The P̃! -Theorem

M. Mainardis, University of Udine, U. Meierfrankenfeld, Michigan State University G. Parmeggiani, University of Padova B. Stellmacher, University of Kiel

In this paper we prove a result that is used in the investigation of finite K_{p}-groups of local characteristic p. It is part of an attempt to revise a major part of the classification of the finite simple groups. An overview of this revision can be found in [MSS].

More precisely, the P̃!-Theorem proved in this paper together with the P!-Theorem in [PPS] show that under certain hypotheses there exist parabolic parabolic subgroups P and P̃ (containing a common Sylow psubgroup) in a K_{p}-group H of local characteristic p that behave like the two minimal parabolic subgroups of a group of Lie type in characteristic p that correspond to the end node and its neighbor in the Dynkin diagram. They also establish that the remaining part of a hypothetical Dynkin diagram for H can be found in a single maximal p-local subgroup, which is called C̃ further below.

Moreover, as it is outlined in [MSS, 2.4.9], these two theorems allow to restrict the structure of C̃ and consequently that of the missing part of the diagram.

To get started we need some definitions. Let H be a finite group and p a fixed prime. Then H is of characteristic p if

C_{H}(O_{p}(H)) ≤ O_{p}(H),

and H is of local characteristic p if every p-local subgroup of H is of characteristic p. Moreover, H is a K_{p}-group if the simple sections of plocal subgroups are ”known” simple groups^{1}.

1Which means, they are groups of prime order, groups of Lie type, alternating groups or one of the 26 sporadic groups.

For any p-local subgroup L ≤ H let Y_{L} be the largest elementary abelian normal p-subgroup of L satisfying

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

For elementary properties of this subgroup see [MSS] and [PPS].

By L_{H}(X) we denote the set of subgroups L ≤ H containing a given subgroup X and satisfying C_{H}(O_{p}(L)) ≤ O_{p}(L), and by M_{H}(X) the set of maximal p-local subgroups containing X.

We fix S ∈ Syl_{p}(H) and C̃ ∈ M_{H}(N_{H}(Ω_{1}Z(S))) and put

Q := O_{p}(C̃) and X^{∘} := ⟨Q^{h} | h ∈ H with Q^{h} ≤ X⟩ for X ≤ H.

By P_{H}(S) we denote the set of subgroups P ≤ H such that P ≰ N_{H}(S), O_{p}(P) ≠ 1 and S is contained in a unique maximal subgroup of P. The elements of P_{H}(S) are called the minimal parabolic subgroups of H containing S.

Let K ≅ SL_{n}(p^{k}) or Sp_{2n}(p^{k}) and V be an irreducible GF (p)K-module. Set F = End_{K}(V). Then V is a natural SL_{n}(p^{k})-module for K if K ≅ SL_{n}(p^{k}) and dim_{F} V = n, and V is a natural Sp_{2n}(p^{k})-module for K, if K ≅ Sp_{2n}(p^{k}), dim_{F} V = 2n, and K leaves invariant a non-degenerate symplectic form s on the F -space V.

Note that a natural module for SL_{2}(p^{k}) is unique up to isomorphism. For SL_{n}(p^{k}), n > 2, and Sp_{4}(2^{k}) there are two isomorphism classes of natural modules. The second class can be obtained from the first one by applying a graph automorphism.

By a k-dimensional subspace of V we mean an k-dimensional F - subspace of V.

For K ≅ Sp_{2n}(p^{k}) a k-dimensional subspace U of V is singular if the symplectic form s restricted to U is zero. Since s is unique (up to scalar multiplication by elements of F), the definition of a singular subspace does not depend on the choice of s.

For K ≅ SL_{n}(p^{k}) any subspace of V is called singular.

In this paper we investigate finite K_{p}-groups of local characteristic p satisfying in addition:

C_{H}(x) ≤ C̃ for every 1 ≠ x ∈ C_{H}(Q) (Q-uniqueness).

We prove:

P̃ ! -Theorem. Let H be a LK_{p}-group of local characteristic p that satisfies Q-uniqueness. Suppose that there exists P ∈ P_{H}(S) such that P ≰ C̃ and Y_{M} ≤ Q for every M ∈ M_{H}(P). Then one of the following holds:

(a) There exists at most one P̃ ∈ P_{H}(S) such that P̃ ≰ N_{H}(P^{∘}) and ⟨P, P̃⟩ ∈ L_{H}(P). Moreover, if such P̃ exists and M_{1} := ⟨P, P̃⟩^{∘}C_{S}(Y_{P}), then

(a_{1}) M_{1}/C_{M_{1}}(Y_{M_{1}}) ≅ SL_{3}(p^{n}), Sp_{4}(p^{n}), or Sp_{4}(2)′ (and p = 2), and

(a_{2}) [Y_{M_{1}}, M_{1}] is the corresponding natural module for M_{1}/C_{M_{1}}(Y_{M_{1}}).

(b) There exist at least two P̃_{1}, P̃_{2} ∈ P_{H}(S) such that P̃_{i} ≰ N_{H}(P^{∘}) and ⟨P, P̃_{i}⟩ ∈ L_{H}(P), i = 1, 2. Moreover, for any such P̃_{i} and M_{i} := ⟨P, P̃_{i}⟩^{∘}C_{S}(Y_{P}), i = 1, 2 :

(b_{1}) p = 3 or 5 and O^{p′}(M_{1} ∩ M_{2}) = P,

(b_{2}) M_{i}/O_{p}(M_{i}) ≅ SL_{3}(p),

(b_{3}) O_{p}(M_{i})/Z(O_{p}(M_{i})) and Z(O_{p}(M_{i})) are natural SL_{3}(p)-modules for M_{i}/O_{p}(M_{i}) dual to each other.

The P̃! -Theorem is a corollary of the following more general result on amalgams. By 3Sp_{4}(2)′ we denote a non-split central extension of a group of order 3 by Sp_{4}(2)′, and by 3Sp_{4}(2) a group that has 3Sp_{4}(2)′ as a subgroup of index 2 and Sp_{4}(2) as a factor group.

Theorem 1. Let G be group generated be two finite subgroups M_{1} and M_{2}. Set B := M_{1} ∩ M_{2} and \overline{M}_{i} := M_{i}/C_{M_{i}}(Y_{M_{i}}), and suppose that for i = 1, 2 the following hold:

(1) Syl_{p}(M_{1})∩Syl_{p}(M_{2}) = Syl_{p}(B), and M_{1} and M_{2} are of characteristic p.

(2) No non-trivial normal subgroup of G is contained in B.

(3) \overline{M}_{i} ≅ SL_{3}(q_{i}), Sp_{4}(q_{i}), q_{i} = p^{n_{i}}, or Sp_{4}(2)′ (and q_{i} = p = 2).

(4) [Y_{M_{i}}, O^{p}(M_{i})] is a natural module for \overline{M}_{i}, and Z(M_{i}) = 1.

(5) There exists a 2-dimensional singular subspace W in [Y_{M_{i}}, O^{p}(M_{i})] such that O^{p′}(N_{\overline{M}_{i}}(W)) ≤ \overline{B}.

(6) C_{M_{i}}(Y_{M_{i}}) = O_{p}(M_{i}), or q_{i} = 2 and

M_{i}/O_{2}(M_{i}) ≅ 3Sp_{4}(2) or 3Sp_{4}(2)′.

Then one of the following holds for i = 1, 2:

(a) p = 2, Y_{M_{i}} = O_{2}(M_{i}), M_{i}/O_{2}(M_{i}) ≅ Sp_{4}(2)′ or Sp_{4}(2), and |Y_{M_{i}}| = 2^{4} or 2^{5}.

(b) q := q_{1} = q_{2}, p = 3 or q = 5, M_{i}/O_{p}(M_{i}) ≅ SL_{3}(q), and O_{p}(M_{i})/Y_{M_{i}} and Y_{M_{i}} are natural SL_{3}(q)-modules for M_{i}/O_{p}(M_{i}) dual to each other.

Since Theorem 1 does not depend on the hypothesis of the P̃!-Theorem, it may also be useful in more general situations; for example, when the condition Y_{M} ≤ Q is not satisfied.

We also want to remark that Theorem 1 is in the same vein as the (much more general) main result of [ST]. Unfortunately, the hypotheses there are not compatible with the situation here, our Hypothesis (5) being the reason.

1 Elementary Properties

Throughout this section H is a finite group of local characteristic p satisfying Q-uniqueness (with the notation given in the introduction), and X is an arbitrary finite group.

1.1 Let X be of characteristic p, S ∈ Syl_{p}(X) and P ∈ P_{X}(S). Then the following hold:

(a) Ω_{1}(Z(S)) ≤ Y_{X}.

(b) X = N_{X}(S)⟨P_{X}(S)⟩.

(c) For every normal subgroup N of P either O^{p}(P) ≤ N or S ∩ N ≤ O_{p}(P).

(d) For every normal subgroup T of S either O^{p}(P) = [O^{p}(P), T] or T ≤ O_{p}(P).

Proof. See (1.2) (c) and (1.3) (a), (b), (c) of [PPS]. □

1.2 Let X be of characteristic p, S ∈ Syl_{p}(X), Y_{X} ≤ N ⊴ X, and V := ⟨Ω_{1}(Z(S))^{X}⟩. Then the following hold:

(a) Y_{X} ≤ Y_{N}.

(b) Y_{X} = Y_{N} if C_{S}(Y_{X}) ≤ N.

(c) V = C_{V} (X)[V, O^{p}(X)] and V ≤ Y_{X}.

Proof. Recall that Y_{N} is the unique maximal elementary abelian normal p-subgroup of N satisfying O_{p}(N/C_{N}(Y_{N})) = 1. Hence, Y_{N} is characteristic in N and thus normal in X.

Set X̃ := X/C_{X}(Y_{X}). Then 1 = O_{p}(Ñ) ≅ O_{p}(N/C_{N}(Y_{X})), so Y_{X} ≤ Y_{N}. This is (a).

Assume now that C_{S}(Y_{X}) ≤ N, and let D ≤ S such that

DC_{X}(Y_{N})/C_{X}(Y_{N}) = O_{p}(X/C_{X}(Y_{N})).

Since Y_{X} ≤ Y_{N} we have C_{X}(Y_{N}) ≤ C_{X}(Y_{X}). Thus DC_{X}(Y_{X})/C_{X}(Y_{X}) ≤ O_{p}(X/C_{X}(Y_{X})) = 1 and D ≤ C_{S}(Y_{X}) ≤ N. It follows that

DC_{X}(Y_{N})/C_{X}(Y_{N}) ≅ D/C_{D}(Y_{N}) ≅ DC_{N}(Y_{N})/C_{N}(Y_{N}) ≤ O_{p}(N/C_{N}(Y_{N})),

and thus D ≤ C_{S}(Y_{N}) and O_{p}(X/C_{X}(Y_{N})) = 1. Hence Y_{N} ≤ Y_{X}, and (a) implies (b).

A proof for the first part of (c) can be found in [CD], Lemma 2.5, the second part follows from 1.1 (a). □

1.3 The following hold:

(a) Q = Q^{h} and h ∈ C̃ for every h ∈ H with Q^{h} ≤ C̃.

(b) If T is a p-subgroup containing Q, then N_{H}(T) ≤ C̃.

Proof. For (a) see (1.6) of [PPS]; (b) is a direct consequence of (a).

