} is a component and since A_{1} ≰ A_{2}, we get ( see for example 3.3) [A_{1}, A_{2}] = 1. Clearly (b) holds in this case. So we may assume that for i = 1 and i = 2, A_{i} is an r_{i} group for some prime p. Hence A_{i} ≤ O_{r_{i}}(G). If r_{1} ≠ r_{2} we get again get [A_{1}, A_{2}] = 1. If r_{1} = r_{2} ≠ p, then G is a p′ group and (b) holds. Suppose finally that r_{1} = r_{2} = p. Then A_{i} = S_{i} and G = ⟨S_{1}, ⟩S_{2}. So (b) holds in all cases.

(c) By (a) and (b) both ⟨S ∩ A_{1}, S ∩ A_{2}⟩ and ⟨A_{1}, A_{2}⟩ ∩ S are Sylow p-subgroups of ⟨A_{1}, A_{2}⟩. Since the first of these groups is contained in the second, they are equal. □

Lemma 5.6 [point stabilizers and subnormal subgroups] Let G be a finite group of local characteristic p, A_{1} and A_{2} subnormal subgroups of G, A = ⟨A_{1}, A_{2}⟩, and S ∈ Syl_{p}(G).

(a) [a] O^{p′}(A_{i} ∩ P_{G}(S) = P_{A_{i}}(S ∩ A_{i}).

(b) [b] Let S ∈ Syl_{p}(G). Then

P_{A}(S ∩ A) = ⟨P_{A_{1}}(S ∩ A_{1}), P_{A_{2}}(S ∩ A_{2})⟩.

(c) [c] For i = 1, 2 let P_{i} be a point stabilizer of A_{i}. Then ⟨P_{1}, P_{2}⟩ contains a point stabilizer of A.

Proof: (a) Let L = A_{i}. By the Kieler Lemma 5.2, C_{L}(Ω_{1}Z(S)) = C_{L}(Ω_{1}Z(S ∩ L). Thus

P_{L}(S ∩ L) = O^{p′}(C_{L}(Ω_{1}Z(S))) = O^{p′}(L ∩ P_{G}(S).

(b) Let T be a Sylow p-subgroup of P_{A}(S ∩ A). By (a) T ∩ A_{i} ≤ P_{A_{i}}(S ∩ A_{i}). Hence by 5.5

T = ⟨T ∩ A_{1}, T ∩ A_{2}⟩ ≤ ⟨P_{A_{1}}(S ∩ A_{1}), P_{A_{2}}(S ∩ A_{2})⟩.

So (b) holds.

(c) Let H = ⟨P_{1}, P_{2}⟩. Note that P_{i} contains a Sylow p-subgroup S_{i} of A_{i} and so by 5.5, H contains a Sylow p-subgroup T of A. Then T ∩ A_{i} = S_{i}^{g_{i}} for some g_{i} ∈ H ∩ A_{i} and so P_{A_{i}}(T ∩ A_{i})) = P_{i}^{g_{i}} ≤ H. Hence by (b)

P_{A}(T) = ⟨P_{A_{1}}(T ∩ A_{1}), P_{A_{2}}(T ∩ A_{2}) ≤ H.

So P_{A}(T) is a point stabilizer of A contained in H. □

The following example show that under the assumtions of the previous lemma one might have:

C_{A}(Ω_{1}Z(S ∩ A) ≠ ⟨C_{A_{1}}(Ω_{1}Z(S ∩ A_{1})), C_{A_{2}}(Ω_{1}Z(S ∩ A_{2}))⟩

Indeed let q be a power of p with q > 2, D = Ω_{4}^{+}(q) and V the natural module for D. Then D = D_{1} ∘ D_{2} with D_{i} ≅ Sl_{2}(q). Let G = V ⋊ D and A_{i} = V D_{i}. The A_{i} is normal in G and G = A = A_{1}A_{2}. Let S ∈ Syl_{p}(G). Then it is easy to verify that C_{A_{i}}(Ω_{1}Z(S ∩ A_{i})) = S ∩ A_{i} and so

C_{A_{1}}(Ω_{1}Z(S ∩ A_{1})), C_{A_{2}}(Ω_{1}Z(S ∩ A_{2}))⟩ = S

On the otherhand C_{G}(Ω_{1}Z(S)) is cylic of order q − 1.

References

[BBSM] The Big Book of Small Modules