□

1.4 Let L ∈ L_{H}(S). Then [C_{L}(Y_{L}), L^{∘}] ≤ O_{p}(L).

Proof. See 2.4.2 (dc) in [MSS]. □

1.5 Let L ∈ L_{H}(S), P ∈ P_{L}(S) and L_{0} := L^{∘}C_{S}(Y_{L}). Then one of the following holds:

(a) P ≤ L_{0}S,

(b) P ≤ N_{L}(S ∩ L_{0}) ∩ N_{L}(L_{0}) ∩ C̃,

(c) [O^{p}(P), L^{∘}] ≤ O_{p}(L) and P ≤ C̃.

Proof. If O^{p}(P) ≤ C_{L}(Y_{L}), then 1.4 and 1.1 (a) give (c). Thus, we may assume that O^{p}(P) ≰ C_{L}(Y_{L}). Then 1.1 (c) yields C_{S}(Y_{L}) ≤ O_{p}(P), in particular C_{S}(Y_{L}) = C_{P} (Y_{L}) ∩ O_{p}(P). Hence, C_{S}(Y_{L}) is normal in P, and P ≤ N_{L}(L_{0}). Now S ∩ L_{0} is normal in S, and 1.1 (d) shows that either S ∩ L_{0} ≤ O_{p}(P) or O^{p}(P) = [O^{p}(P), S ∩ L_{0}].

In the first case S ∩ L_{0} = O_{p}(P) ∩ L_{0}, so S ∩ L_{0} is normal in P. As Q ≤ S ∩ L_{0}, 1.3 (b) implies P ≤ C̃, and (b) holds.

In the second case O^{p}(P) ≤ L_{0} since P ≤ N_{L}(L_{0}), and (a) holds. □

1.6 Let V be an irreducible GF (p)X-module and F := End_{X}(V). Then the direct sum V ⊕ V contains exactly |F | + 1 GF (p)X-submodules isomorphic to V.

Proof. This follows from Theorem 3.5.6 in [Go]. □

Let L = SL_{2}(q), q = p^{n}, and V be a natural GF (p)L-module. Set F := End_{L}(V). Since L ≅ Sp_{2}(q) there exists an L-invariant non-degenerated symplectic form

s : V × V → F.

Let {v, w} be a basis of the F -space V such that s(v, w) = 1, and set

Ω_{0} := {(μ(v), v) | μ ∈ F } and Ω_{1} = {(μ(v), w) | μ ∈ F^{♯}}.

Note that s(μ(v), w) = μ, so s(Ω_{1}) ∪ {0} = F. Note further that Ω := Ω_{0} ∪ Ω_{1} ∪ {(v, 0), (0, 0)} is a set of representatives for the L-orbits on V × V.

We consider F as a trivial GF (p)L-module.

1.7 Let C be any trivial GF (p)L-module and α : V ×V → C be L-invariant and GF (p)-bilinear. Then there exists a GF (p)L-module homomorphism β : F → C such that β(s(x, y)) = α(x, y) for all x, y ∈ V.

Proof. For every μ ∈ F the pair (μ(v), v + w) is in the same L-orbit as (μ(v), w). It follows that

α(μ(v), w) = α(μ(v), v + w) = α(μ(v), v) + α(μ(v), w),

and α(μ(v), v) = 0. Moreover, the linearity of α gives α(v, 0) = α(0, 0) = 0. We define

β : F = s(Ω_{1}) ∪ {0} → C with s(μ(v), w) ↦ α(μ(v), w) and 0 ↦ 0,

so βs|_{Ω} = α|_{Ω}. As s and α are both L-invariant and Ω is a set of representatives for the L-orbits of V × V, this shows that βs = α. Moreover, it is easy to check that β is a GF (p)L-module homomorphism from F into C. □

2 Properties of Sp_{4}(q) and SL_{3}(q)

In this section G ∈ {SL_{3}(q), Sp_{4}(q)}, where q is a power of the prime p, and V is a natural GF (p)G-module. We fix a 2-dimensional singular subspace W of V and we set P := O^{p′}(N_{G}(W)).

The next two Lemmata give properties of G we assume the reader to be familiar with:

2.1 Let G = SL_{3}(q) and S ∈ Syl_{p}(P). Then |P | = q^{2}|SL_{2}(q)|, and the following hold:

(a) G is 2-transitive on the 1-dimensional and on the 2-dimensional subspaces of V.

(b) [V, O_{p}(P)] = W and C_{G}(W) = O_{p}(P).

(c) P/O_{p}(P) ≅ SL_{2}(q) and Syl_{p}(P) ⊆ Syl_{p}(G).

(d) W and O_{p}(P) are natural SL_{2}(q)-modules for P/O_{p}(P).

(e) |Z(S)| = |C_{V} (S)| = q and Z(S) = C_{G}(W) ∩ C_{G}(V/C_{V} (S)).

(f) G is generated by three conjugates of Z(S).

(g) If A ≤ S such that |[V, A]| = q or |V/C_{V} (A)| = q, then A is conjugate to a subgroup of Z(S).

(h) N_{G}(W) is the unique maximal subgroup of G containing P. □

2.2 Let G = Sp_{4}(q), S ∈ Syl_{p}(P) and A be a minimal normal subgroup of N_{G}(W). Then |P | = q^{3}|SL_{2}(q)|, and the following hold:

(a) N_{G}(W) is a maximal subgroup of G.

(b) P/O_{p}(P) ≅ SL_{2}(q), Syl_{p}(P) ⊆ Syl_{p}(G), and O_{p}(P) is elementary abelian.

(c) [W, O_{p}(P)] = 0, and [v, O_{p}(P)] = W for every v ∈ V \ W.

(d) W and V/W are natural SL_{2}(q)-modules for P/O_{p}(P).

(e) Either (e_{1}) A = O_{p}(P) and p ≠ 2, or (e_{2}) p = 2, |A| = q and O_{2}(P)/A is a natural SL_{2}(q)-module for

P/O_{2}(P), or (e_{3}) q = 2, |A| = 4 and [v, A] = W for every v ∈ V \ W.

(f) There exists g ∈ G with W ∩ W^{g} = 0. Moreover,

X ≰ N_{G}(W^{g}) and G = ⟨P^{g}, X⟩

for any such g and for any 1 ≠ X ≤ O_{p}(P).

(g) A ≰ O_{p}(C_{G}(x)) for every x ∈ W^{♯}.

(h) N_{G}(W) is the unique maximal subgroup of G containing P. □ 2.3 Let G = SL_{3}(q) and g ∈ G \ N_{G}(W). Then O_{p}(P) ∩ O_{p}(P^{g}) = 1, and there exists S ∈ Syl_{p}(P) such that O_{p}(P) ∩ P^{g} = Z(S); in particular |O_{p}(P)/O_{p}(P) ∩ P^{g}| = q.

Proof. Let V_{0} = W ∩ W^{g}. Then V_{0} is 1-dimensional, and by 2.1 (a) and (e) there exists S ∈ Syl_{p}(P) such that V_{0} = C_{V} (S). Hence 2.1 (b) and (e) give

Z(S) ≤ O_{p}(P) ∩ C_{G}(V/V_{0}) ≤ O_{p}(P) ∩ N_{G}(W^{g}),

so Z(S) ≤ O_{p}(P)∩P^{g}. On the other hand O_{p}(P)∩O_{p}(P^{g}) ≤ C_{G}(W W^{g}) = 1, so by 2.1 (c) and (e) Z(S) = O_{p}(P) ∩ P^{g}. □ 2.4 Let G = Sp_{4}(q) and B ≤ O_{p}(P) be such that |V/C_{V} (B)| = q or |[V, B]| = q. Then the following hold:

(a) [V, A] = W for every non-trivial normal subgroup A of P in O_{p}(P).

(b) |V/C_{V} (B)| = |[V, B]| = q and |B| ≤ q.

Proof. By 2.2 (d) P acts irreducible on W and by 2.2 (c) [V, O_{p}(P)] = W. This gives (a).

Note that [V, B] is perpendicular to C_{V} (B) with respect to the symplectic form on V. Hence, the dimension formula gives

dim V = dim C_{V} (B) + dim [V, B]

and the first part of (b). Then |B/C_{B}(v)| ≤ q for v ∈ V \ C_{V} (B), and V = C_{V} (B) + V_{0}, where V_{0} is the subspace generated by v. This gives C_{B}(v) = 1 and |B| ≤ q. □

The next result follows from Theorem A in [CD]:

2.5 Let Y be a faithful GF (p)G-module and 1 ≠ A an elementary abelian p-subgroup of G. Suppose that G = SL_{3}(q) and |Y/C_{Y} (A)| ≤ |A|. Then [Y, G]/C_{[Y,G]}(G) is a natural module or the direct sum of two isomorphic natural modules for G. □

2.6 Let M be a finite group of characteristic p with subgroups Y ≤ D ≤ M such that D is a normal p-subgroup of M and V_{1} := Φ(D) ≤ Ω_{1}Z(O_{p}(M)). Suppose that for Ṽ = D/Z(D) and P_{1} := O^{p′}(N_{M}(Y ∩ V_{1})):

(a) M/O_{p}(M) = SL_{3}(q), and V_{1} is a natural module for M/O_{p}(M).

(b) D = ⟨Y^{M}⟩, Y ≰ Z(D) and C_{D}(y) = C_{D}(Y) for all y ∈ Y \ V_{1}.

(c) Y ∩ V_{1} is 2-dimensional subspace of V_{1}, and [Y, P_{1}] ≤ Y ∩ V_{1}.

(d) |Ṽ/CṼ (T)| ≤ |T/O_{p}(M)| where T := O_{p}(C_{M}(z)), 1 ≠ z ∈ V_{1}.

Then Ṽ is a natural SL_{3}(q)-module dual to V_{1}.

Proof. Set \overline{M} := M/O_{p}(M), so \overline{M} = SL_{3}(q) and by 2.1 (c)

\overline{P}_{1}/O_{p}(\overline{P}_{1}) ≅ SL_{2}(q) and Syl_{p}(P_{1}) ⊆ Syl_{p}(M).

By (b) and (c) [D, O_{p}(M)] ≤ V_{1} and Ṽ = [Ṽ, \overline{M}]Ỹ; in particular Ṽ is a GF (p)\overline{M}-module. As Ỹ is centralized by a Sylow p-subgroup of \overline{M}, Gaschütz’ Theorem implies that Ṽ = [Ṽ, \overline{M}]CṼ (\overline{M}).

Let V_{1} ≤ U ≤ D be such that U is M-invariant and Ũ ≠ 1, and pick y ∈ Y \V_{1}. By (c) [y, U] is a P_{1}-submodule of V_{1}, so by (a) either [y, U] = 1 or [y, U] = Y ∩ V_{1}. In the first case (b) yields U ≤ C_{D}(Y) and then U ≤ Z(D), which contradicts Ũ ≠ 1. Hence, we have

(∗)  [y, U] = Y ∩ V_{1};

in particular CṼ (\overline{M}) = 1 and Ṽ = [Ṽ, \overline{M}].

Assume now, in addition, that Ũ is an irreducible \overline{M}-module. We apply 2.5. Then Ũ is a natural SL_{3}(q)-module for \overline{M}. Moreover, since U/C_{U}(y) and [y, U] = Y ∩ V_{1} are isomorphic P_{1}-modules, the module Ũ is dual to V_{1}. It remains to prove that Ũ = D̃.

Assume that D̃ ≠ Ũ. Then, again by 2.5, Ṽ is the direct sum of two natural modules isomorphic to Ũ. Thus Ṽ/CṼ (P_{1}) is the direct sum of two natural modules for P_{1}/O_{p}(P_{1}). By 1.6 there are exactly q+1 \overline{M}-submodules Ũ_{0}, …, Ũ_{q} isomorphic to Ũ in Ṽ and also q + 1 irreducible P_{1}-submodules in Ṽ/CṼ (P_{1}), so Ũ_{i}CṼ (P_{1})/CṼ (P_{1}), i = 0, …, q, are these P_{1}-submodules.

Let D_{y} := C_{D}(y). As seen above D/D_{y} is a natural P_{1}-module. Hence, also D̃_{y} involves a natural P_{1}-module. Moreover, an application of the Three-Subgroups Lemma shows that CṼ (P_{1}) ≤ D̃_{y}. Thus, there exists k ∈ {0, …, q} such that D̃_{y} = (Ũ_{k} ∩ D̃_{y})CṼ (P_{1}). Then

Ũ_{k} = (Ũ_{k} ∩ D̃_{y})CŨ_{k}(P_{1}) ≤ D̃_{y},

which contradicts (∗). □

2.7 Let L be a finite group, p an odd prime, and D a normal p-subgroup of L such that Z(D) = Ω_{1}(Z(D)) =: V, and let A be the set of all normal subgroups A of L satisfying A ≤ D, |A| = q^{3} and C_{D}(A)′ = A. Suppose that

(i) L/O_{p}(L) ≅ SL_{2}(q), q = p^{n},

(ii) V is a natural module and D/D′ is the direct sum of two natural modules for L/O_{p}(L),

(iii) |A| ≥ 2, [D′, L] ≤ V and |D′| = q^{4}.

Then one of the following holds:

(a) p ≠ 3 and |A| = 3.

(b) p = 3 and |A| = q.

Proof. Set R := D′ and \overline{D} := D/R. Note that by (ii) any non-trivial proper L-invariant subgroup of \overline{D} has order q^{2}. Also

(1)  A = C_{D}(A)′ ≤ R for everyA ∈ A.

By (ii) [R, D, D] ≤ [V, D] = 1, so the Three Subgroups Lemma gives [R, R] = 1. Thus

(2)  R is abelian.

Let 1 ≠ x ∈ A \ V. By (1) and (iii) L normalizes [x, D] and thus [x, D] = V and |D/C_{D}(x)| = q^{2}. On the other hand, by (1) and (2)

R ≤ C_{D}(A) ≤ C_{D}(x) and C_{D}(A) ≠ R

since A = C_{D}(A)′. Now (ii) implies that

C_{D}(A) = C_{D}(x) for every x ∈ A \ V.

According to (iii) there exists B ∈ A such that A ≠ B. As A = C_{D}(x)′ for every x ∈ A \ V and B = C_{D}(x)′ for every x ∈ B \ V, we get

(3)  V = A ∩ B, R = AB and \overline{D} = \overline{C_{D}(A)} × \overline{C_{D}(B)}.

Moreover \overline{C_{D}(A)} and \overline{C_{D}(B)} are natural L/O_{p}(L)-modules.

Let K := End_{L}(V) and R̃ := R/V, so R̃ = Ã × B̃. By (ii) K ≅ GF (q). From now on we will view V as a vector space over K; in particular addition replaces multiplication. Moreover, we write kv rather than k(v) for k ∈ K and v ∈ V.

There exist GF (p)L-isomorphisms

ϕ_{A} : V → \overline{C_{D}(A)} and ϕ_{B} : V → \overline{C_{D}(B)}.

Let r ∈ R and x, y ∈ D. Then [\overline{x}, \overline{y}] := { \atop [x, y]} and [\overline{x}, r̃] := [x, r] are welldefined by (2). Let z ∈ A. Since [z̃, L] = 1, the map

μ_{A}^{∗}(z̃) : V → V with v ↦ [ϕ_{B}(v), z̃]

is L-invariant. Thus μ_{A}^{∗}(z̃) ∈ K.

If μ_{A}^{∗}(z̃) = 0, then [z, C_{D}(B)] = 1 and thus by (3) z ≤ Z(V), so z̃ = 1. Since

μ_{A}^{∗} : Ã → K, z̃ ↦ μ_{A}^{∗}(z̃)

is a group homomorphism, we conclude that it is injective. Hence, as both Ã and K have order q, μ_{A}^{∗} is an isomorphism. Let μ_{A} be its inverse. Similarly define μ_{B}.

Let k, h ∈ K and v, w ∈ V. Then

(4)  kv = [ϕ_{B}(v), μ_{A}(k)] = [ϕ_{A}(v), μ_{B}(k)]

and

(5)  [ϕ_{A}(v)ϕ_{B}(w), μ_{A}(k)μ_{B}(h)] = kw + hv,

(recall that we write V additively).

For every z ∈ R \ A there exist k ∈ K and h ∈ K^{♯} such that z̃ = μ_{A}(k)μ_{B}(h). Let z_{(h,k)} ∈ R be such that z̃_{(h,k)} = μ_{A}(−kh^{−1})μ_{B}(−1). Then (2) and (5) yield

C_{D}(z) = C_{D}(z_{(h,k)}).

Hence

{C_{D}(z) | z ∈ R \ V } = {C_{D}(A)} ∪ {C_{D}(μ_{A}(t)μ_{B}(−1)) | t ∈ K}.

For t ∈ K set D_{t} := C_{D}(μ_{A}(t)μ_{B}(−1)) and B_{t} := Z(D_{t}). From (5) we obtain:

(6) \overline{D}_{t} = \{ϕ_{A}(tv)ϕ_{B}(v) | v ∈ V \} and B̃_{t} = \{μ_{A}(tk)μ_{B}(−k) | k ∈ K\}.

In the following we determine for which t ∈ K actually B_{t} ∈ A. As B_{0} = B ∈ A we can assume that t ≠ 0.

We investigate the L-invariant GF (p)-bilinear form κ on \overline{D} induced by the commutator mapping

κ : \overline{D} × \overline{D} → R̃ with (\overline{x}, \overline{y}) ↦ [\overline{x}, \overline{y}] for all x, y ∈ D.

As R̃ = Ã × B̃, the projection mappings

π_{A} : R̃ → Ã and π_{B} : R̃ → B̃

give rise to L-invariant GF (p)-bilinear forms κπ_{A} and κπ_{B} satisfying

[\overline{x}, \overline{y}] = [\overline{x}, \overline{y}]π_{A}[\overline{x}, \overline{y}]π_{B},

so κπ_{A}μ_{A}^{∗} and κπ_{B}μ_{B}^{∗} are GF (p)-bilinear forms with values in K.

We now restrict κ to \overline{C_{D}(A)} × \overline{C_{D}(A)}, \overline{C_{D}(B)} × \overline{C_{D}(B)} and \overline{C_{D}(A)} × \overline{C_{D}(B)}, respectively. Let s be an L-invariant non-degenerate symplectic form on V. According to 1.7 there exist GF (p)L-module homomorphisms

λ_{A}, λ_{B}, ρ_{A}, ρ_{B} : K → K

such that for all v, w ∈ V

(7)  [ϕ_{A}(v), ϕ_{A}(w)] = μ_{A}(λ_{A}(s(v, w))),
(8)  [ϕ_{B}(v), ϕ_{B}(w)] = μ_{B}(λ_{B}(s(v, w))),
(9)  [ϕ_{A}(v), ϕ_{B}(w)] = [ϕ_{A}(v), ϕ_{B}(w)]π_{A}[ϕ_{A}(v), ϕ_{B}(w)]π_{B}
= μ_{A}(ρ_{A}(s(v, w)))μ_{B}(ρ_{B}(s(v, w))).

It follows that

(10)  \begin{aligned}[ϕ_{B}(v), ϕ_{A}(w)] & = [ϕ_{A}(w), ϕ_{B}(v)]^{−1} \\ & = (μ_{A}(ρ_{A}(s(w, v)))μ_{B}(ρ_{B}(s(w, v))))^{−1} \\ & = (μ_{A}(ρ_{A}(s(w, v))))^{−1}(μ_{B}(ρ_{B}(s(w, v))))^{−1} \\ & = μ_{A}(ρ_{A}(−s(w, v)))μ_{B}(ρ_{B}(−s(w, v))) \\ & = μ_{A}(ρ_{A}(s(v, w)))μ_{B}(ρ_{B}(s(v, w))). \end{aligned}

Let x, y, z ∈ D. The Jacobi identity

[x, y^{−1}, z]^{y}[y, z^{−1}, x]^{z}[z, x^{−1}, y]^{x} = 1

yields

(11)  [\overline{y}, \overline{x}, \overline{z}] + [\overline{z}, \overline{y}, \overline{x}] + [\overline{x}, \overline{z}, \overline{y}] = 0

since V ≤ Z(D) and R̃ ≤ Z(D̃).

Let v, w ∈ V and k ∈ K^{♯}. We will compute all three triple commutators for \overline{x} = ϕ_{A}(v), \overline{y} = ϕ_{A}(kw) and \overline{z} = ϕ_{B}(w). From (9) we get

\begin{aligned}[ϕ_{A}(kw), ϕ_{B}(w)] & = μ_{A}(ρ_{A}(s(kw, w)))μ_{B}(ρ_{B}(s(kw, w))) \\ & = μ_{A}(ρ_{A}(0))μ_{B}(ρ_{B}(0)) = 1. \end{aligned}

Thus, (5), (7), (9) and (10) give

[ϕ_{A}(v), ϕ_{A}(kw), ϕ_{B}(w)] = [μ_{A}(λ_{A}(s(v, kw))), ϕ_{B}(w)]
= −λ_{A}(s(v, kw))w = λ_{A}(s(kw, v))w,
[ϕ_{B}(w), ϕ_{A}(v), ϕ_{A}(kw)] = [μ_{A}(ρ_{A}(s(w, v)))μ_{B}(ρ_{B}(s(w, v))), ϕ_{A}(kw)]
= −ρ_{B}(s(w, v))kw,
[ϕ_{A}(kw), ϕ_{B}(w), ϕ_{A}(v)] = 0.

Now (11) yields:

(12) λ_{A}(s(kw, v))w = ρ_{B}(s(w, v))kw for every v, w ∈ V and k ∈ K.

Choosing v, w ∈ V such that s(w, v) = k_{2} one gets from (12)

λ_{A}(k_{1}k_{2}) = ρ_{B}(k_{2})k_{1} for all k_{1}, k_{2} ∈ K.

A similar argument (with the roles of A and B reversed) gives

λ_{B}(k_{1}k_{2}) = ρ_{A}(k_{2})k_{1} for all k_{1}, k_{2} ∈ K.

In particular, with ℓ_{A} := λ_{A}(1) and ℓ_{B} := λ_{B}(1) the cases k_{1} = 1 or k_{2} = 1, respectively, yield:

(13) ρ_{B}(k) = λ_{A}(k) = ℓ_{A}k and ρ_{A}(k) = λ_{B}(k) = ℓ_{B}k for all k ∈ K.

We now determine for which t the subgroup B_{t} is in A. Let v, w ∈ V and t ∈ K^{#}. Put k := s(v, w) and recall that μ_{A}, μ_{B} are homomorphisms from the additive group of K into the multiplicative group Ã or B̃, respectively. It follows from (7) – (10) and (13) that

[ϕ_{A}(tv)ϕ_{B}(v), ϕ_{A}(tw)ϕ_{B}(w)] =
[ϕ_{A}(tv), ϕ_{A}(tw)][ϕ_{A}(tv), ϕ_{B}(w)][ϕ_{B}(v), ϕ_{A}(tw)][ϕ_{B}(v), ϕ_{B}(w)] =

μ_{A}(λ_{A}(s(tv, tw)))μ_{A}(ρ_{A}(s(tv, w)))μ_{B}(ρ_{B}(s(tv, w)))μ_{A}(ρ_{A}((s(v, tw)))

μ_{B}(ρ_{B}(s(v, tw)))μ_{B}(λ_{B}(s(v, w)))
= μ_{A}(λ_{A}(t^{2}k) + 2ρ_{A}(tk))μ_{B}(2ρ_{B}(tk) + λ_{B}(k))
= μ_{A}(k(t^{2}ℓ_{A} + 2tℓ_{B}))μ_{B}(k(2tℓ_{A} + ℓ_{B})).

For ℓ := \frac{ℓ_{B}}{ℓ_{A}} we get

(14)  \begin{matrix}|\{[ϕ_{A}(tv)ϕ_{B}(v), ϕ_{A}(tw)ϕ_{B}(w)] | v, w ∈ V \}| = q ⟺ \\ t ∉ \{−2ℓ, −\frac{1}{2}ℓ\}. \end{matrix}

By (6)

(15)  D′_{t} ≤ B_{t} ⟺ t^{2}ℓ_{A} + 2tℓ_{B} = (−t)(ℓ_{B} + 2tℓ_{A}).

The equation on the right hand is equivalent to

(16)  3t = −3ℓ,

since we are assuming that t ≠ 0. Hence, according to (14)

D′_{t} = B_{t} ⟺ 3t = −3ℓ and t ∉ \{−2ℓ, −\frac{1}{2}ℓ\}.

If p ≠ 3, then (16) has a unique solution, namely t = −ℓ, so A = {A, B, B_{−ℓ}} and (a) holds.

Assume now that p = 3. Then every t ∈ K^{♯} satisfies (16), and every t ≠ ℓ satisfies t ∉ \{−2ℓ, −\frac{1}{2}ℓ\}. Hence, there are q − 2 elements in A different from A and B, and (b) holds. □

2.8 Let L be as in 2.7. Suppose in addition that L is a subgroup of the finite group M, and

(i) M/O_{p}(M) ≅ SL_{3}(q)

(ii) Z(O_{p}(M)) and O_{p}(M)/Z(O_{p}(M)) are natural SL_{3}(q)-modules dual to each other.

Then q = 5 or p = 3.

Proof. We may assume that p ≠ 3. Then by 2.7 |A| = 3. A comparison of orders shows that L contains a Sylow p-subgroup of M and D = O_{p}(L) (D as in 2.7). Hence there exists L ≤ R ≤ M such that L = O^{p′}(R) and R/O_{p}(M) is a maximal parabolic subgroup of M/O_{p}(M).

Let Y = Z(O_{p}(M)). By our hypothesis O_{p}(M) = C_{D}(Y) and O_{p}(M)′ = Y, so Y ∈ A. Let A and B be the other two elements of A, and let R_{0} be the unique subgroup of index 2 in R with L ≤ R_{0}, so K := R_{0}/L is cyclic of order \frac{q−1}{2}. Since R normalizes Y and acts on A, R_{0} normalizes A and B.

Let D̃ := D/Z(D). Then Ỹ, Ã, and B̃ are three different GF (p)Ksubmodules of order q in D̃′, so the GF (p)K-modules W_{1} := Ỹ and W_{2} := D̃′/Ỹ are isomorphic.

Let F_{i} = End_{K}(W_{i}), i = 1, 2, and note that K is embedded in F_{i} since K is abelian. Then there exists a field isomorphism

π : F_{2} → F_{1} with π|_{K} = id.

On the other hand, by our hypothesis W_{2} is a submodule of the module O_{p}(M)/Y, which is dual to Y. Thus, there exists a field isomorphism

μ : F_{1} → F_{2} with k ↦ k^{−1} for k ∈ K.

Hence, σ := μπ is an automorphism of F_{1} that inverts the elements of K.

As every automorphism of F_{1} is a power of the Frobenius automorphism x ↦ x^{p}, there exists 0 ≤ m < n (where q = p^{n}) such that f^{σ} = f^{p^{m}} for f ∈ F_{1} and

k^{σ} = k^{−1} = k^{p^{m}} for k ∈ K.

It follows that k^{p^{m}+1} = 1 and thus

|K| = \frac{q − 1}{2} | p^{m} + 1.

This shows that

q − 1 = p^{n} − 1 ≤ 2p^{m} + 2 and p^{m}(p^{n−m} − 2) ≤ 3,

so m = 0 and q = 5. □

2.9 Let G be a finite group, and let L, M_{1}, and M_{2} be subgroups of G such that L ≤ M_{1} ∩ M_{2}. Suppose that the following hold:

(a) L satisfies 2.7.

(b) M_{1} and M_{2} satisfy 2.8 in place of M.

(c) O_{p}(M_{1}) ≠ O_{p}(M_{2}), and C_{G}(O_{p}(M_{i})) ≤ O_{p}(M_{i}), i = 1, 2.

(d) C_{G}(L/D)/D is a p′-group (D as in 2.7).

Then q = 3 or 5.

Proof. Let T ∈ Syl_{p}(L) and set

U := N_{G}(L) ∩ N_{G}(T), U_{0} := U ∩ L, U_{i} := U ∩ M_{i}.

Moreover, let A and V be as in 2.7. We use a similar approach as in 2.8 getting T ∈ Syl_{p}(M_{i}), D = O_{p}(M_{1})O_{p}(M_{2}), Z(O_{p}(M_{i})) ∈ A, and Z(O_{p}(M_{1}))Z(O_{p}(M_{2})) = D′.

Observe that U_{i}/T ≅ C_{q−1} × C_{q−1} and that U_{1} and U_{2} centralize U_{0}/T. Note further that N_{G}(L)/C_{G}(L/D) is a subgroup of Aut(L_{2}(q)). Thus, the structure of Aut(L_{2}(q)) together with the hypothesis on C_{G}(L/D) shows that ⟨U_{1}, U_{2}⟩/T is a p′-group. Now the Theorem of Schur-Zassenhaus gives a complement X for T in ⟨U_{1}, U_{2}⟩.

According to 2.7 and 2.8 we may assume that p = 3 and |A| = q. Set D̃ = D/D′. By 1.6 there are exactly q + 1 natural L/D-submodules in D̃. On the other hand, for every A ∈ A the factor group C_{D}(A)/D′ is one of these submodules. Hence there exists a unique D′ ≤ W_{0} ≤ D such that W̃_{0} is a natural L/D-module and W′_{0} ∉ A. In particular X normalizes W_{0}.

Assume first that |X| = (q − 1)^{2}. Then U_{1} = U_{2} and, similar as in 2.8, the GF (p)U_{1}-modules Z(O_{p}(M_{1}))/V and D′/Z(O_{p}(M_{1})) are isomorphic and dual to each other with C_{U_{1}}(Z(M_{1})/V) = U_{0}. As |U_{1}/U_{0}| = q − 1, an argument as in 2.8 gives

q − 1 = p^{n} − 1 | p^{k} + 1 for some 0 ≤ k < n,

and n = 1, k = 0 and q = 3.

Assume now that |X| > (q − 1)^{2}. Since XL/C_{XL}(W̃_{0}) ≅ GL_{2}(q) we conclude that C_{X}(W̃_{0}) ≠ 1 and X′ ≤ C_{X}(W̃_{0}). On the other hand

C_{X}(D̃) = C_{X}(D) ≤ C_{X}(O_{p}(M_{1})) = 1

since X is a p′-group. It follows that also W_{1} := [D, C_{X}(W̃_{0})]D′ ≠ D′, so W̃_{1} is also a natural L/D-module. In particular A := W′_{1} ∈ A.

Now with the same argument as above C_{X}(W̃_{1}) ≠ 1 and X′ ≤ C_{X}(W̃_{1}), so

X′ ≤ C_{X}(W̃_{0}) ∩ C_{X}(W̃_{1}) = C_{X}(D̃) ≤ C_{X}(O_{p}(M_{1})) = 1,

and X is abelian.

If A = Z(O_{p}(M_{1})), then W_{1} = O_{p}(M_{1}) and

C_{X}(W̃_{1}) = C_{X}(O_{p}(M_{1})) = 1,

a contradiction. With the same argument A ≠ Z(O_{p}(M_{2})), so there are three X-submodules of order q in D′/V, namely A/V, Z(O_{p}(M_{1}))/V, and Z(O_{p}(M_{2}))/V. Hence, these submodules are isomorphic.

If X normalizes Z(O_{p}(M_{2})), then Z(O_{p}(M_{1}))/V and D′/Z(O_{p}(M_{1})) are isomorphic GF (p)U_{1}-modules and dual to each other, so as above q = 3. Thus we may assume that there exists x ∈ X such that Z(O_{p}(M_{2}))^{x} ≠ Z(O_{p}(M_{2})). Now Z(O_{p}(M_{2}))/V and D′/Z(O_{p}(M_{2}))^{x} are isomorphic and dual to each other as GF (p)U_{2}-modules, and again q = 3. □

3 The Local P̃! -Theorem

In this section we assume that H is a K_{p}-group of local characteristic p, which satisfies Q-uniqueness. Moreover, we assume that there exists P ∈ P_{H}(S) such that P ≰ C̃ and Y_{M} ≤ Q for every M ∈ M_{H}(P).

Notation. For L ∈ L_{H}(S) set L_{0} := L^{∘}C_{S}(Y_{L}).

We first collect two results from [PPS]:

3.1 P_{0}/O_{p}(P_{0}) ≅ SL_{2}(p^{m}), and Y_{P} is a natural SL_{2}(p^{m})-module for P_{0}/O_{p}(P_{0}).

Proof. This follows from the P!-Theorem in [PPS]. □

3.2 Let L ∈ L_{H}(P) and \overline{L} := L/C_{L}(Y_{L}). Then the following hold:

(a) F^{∗}(\overline{L}) = [\overline{L}_{0}, \overline{L}_{0}], \overline{L}_{0} ≅ SL_{n}(p^{m}), Sp_{2n}(p^{m}) or Sp_{4}(2)′ (and p = 2),

(b) [Y_{L}, L_{0}] is the corresponding natural module, and Z(L) = 1.

(c) Y_{P} is a 2-dimensional (singular) subspace of [Y_{L}, L_{0}].

(d) Either C_{L_{0}}(Y_{L}) = O_{p}(L_{0}), or p = 2 and L_{0}/O_{2}(L_{0}) ≅ 3Sp_{4}(2)′.

(e) P_{0} ≤ L_{0} and S ∩ P_{0} ∈ Syl_{p}(L_{0}), or \overline{L}_{0} ≅ Sp_{4}(2)′ and \overline{L}_{0}\overline{C_{S}(Y_{P})} ≅ Sp_{4}(2).

Proof. Claims (a) – (d) follow from the Corollary in [PPS]. For the proof of (e) set P_{1} := P_{0} ∩ L_{0} and S_{0} := S ∩ L_{0}. Since P^{∘} ≤ P_{1}

P^{∘}C_{S}(Y_{P}) = P_{0} = P_{1}C_{S}(Y_{P}).

From (a), (b) and (c) we get that |S_{0}/C_{S_{0}}(Y_{P})| = p^{m}. On the other hand, C_{S_{0}}(Y_{P}) = C_{S}(Y_{P}) ∩ P_{1} and thus

P_{1}/C_{S_{0}}(Y_{P}) ≅ P_{0}/C_{S}(Y_{P}) ≅^{3.1} SL_{2}(p^{m}),

so S_{0} ≤ P_{1}.

If C_{S}(Y_{P}) = C_{S_{0}}(Y_{P}), then the first possibility of (e) holds. Thus we may assume that there exists t ∈ C_{S}(Y_{P}) \ S_{0}; in particular \overline{t} ∉ \overline{L}_{0}. This element induces an automorphism in \overline{L}_{0} that centralizes a 2-dimensional singular subspace of [Y_{L}, L_{0}].

An inspection of the automorphisms of SL_{3}(p^{m}) and Sp_{4}(p^{m}) having this property shows that either \overline{t} ∈ \overline{L}_{0}, which is not the case, or \overline{L}_{0} ≅ Sp_{4}(2)′ and the second possibilty of (e) holds. □

3.3 N_{H}(Y_{P})^{∘} = P^{∘} and P^{∘} is normal in N_{H}(Y_{P}).

Proof. By 3.1, P is acts transitively on Y_{P}, so the claim follows from (2.4.2)(dd) in [MSS]. □

3.4 Let L ∈ L_{H}(P) and S_{1} := L ∩ S. Then

N_{L}(S_{1}) ≤ N_{L}(Y_{P}) ≤ N_{L}(P^{∘}).

In particular, for every P̃ ∈ P_{L}(S) either P̃ ≤ N_{L}(P^{∘}) or P̃ ≤ L_{0}S.

Proof. The structure of \overline{L}_{0} and [Y_{L}, L_{0}] described in 3.2 shows that Y_{P} is the unique S_{1}-invariant 2-dimensional singular subspace of [Y_{L}, L_{0}]. Together with 3.3 we get that N_{L}(S_{1}) ≤ N_{L}(Y_{P}) ≤ N_{L}(P^{∘}). Now 1.5 yields the second part of the claim. □

3.5 Local P̃! -Theorem. Let L ∈ L_{H}(P) be such that P^{∘} ≠ L^{∘}. Then there exists a unique P̃ ∈ P_{L}(S) such that P̃ ≰ N_{L}(P^{∘}). Moreover, for U := ⟨P, P̃⟩, U^{∗} := U^{∘}C_{S}(Y_{P}) and \overline{U}^{∗} := U^{∗}/C_{U^{∗}}(Y_{U}) the following hold:

(a) U = U^{∗}S and \overline{U}^{∗} ≅ SL_{3}(p^{m}), Sp_{4}(p^{m}) or Sp_{4}(2)′ (and p = 2).

(b) Z(U^{∗}) = 1, and [Y_{U}, U^{∗}] is a corresponding natural module for \overline{U}^{∗}.

(c) Y_{P} is a 2-dimensional singular subspace of [Y_{U}, U^{∗}].

(d) C_{U^{∗}}(Y_{U}) = O_{p}(U^{∗}) or p = 2 and U^{∗}/O_{2}(U^{∗}) ≅ 3Sp_{4}(2)′ or 3Sp_{4}(2).

(e) P_{0} ≤ U^{∗} and S ∩ P_{0} ∈ Syl_{p}(U^{∗}).

Proof. Set S_{1} := S ∩ L_{0}. It is evident that P^{∘} ≤ L^{∘}. Thus, if L_{0} ≤ P, then L^{∘} = P^{∘}, which contradicts the hypothesis. Hence, we have L_{0} ≰ P, and the structure of \overline{L}_{0} and [Y_{L}, L_{0}] described in 3.2 shows:

(∗) There exists a unique minimal parabolic subgroup R ∈ P_{L_{0}}(S_{1}) such that R ≰ N_{L_{0}}(Y_{P}).

Pick P̃ ∈ P_{L}(S) with P̃ ≰ N_{L}(P^{∘}). We first prove the uniqueness of P̃. Let X be the unique maximal subgroup of P̃ containing S, and set P̃_{1} := P̃ ∩ L_{0}. By 3.4 and 1.1 (b) P̃ = SP̃_{1},

P̃_{1} = NP̃_{1}(S_{1})⟨P̂ | P̂ ∈ PP̃_{1}(S_{1})⟩ and NP̃(S_{1}) ≤ NP̃(P^{∘}) ≤ X.

Hence, there exists U ∈ PP̃_{1}(S_{1}) such that U ≰ X, and (∗) implies U = R, consequently RS = P̃. This shows that P̃ is uniquely determined.

We now apply 3.2 with U and \overline{U} = U/C_{U}(Y_{U}) in place of L and \overline{L}. Note that by 3.2 (e) C_{S}(Y_{U}) ≤ U_{0} ≤ U^{∗} and that U_{0} and U^{∗} are both normal in U. Hence 1.2 yields Y_{U} = Y_{U_{0}} = Y_{U^{∗}}. Note further that by 3.2 (e) either U_{0} = U^{∗} or \overline{U}_{0} ≅ Sp_{4}(2)′ and \overline{U}^{∗} ≅ Sp_{4}(2). Then the claims (a) – (e) follow from the corresponding claims in 3.2 since \overline{U} has Lie rank 2. □

3.6 Suppose that there exist two different P̃_{1}, P̃_{2} ∈ P_{H}(S) such that

P̃_{i} ≰ N_{H}(P^{∘}) and O_{p}(⟨P̃_{i}, P ⟩) ≠ 1 (i = 1, 2).

Set M_{i} := ⟨P, P̃_{i}⟩^{∘}C_{S}(Y_{P}) and G := ⟨M_{1}, M_{2}⟩. Then G together with M_{1} and M_{2} satisfies the hypothesis of Theorem 1.

Proof. Note that 3.5 applies to M_{i} in place of U^{∗}, for i = 1 and 2. The Hypotheses (1), (3), (4), and (6) of Theorem 1 follow from (2.5) (e), (a), (b), and (d), respectively.

Assume that there exists 1 ≠ N ⊴ G such that N ≤ B. Since B has characteristic p, we get O_{p}(N) ≠ 1 and thus O_{p}(G) ≠ 1. Since S normalizes G we get that GS ∈ L_{H}(P), and 3.5 applies to GS.

In particular, there exists a unique P̃ ∈ P_{GS}(S) not normalizing P^{∘}. This contradicts P̃_{1} ≠ P̃_{2}, and this contradiction shows Hypothesis (2) of Theorem 1.

By 3.5 (e) P_{0} ≤ M_{1} ∩ M_{2}. Hence 3.5 (a), (c) and (e) give Hypothesis (5) of Theorem 1.

4 The Coset Graph

In this section we assume that the Hypothesis of Theorem 1 holds. We will apply the amalgam method to the group G := ⟨M_{1}, M_{2}⟩ and the pair of subgroups M_{1} and M_{2} using the standard notation (see for example [PPS] or [ST]). For the convenience of the reader we will repeat some of the notation.

Let Γ be the coset graph of G with respect to the subgroups M_{1} and M_{2}; so the vertices are the right cosets of M_{1} and M_{2} in G, and two vertices are adjacent if and only if they are different and have non-empty intersection. Then G acts by right multiplication on Γ, and the vertex stabilizers in G are conjugate to M_{1} or M_{2}, while the edge stabilizers are conjugate to M_{1} ∩M_{2}. By d( , ) we denote the usual distance metric on Γ.

For a finite group L define Z_{L} := ⟨Ω_{1}(Z(T)) | T ∈ Syl_{p}(L)⟩.

For every δ ∈ Γ define

Δ(δ) := {λ | d(δ, λ) = 1}, G_{δ} := {g ∈ G | δ^{g} = δ},
Y_{δ} := Y_{G_{δ}}, Z_{δ} := Z_{G_{δ}},
Q_{δ} := O_{p}(G_{δ}),  V_{δ}^{(i)} := ⟨Z_{λ} | d(λ, δ) = i⟩,
V_{δ} := V_{δ}^{(1)}, D_{δ} := C_{Q_{δ}}(V_{δ}),
\overline{G}_{δ} := G_{δ}/C_{G_{δ}}(Z_{δ}),

and

b := min{d(δ, λ) | δ, λ ∈ Γ and Z_{δ} ≰ Q_{λ}}.

For adjacent vertices α and β define

G_{αβ}^{∗} := O^{p′}(G_{α} ∩ G_{β}), Q_{αβ} := O_{p}(G_{αβ}^{∗}), Z_{αβ} := Z_{G_{αβ}^{∗}}.

A pair (α, α′) of vertices is called critical if Z_{α} ≰ Q_{α′} and d(α, α′) = b. If (α, α′) is a critical pair, then we fix a path of length b from α to α′ and denote its vertices by

(α, α + 1, ..., α + b) or (α′ − b, , ..., α′ − 1, α′),

so α + i = α′ − (b − i).

The first lemma mostly phrases the hypothesis of Theorem 1 in terms of this new setup and notation.

4.1 Let α and β be adjacent vertices in Γ. Then the following hold:

(a) G_{α} is of characteristic p, and Syl_{p}(G_{αβ}^{∗}) = Syl_{p}(G_{α}) ∩ Syl_{p}(G_{β}).

(b) No non-trivial subgroup of G_{αβ}^{∗} is normal in both G_{α} and G_{β}.

(c) \overline{G}_{α} ≅ SL_{3}(q_{α}), Sp_{4}(q_{α}) or Sp_{4}(2)′ (and q_{α} = 2).

(d) Z_{α} = [Y_{α}, O^{p}(G_{α})] is a natural module for \overline{G}_{α} and Z(G_{α}) = 1.

(e) Z_{αβ} = Z_{α} ∩ Z_{β}, |Z_{αβ}| = q_{α}^{2} = q_{β}^{2}, and O^{p′}(N_{\overline{G}_{α}}(Z_{αβ})) = \overline{G}_{αβ}^{∗}.

(f) q := q_{α} = q_{β}.

(g) C_{G_{α}}(Z_{α}) = Q_{α}, or p = q_{α} = 2 and G_{α}/Q_{α} ≅ 3Sp_{4}(2) or 3Sp_{4}(2)′.

(h) N_{G_{α}}(Z_{αβ}) is the unique maximal subgroup of G_{α} containing G_{αβ}^{∗}.

Proof. The pair {G_{α}, G_{β}} is conjugate to {M_{1}, M_{2}}, so the hypothesis of Theorem 1 applies to both G_{α} and G_{β}.

From 1.2 (c) we get that Z_{α} = [Z_{α}, O^{p}(G_{α})]Ω_{1}(Z(G_{α})) and Z_{α} ≤ Y_{α}. Now Hypothesis (4) of Theorem 1 yields Z(G_{α}) = 1 and Z_{α} = [Y_{α}, O^{p}(G_{α})]. In particular, C_{G_{α}}(Y_{α}) ≤ C_{G_{α}}(Z_{α}), and the action of G_{α} on [Y_{α}, O^{p}(G_{α})] given in Hypothesis (4) of Theorem 1 shows that C_{G_{α}}(Y_{α}) = C_{G_{α}}(Z_{α}).

With these remarks in mind the statements (a), (b), (c), (d) and (g) follow from the Hypotheses (1), (2), (3), (4) and (6) of Theorem 1.

By Hypothesis (5) of Theorem 1 there exists a 2-dimensional singular subspace W_{α} ≤ Z_{α} with O^{p′}(N_{\overline{G}_{α}}(W_{α})) ≤ \overline{G_{α} ∩ G_{β}}. By 2.1 (d) or 2.2 (d) N_{G_{α}}(W_{α}) acts irreducibly on W_{α}, so

W_{α} ≤ Z_{αβ} ≤ Z_{α} ∩ Z_{β}.

With a symmetric argument there also exists a 2-dimensional singular subspace W_{β} ≤ Z_{β} such that O^{p′}(N_{\overline{G}_{β}}(W_{β})) ≤ \overline{G_{α} ∩ G_{β}} and

W_{β} ≤ Z_{αβ} ≤ Z_{α} ∩ Z_{β}.

Suppose that W_{α} ≠ Z_{α} ∩ Z_{β}. Let T ∈ Syl_{p}(G_{αβ}^{∗}). If \overline{G}_{α} ≅ SL_{3}(q_{α}), then clearly C_{T} (Z_{α} ∩ Z_{β}) = Q_{α} since Z_{α} is 3-dimensional. If G_{α} ≅ Sp_{4}(q_{α}) or Sp_{4}(2)′, then C_{T} (Z_{α} ∩ Z_{β}) = Q_{α} follows from 2.2 (d). Thus C_{T} (Z_{α} ∩ Z_{β}) = Q_{α} holds in both cases. Now (b) implies that C_{T} (Z_{α} ∩ Z_{β}) ≠ Q_{β}. Consequently, the above argument shows that W_{β} = Z_{α} ∩ Z_{β}. But now the irreducibility of W_{α} as a G_{αβ}^{∗}-module yields W_{α} = W_{β} = Z_{α} ∩ Z_{β}, a contradiction.

We have shown that W_{α} = Z_{α} ∩ Z_{β}. Then with a symmetric argument also W_{β} = Z_{α} ∩ Z_{β} and thus

W_{α} = Z_{αβ} = W_{β},

in particular q_{α} = q_{β} and G_{αβ}^{∗} ≤ N_{G_{α}}(W_{α}). Hence, (e), (f) and (h) hold. □

In the following we use the parameter q as defined in 4.1 (f). Observe that Q_{α}Q_{β} ≤ Q_{αβ}. Thus, because of 4.1 (c), (d), (e), properties of the action of Q_{α}Q_{β} on Z_{α} and Z_{β} are given in 2.1 or 2.2, respectively. This fact will be used frequently.

4.2 Let (α, α′) be a critical pair. Then

(a) [Z_{α}, Z_{α′}] ≤ Z_{α} ∩ Z_{α+1}, and

(b) [Z_{α}, Z_{α′}] ≠ 1; in particular (α′, α) is also a critical pair.

Proof. As Z_{α′} ≤ Q_{α+1}, claim (a) follows from 4.1 and 2.1 (b) or 2.2 (c), respectively. Claim (b) is a consequence of 4.1 (g). □

4.3 Let α, β ∈ Γ be adjacent. Then Q_{α}Q_{β} = Q_{αβ}, or

(∗) q = 2, |Q_{αβ}/Q_{α}Q_{β}| = 2, and \overline{G}_{δ} ≅ Sp_{4}(2) for every δ ∈ Γ.

Proof. Set A := Q_{α}Q_{β}. Since A is a normal p-subgroup of G_{αβ}^{∗} we have A ≤ Q_{αβ}. By 4.1 (b) A ≰ Q_{α} or A ≰ Q_{β}, and we choose our notation such that A ≰ Q_{α}.

Assume that A ≠ Q_{αβ}. Then 2.1 (d) shows that \overline{G}_{α} ≅ S̸L_{3}(q), and 2.2 (e) implies that either

q = 2 and |Q_{αβ}/A| = 2, or p = 2 and |Q_{αβ}/A| = q^{2}.

In the first case computing |Q_{αβ}/A| in \overline{G}_{β} shows (∗).

Hence, we may assume that we are in the second case. Then either

(I) \overline{G}_{β} ≅ Sp_{4}(q) and |A/Q_{β}| = q, or

(II) \overline{G}_{β} ≅ SL_{3}(q) and Q_{α} ≤ A = Q_{β}.

From 2.2 (d) and |Q_{αβ}/A| = q^{2} we get J(A) ≤ Q_{α}. Similarly, (I) and (II) both imply J(A) ≤ Q_{β}. Thus J(Q_{α}) = J(A) = J(Q_{β}), a contradiction.

□

4.4 Let α, β ∈ Γ be adjacent. Suppose that X ≤ G_{α} such that X ≰ N_{G_{α}}(Z_{αβ}) and A ≤ Z_{β} such that A ≰ Y_{α}. Then [A, X] ≰ Y_{α}.

Proof. Assume that [A, X] ≤ Y_{α}. Then X normalizes AY_{α} and thus also [AY_{α}, Q_{α}] = [A, Q_{α}]. Now 4.3, 2.1 (b) and 2.2 (c) and (e) show that [A, Q_{α}] = Z_{αβ} and X ≤ N_{G_{α}}(Z_{αβ}), a contradiction. □

4.5 Let α, β ∈ Γ be adjacent such that \overline{G}_{α} ≅ \overline{G}_{β} ≅ SL_{3}(q), and let x ∈ Q_{α} be such that [x, G_{α}] = Z_{α}. Suppose that V_{β} ≰ Q_{α}. Then x ∈ Z(Q_{α}).

Proof. Assume that x ∉ Z(Q_{α}). We set Y := [Q_{α}, G_{αβ}^{∗}]C_{Q_{α}}(x) and Q̃_{α} := Q_{α}/C_{Q_{α}}(x). Note that C_{Q_{α}}(x) = C_{Q_{α}}(xZ_{α}), so C_{Q_{α}}(x) is normal in G_{α}. The mapping

Q̃_{α} → Z_{α} such that yC_{Q_{α}}(x) ↦ [y, x]

shows:

(1) Z_{α} and Q̃_{α} are isomorphic GF (p)\overline{G}_{α}-modules, in particular

(2) Ỹ is contained in every non-trivial G_{αβ}^{∗}-invariant subgroup of Q̃_{α}.

On the other hand, \overline{G}_{β} ≅ SL_{3}(q) and thus by 4.3 and 2.1 (d) \overline{Q}_{α} = \overline{Y}. It follows that Q_{α} = (Q_{α}∩Q_{β})Y. On the other hand, by (1) Q_{α}∩Q_{β} ≰ C_{Q_{α}}(x) since |Q_{α}/Q_{α} ∩ Q_{β}| ≤ q^{2}, so (2) implies that Q_{α} = (Q_{α} ∩ Q_{β})C_{Q_{α}}(x). In particular

(3) [Q_{α}, V_{β}] ≤ [Q_{β}, V_{β}]C_{Q_{α}}(x).

Again, as \overline{G}_{β} ≅ SL_{3}(q), we get that Z_{β}/Z_{αβ} is a G_{αβ}^{∗}-invariant subgroup of order q. But Q_{αβ}/Q_{α} is an irreducible G_{αβ}^{∗}-module of order q^{2}, so Z_{β} ≤ Q_{α}, and (2) yields Z_{β} ≤ C_{Q_{α}}(x).

In particular Z_{α} ≤ Q_{β} and thus V_{β} ≤ Q_{β}. By 2.1 (b) [Z_{α}, Q_{β}] ≤ Z_{αβ} ≤ Z_{β}, so [V_{β}, Q_{β}] ≤ Z_{β} ≤ C_{Q_{α}}(x). Now (3) shows that V_{β} centralizes Q̃_{α}.

Since V_{β} ≰ Q_{α} we conclude that \overline{G}_{α} = ⟨\overline{V} _{β}^{\overline{G}_{α}}⟩, and Q̃_{α} is a central \overline{G}_{α}- module. This contradicts (1). □

4.6 Suppose that δ ∈ Γ with \overline{G}_{δ} ≅ SL_{3}(q). Then G_{δ} acts transitively on set of pairs (β, γ) with β, γ ∈ Δ(δ) and Z_{βδ} ≠ Z_{γδ}.

Proof. It is easy to see that G_{βδ}^{∗} acts transitively on the 2-dimensional subspaces of Z_{δ} that are distinct from Z_{βδ}. Now let γ, γ′ ∈ Δ(δ) be such that Z_{γδ} = Z_{γ′δ} ≠ Z_{βδ}. By 4.1 (h) there exists x ∈ N_{G_{δ}}(Z_{γδ}) such that γ^{x} = γ′. On the other hand

N_{G_{δ}}(Z_{γδ}) = (G_{βδ}^{∗} ∩ N_{G_{δ}}(Z_{γδ}))G_{γ′δ}^{∗},

so this x can be chosen in G_{βδ}^{∗}, and the result follows. □

5 The Discussion of the Amalgam Problem.

In this section we adopt the hypothesis and notation of Section 4. Moreover, (α, α′) is always a critical pair, and

Λ(α, α′) := {μ ∈ Δ(α) | Z_{α′} ≰ N_{G_{α}}(Z_{αμ})}.

5.1 The following hold:

(a) Λ(α, α′) ≠ ∅.

(b) ⟨Z_{α′}, G_{αμ}^{∗}⟩ = G_{α} for every μ ∈ Λ(α, α′).

Proof. By 4.2 (b) Z_{α′} ≰ Q_{α}, so Z_{α′} does not stabilize every 2-dimensional subspace of Z_{α}. Hence, 4.1 (e) gives (a).

Claim (b) follows from 4.1 (h). □

5.2 Suppose that α − 1 ∈ Λ(α, α′) with Z_{α−1} ∩ Q_{α′−1} ≠ Z_{α−1α}. Then the following hold:

(a) \overline{G}_{α′} ≅ SL_{3}(q).

(b) |[Z_{α}, Z_{α′}]| = q.

(c) Z_{α′α′−1} ≰ Z_{α} and Z_{α′α′−1} ≤ V_{α}.

(d) If Z_{α−1} ≤ Q_{α′−1}, then \overline{G}_{α−1} ≅ SL_{3}(q).

Proof. Set A := Z_{α−1}∩Q_{α′−1}. Since A ≰ Z_{αα−1} and Z_{α′} ≰ N_{G_{α}}(Z_{αα−1}), 4.4 implies [A, Z_{α′}] ≰ Z_{α}, in particular [Z_{α}, Z_{α′}] < [AZ_{α}, Z_{α′}]. As AZ_{α} ≤ Q_{α′−1} ≤ Q_{α′−1α′}, we get from 2.1 (b) and 2.2 (c) that [AZ_{α}, Z_{α′}] ≤ Z_{α′−1α′}. Since Z_{α′−1α′} is a 2-dimensional subspace of Z_{α′} this implies:

Z_{α′α′−1} = [AZ_{α}, Z_{α′}], |[Z_{α}, Z_{α′}]| = q and Z_{α′α′−1} ≰ Z_{α}.

This gives (b) and the first part of (c). As AZ_{α} ≤ V_{α} and V_{α} is normal in G_{α}, we get Z_{α′α′−1} = [AZ_{α}, Z_{α′}] ≤ V_{α} and the second part of (c).

Suppose now that \overline{G}_{α′} ≅ Sp_{4}(q) or Sp_{4}(2)′. Let X := Z_{α′} ∩ Q_{α} = C_{Z_{α′}} (Z_{α}). Since |[Z_{α}, Z_{α′}]| = q we conclude from 2.4 that |Z_{α′}/X| = q and |C_{AZ_{α}}(X)Q_{α′}/Q_{α′}| ≤ q. As |Z_{α}Q_{α′}/Q_{α′}| = q and A ≰ Z_{α}Q_{α′}, this shows that [A, X] ≠ 1. Note that [A, X] ≤ Z_{α−1α} ≤ Z_{α}.

If [A, X] ≰ [Z_{α}, Z_{α′}] then

Z_{α′α′−1} = [AZ_{α}, Z_{α′}] = [A, X][Z_{α}, Z_{α′}] ≤ Z_{α},

a contradiction. Thus [Z_{α}, Z_{α′}] = [A, X] ≤ Z_{α−1α} and so Z_{α′} normalizes Z_{α−1α}, again a contradiction. This shows that \overline{G}_{α′} ≅ SL_{3}(q), and (a) holds.

Set Y := A ∩ (Z_{α}Q_{α′}). Then by 2.1 (d) |A/Y | ≤ q. Since [Y, Z_{α′}] ≤ Z_{α}, 4.4 implies Y = Z_{α−1α} and thus |A/Z_{α−1α}| ≤ q. This rules out the case A = Z_{α−1} and \overline{G}_{α−1} ≅ Sp_{4}(q), Sp_{4}(2)′, so (d) holds. □

5.3 Suppose that \overline{G}_{δ} ≅ Sp_{4}(q) or Sp_{4}(2)′ for some δ ∈ Γ. Then b = 1.

Proof. We say that G_{δ} is of symplectic type if \overline{G}_{δ} ≅ Sp_{4}(q) or Sp_{4}(2)′. First we will show that there exists a critical pair (α, α′) such that

(∗) G_{α} is of symplectic type, and either G_{α′} is of symplectic type or |[Z_{α}, Z_{α′}]| = q^{2}.

Let α − 1 ∈ Λ(α, α′). Suppose first that b is even. Then G_{α} is conjugate to G_{α′}, so (∗) holds if G_{α} is of symplectic type. Assume that \overline{G}_{α} ≅ SL_{3}(q). Then by our hypothesis G_{α−1} is of symplectic type, and by 5.2 (α−1, α′ −1) is critical. Hence (α − 1, α′ − 1) satisfies (∗).

Suppose now that b is odd. Since also (α′, α) is critical we may assume that G_{α′} is of symplectic type. Thus by 5.2 Z_{α−1} ∩ Q_{α′−1} = Z_{α−1α}, in particular (α − 1, α′ − 1) is critical. Since b is odd, we get that G_{α−1} is conjugate to G_{α′}, so G_{α−1} is of symplectic type and |Z_{α−1}/Z_{α−1α}| = q^{2}. It follows that |Z_{α−1}Q_{α′−1}/Q_{α′−1}| = q^{2} and either G_{α′−1} is of symplectic type or |[Z_{α′−1}, Z_{α−1}]| = q^{2}, so (α − 1, α′ − 1) satisfies (∗).

Choose a critical pair (α, α′) that satisfies (∗). Let D be a three dimensional subspace of Z_{α} containing Z_{α} ∩ Q_{α′}, and let Σ be the set of 1-dimensional subspaces E of D with E ≰ Z_{αα+1}. For E ∈ Σ pick a 2dimensional singular subspace W_{E} of Z_{α} with W_{E} ∩ D = E. Note that Z_{α} = W_{E} × Z_{αα+1}.

By 2.2 (f) Z_{α′} does not normalize W_{E}. Pick μ_{E} ∈ Δ(α) with Z_{αμ_{E}} = W_{E}. Then μ_{E} ∈ Λ(α, α′), so by (∗) and 5.2 (μ_{E}, α′ − 1) is critical.

Assume that b > 1. Then [Z_{μ_{E}}, Z_{α′−1}] ≤ Z_{α′−1} ≤ Q_{α′}, and

[Z_{μ_{E}}, Z_{α′−1}] ≤ Z_{αμ_{E}} ∩ Q_{α′} ≤ Z_{αμ_{E}} ∩ D = E.

Hence E = [Z_{μ_{E}}, Z_{α′−1}] ≤ Z_{α′−2α′−1}.

Since this is true for all E ∈ Σ we get that D = ⟨Σ⟩ ≤ Z_{α′−2α′−1}. But D has order q^{3} while Z_{α′−2α′−1} has only order q^{2}. This contradiction shows that b = 1. □

5.4 b ≤ 2.

Proof. Suppose b ≥ 3. Then by 5.3 \overline{G}_{δ} ≅ SL_{3}(q) for all δ ∈ Γ. We will first show that there exists a critical pair (α, α′) such that

(∗)  Z_{α′−1α′} ≤ V_{α}.

Let α−1 ∈ Λ(α, α′). If (α−1, α′−1) is not critical, then 5.2 (c) implies (∗) for (α, α′). Suppose that (α − 1, α′ − 1) is critical. Since [Z_{α−1}, Z_{α′−1}] ≤ Z_{α−1α} and Z_{α′} does not normalize Z_{α−1α} we have [Z_{α}, Z_{α′}] ≰ [Z_{α−1}, Z_{α′−1}]. Thus A := [Z_{α}, Z_{α′}][Z_{α−1}, Z_{α′−1}] ≤ Z_{α} has order at least q^{2}. Moreover A ≤ Z_{α′−1}Z_{α′} ≤ V_{α′}, and Z_{α′} centralizes A since b > 1. Thus A ≤ C_{Z_{α}}(Z_{α′}) = Z_{αα+1}, and since Z_{αα+1} has order q^{2} we get Z_{αα+1} = A ≤ V_{α′}, so (∗) holds for (α′, α).

Now choose a critical pair (α, α′) satisfying (∗). Suppose first that b > 3. Then [V_{α}^{(2)}, V_{α}] = 1, so (∗) implies [V_{α}^{(2)}, Z_{α′α′−1}] = 1. Thus 4.4 with

(α′ − 2, α′ − 1, Z_{α′α′−1}, V_{α}^{(2)}) in place of (α, β, A, X)

gives Z_{α′α′−1} = Z_{α′−1α′−2} or V_{α}^{(2)} ≤ N_{G_{α′−2}}(Z_{α′−1α′−2}). Evidently the first

case implies the second one. Thus we have in both cases V_{α}^{(2)} ≤ G_{α′−1} by 4.1 (e). Hence V_{α}^{(2)} ≤ C_{G_{α′−1}}(Z_{α′α′−1}) = Q_{α′α′−1} and [V_{α}^{(2)}, Z_{α′}] ≤ Z_{α′α′−1} ≤ V_{α}. This shows that Z_{α′} and G_{αα−1}^{∗} normalize V_{α−1}V_{α}. Since α−1 ∈ Λ(α, α′) we conclude from 5.1 (b) that V_{α−1}V_{α} is normal in G_{α}. The transitivity of G_{α} on Δ(α) gives V_{α}^{(2)} = V_{α}V_{β} for all β ∈ Δ(α). Conjugation to G_{α+2} yields V_{α+2}^{(2)} = V_{α+2}V_{α+3}, in particular

Z_{α} ≤ V_{α+2}V_{α+3} ≤ Q_{α′},

a contradiction.

We have shown that b = 3. Let (μ, μ′) be any critical pair. Suppose that Z_{μ+1μ+2} = Z_{μ′μ+2}. Then V_{μ} ≤ C_{G_{μ+2}}(Z_{μ′μ+2}) ≤ Q_{μ′μ+2} and thus

[Z_{μ′}, V_{μ}] ≤ Z_{μ′μ+2} = Z_{μ+1μ+2} ≤ Z_{μ+1}.

Let μ − 1 ∈ Λ(μ, μ′). Then 4.4, with (μ, μ − 1, Z_{μ′}, Z_{μ−1}) in place of (α, β, X, A), implies that [Z_{μ−1}, Z_{μ′}] ≰ Z_{μ}. In particular [V_{μ}, Z_{μ′}] ≰ Z_{μ} and thus [Z_{μ′}, V_{μ}] > [Z_{μ′}, Z_{μ}]. Since Z_{μ′} is a natural SL_{3}(q)-module for \overline{G}_{μ′} we get that

[Z_{μ′}, V_{μ}] = Z_{μ′μ+2} and |[Z_{μ′}, Z_{μ}]| = q.

From 4.1 (e), applied to the vertices in Δ(μ), we get [V_{μ}, Q_{μ}] = Z_{μ}. Hence, as [V_{μ}, Z_{μ′}] ≰ Z_{μ},

[V_{μ}, Q_{μ}Z_{μ′}] = Z_{μ}[V_{μ}, Z_{μ′}] = Z_{μ}Z_{μ+1}.

Moreover, |[Z_{μ′}, Z_{μ}]| = q together with 2.4 shows that \overline{Z}_{μ} is a 1-dimensional subspace of \overline{Q}_{μ+1}. As \overline{G}_{μμ+1}^{∗} is transitive on these 1-dimensional subspaces and N_{G_{μ}}(Z_{μ′}Q_{μ}) ≤ N_{G_{μ}}(Z_{μμ+1}), we get with the Frattini argument

N_{G_{μ}}(Z_{μμ+1}) = G_{μμ+1}^{∗}N_{G_{μ}}(Z_{μ′}Q_{μ}).

Since [V_{μ}, Q_{μ}Z_{μ′}] is normalized by N_{G_{μ}}(Z_{μ′}Q_{μ}) while Z_{μ}Z_{μ+1} is normalized by G_{μμ+1}^{∗}, we conclude that N_{G_{μ}}(Z_{μμ+1}) normalizes Z_{μ}Z_{μ+1}. Since Z_{μ}Z_{μ−1} is not normal in G_{μ} and N_{G_{μ}}(Z_{μμ+1}) is a maximal subgroup of G_{μ}, this shows that

N_{G_{μ}}(Z_{μμ+1}) = N_{G_{μ}}(Z_{μ}Z_{μ+1}).

Hence, edge-transitivity also gives

N_{G_{μ+2}}(Z_{μ′}Z_{μ+2}) = N_{G_{μ+2}}(Z_{μ′μ+2}) = N_{G_{μ+2}}(Z_{μ+1μ+2}) = N_{G_{μ+2}}(Z_{μ+1}Z_{μ+2}),

and the transitivity of G_{μ+2} on Δ(μ + 2) yields

Z_{μ′} ≤ Z_{μ′}Z_{μ+2} = Z_{μ+2}Z_{μ+1} ≤ Q_{μ},

a contradiction.

This contradiction shows that Z_{μ+1μ+2} ≠ Z_{μ′μ+2} for all critical pairs (μ, μ′); in particular

(∗∗)  Z_{α+1α+2} ≠ Z_{α′α+2} and Z_{αα+1} ≠ Z_{α+2α+1}.

But then Z_{α+2} = Z_{α+1α+2}Z_{α′α+2}, and (∗) implies that Z_{α+2} ≤ V_{α}. On the other hand, by (∗∗) and 4.6 there exists an element in G_{α+1} that maps (α, α + 2) to (α + 2, α), so also Z_{α} ≤ V_{α+2}. Since b > 2 we get that V_{α+2} is abelian and Z_{α} and Z_{α′} centralize each other. This contradicts 4.2. □

5.5 Suppose that b > 1. Let δ ∈ Γ and γ, γ′ ∈ Δ(δ). Then the following hold:

(a) D_{δ} = Z_{δ}.

(b) (γ, γ′) is critical if and only if Z_{γδ} ≠ Z_{γ′δ}.

Proof. From 5.3 and 5.4 we get that b = 2 and G_{δ}/Q_{δ} ≅ SL_{3}(q) for every δ ∈ Γ. Let β = α + 1.

Since G_{α} = ⟨Z_{α′}^{G_{α}}⟩Q_{α} we get from 4.4, with (Z_{β}, O^{p}(G_{α})) in place of (A, X), that [Z_{β}, ⟨Z_{α′}^{G_{α}}⟩] ≰ Y_{α}. Hence

(1) [V_{α}, Z_{α′}] ≰ Y_{α}.

Assume next that Z_{αβ} = Z_{α′β}. Then V_{α} ≤ C_{G_{β}}(Z_{α′β}) ≤ Q_{α′β} and so

[V_{α}, Z_{α′}] ≤ Z_{α′β} = Z_{αβ} ≤ Z_{α}.

This contradicts (1). Thus

(2) Z_{αβ} ≠ Z_{α′β}.

From 4.6 we conclude that (b) holds for all δ ∈ β^{G}.

Assume that D_{α} = Z_{α}. Then there exists α − 1 ∈ Δ(α) such that (α − 1, β) is a critical pair. By a symmetric argument (b) now also holds for all δ ∈ α^{G} and thus for all δ ∈ Γ. It remains to show that D_{α} = Z_{α}. So we assume:

(3) D_{α} ≠ Z_{α}.

Note that D_{α} ≤ Q_{β} ∩ Q_{α} ≤ Q_{α′β}, so by 2.1 (d) and 4.1 (e) Φ(D_{α}) ≤ Q_{α′}. It follows that [Φ(D_{α}), ⟨Z_{α′}^{G_{α}}⟩] = 1. As G_{α} = ⟨Z_{α′}^{G_{α}}⟩Q_{α} and by 4.1 (d) Z(G_{α}) = 1, we get that Φ(D_{α}) ∩ Z(Q_{α}) = 1 and thus Φ(D_{α}) = 1; i.e.,

(4) D_{α} is elementary abelian.

Let N be the largest normal subgroup of G_{α} in D_{α} such that N ≤ Z_{α}Q_{α′}. Then [N, Z_{α′}] ≤ Z_{α} and thus also [N, O^{p}(G_{α})] = Z_{α}. Hence 4.5 shows that

N ≤ Ω_{1}(Z(Q_{α})) = Y_{α}.

Since |N/C_{N}(Z_{α′})| = q we get from 2.1 (f) N = Z_{α} × C_{N}(G_{α}). Thus, 4.1 (d) implies that N = Z_{α}. In particular, by (3) D_{α} ≰ N. Hence, 2.1 (b) gives [Z_{α′}, D_{α}] = Z_{α′β}. From (2) we get

Z_{α}[D_{α}, Z_{α′}] = Z_{α}Z_{α′β} = Z_{α}Z_{β} ≤ D_{α},

so

(5) V_{α} ≤ D_{α}; in particular, V_{α} is abelian and V_{α} ≤ Q_{β}.

Let α − 1 ∈ Λ(α, α′). Then 4.4, with (α, α − 1, Z_{α′}, Z_{α−1}) in place of (α, β, X, A), implies that Z_{α−1}∩(Z_{α}Q_{α′}) = Z_{αα−1} and thus V_{α}Q_{α′} = Q_{β}Q_{α′}. By symmetry in α and α′,

(6) V_{α}Q_{α′} = D_{α}Q_{α′} = Q_{β}Q_{α′} and Q_{β}Q_{α} = D_{α′}Q_{α} = V_{α′}Q_{α}.

Set Q̃_{α} := Q_{α}/D_{α}. We apply 2.1 (d). Then Q_{αβ}/Q_{β} is a natural module for G_{αβ}^{∗}/Q_{αβ}, so Φ(Q_{α}) ≤ Q_{β}, and Q̃_{α} is elementary abelian. Similarly C_{Q_{αβ}/Q_{β}}(G_{αβ}^{∗}) = 1 and thus CQ̃_{α}(G_{α}) = 1. Moreover, by 2.3 |Q_{α}/Q_{α} ∩ G_{α′}| = q, so [Q_{α} ∩ G_{α′}, Z_{α′}] ≤ Z_{β} ≤ D_{α} yields |Q̃_{α}/CQ̃_{α}(Z_{α′})| ≤ q. Hence 2.5 and 2.1 (f) imply:

(7) Q̃_{α} is a G_{α}-module dual to Z_{α}.

From (7) we get |Q_{α}/D_{α}| = q^{3} and [x, Q_{β}]D_{α}/D_{α} = (Q_{α} ∩Q_{β})/D_{α} for every x ∈ Q_{α} \ Q_{β}. Hence (6) gives

Q_{α} ∩ Q_{β} = [Q_{α} ∩ G_{α′}, D_{α′}]D_{α} = (D_{α′} ∩ Q_{α})D_{α} and Q_{β} = D_{α}D_{α′}.

In particular

Φ(Q_{β}) ≤ D_{α} ∩ D_{α′} ≤^{(4)} Ω_{1}(Z(Q_{β})) =^{1.2} Y_{β} ≤ D_{α} ∩ D_{α′}.

Thus, we get

(8) D_{α} ∩ D_{α′} = Y_{β}, Φ(Q_{β}) ≤ Y_{β} and |Q_{β}/Y_{β}| = q^{6}.

Note that |D_{α}/D_{α} ∩ (Z_{α}Q_{α′})| ≤ q and [D_{α} ∩ (Z_{α}Q_{α′}), Z_{α′}] ≤ Z_{α}. Thus Z_{α′} centralizes a subgroup of index q in W := D_{α}/Z_{α}. From 2.1 (f) we get that W/C_{W} (G_{α}) is a natural SL_{3}(q)-module for \overline{G}_{α}. Since G_{αβ}^{∗} normalizes Z_{β}Z_{α}/Z_{α}, this module is dual to Z_{α}.

Let U be the inverse image of C_{W} (G_{α}) in D_{α}. By 4.5 and (4)

U ≤ Ω_{1}(Z(Q_{α})) = Y_{α}.

On the other hand, Y_{α} ≤ D_{α} since b > 1, so (1) and (6) imply that Y_{α} ≤ Z_{α}Q_{α′}. Since Z(G_{α}) = 1, again 2.1 (f) yields Y_{α} = Z_{α}. We have shown:

(9) D_{α}/Z_{α} is a \overline{G}_{α}-module dual to Z_{α}; in particular Y_{β} = Z_{β} and |Q_{α}| = |Q_{β}| = q^{9}.

By (9) [D_{α}, Q_{α}] = Z_{α}. Pick x ∈ Q_{α} ∩ Q_{β}, so [x, G_{αβ}^{∗}] ≤ D_{α} by (7). Then [Q_{α}, x]Z_{α} is G_{αβ}^{∗}-invariant and |[Q_{α}, x]Z_{α}/Z_{α}| < q^{3}. Hence (9) shows that [Q_{α}, x] ≤ Z_{α}Z_{β}. It follows that [x, Q_{α}, G_{αβ}^{∗}] ≤ Z_{α} and [G_{αβ}^{∗}, x, Q_{α}] ≤ Z_{α}. Thus, also [Q_{α}, G_{αβ}^{∗}, x] ≤ Z_{α}. Since [Q_{α}, G_{αβ}^{∗}]D_{α} = Q_{α} by (7), we conclude that

(10) Q_{α}/Z_{α} is abelian.

From (6) and (10) we get C_{Q_{α}/Z_{α}}(D_{α′}) = C_{Q_{α}/Z_{α}}(Q_{αβ}), so (7) and (9) imply |C_{Q_{α}/Z_{α}}(Q_{αβ})|≤q^{2}. On the other hand, by (9) |D_{α}| = |D_{α′}| = q^{6}, so by (6) |D_{α′} ∩ Q_{α}| = q^{4} and |(D_{α′} ∩ Q_{α})Z_{α}/Z_{α}| = q^{2}. Hence (4) implies

(11) C_{Q_{α}/Z_{α}}(Q_{αβ}) = C_{Q_{α}/Z_{α}}(D_{α′}) = (D_{α′} ∩ Q_{α})Z_{α}/Z_{α}.

Set A := (D_{α′} ∩ Q_{α})Z_{α}. Recall from the above considerations that |A| = q^{5} and |(D_{α′} ∩ Q_{α})/Z_{β}| = q. Let Σ ⊆ Δ(β) be such that α ∈ Σ and for each 2-dimensional subspace W of Z_{β} there exists a unique σ ∈ Σ with Z_{σβ} = W.

Then |Σ| = q^{2} + q + 1, and by (2) and 4.6 any distinct pair of elements in Σ is critical. Let σ ∈ Σ be with σ ≠ α. Then by (11), applied to σ in place of α′, we get

(D_{σ} ∩ Q_{α})Z_{α} = A.

We conclude that for all σ ∈ Σ, D_{σ} ∩ Q_{α} is a subgroup of index q in A. On the other hand, by (8) and (9) the groups (D_{σ} ∩ Q_{α})/Z_{β} have pairwise trivial intersection, so |A/Z_{β}| > q^{2} + q + 1, a contradiction as |A/Z_{β}| = q^{2}.

□

5.6 Suppose that b > 1. Then b = 2, and for every δ ∈ Γ the following hold:

(a) \overline{G}_{δ} ≅ SL_{3}(q), p = 3 or q = 5.

(b) Q_{δ} = V_{δ},

(c) V_{δ}/Z_{δ} is a natural SL_{3}(q)-module dual to Z_{δ}.

Proof. From 5.3, 5.4 and 5.5 we get the following information for every δ ∈ Γ:

(1) b = 2.

(2) G_{δ}/Q_{δ} ≅ SL_{3}(q) and Z_{δ} = D_{δ}.

(3) Let γ, γ′ ∈ Δ(δ). Then (γ, γ′) is critical if and only if Z_{γδ} ≠ Z_{γ′δ}.

We fix a critical pair (α, α′) and set β := α + 1. If |[Z_{α}, Z_{α′}]| = q^{2}, then Z_{αβ} = [Z_{α}, Z_{α′}] = Z_{α′β} which contradicts (3). Thus |[Z_{α}, Z_{α′}]| = q, and by 2.1 (g)

(4) Z_{α′}Q_{α}/Q_{α} is the center of a Sylow p-subgroup of G_{α}/Q_{α} for every critical pair (α, α′).

According to (3) it suffices to show (a) – (c) for δ = α′.

By (4) and 2.3 there exists α − 1 ∈ Δ(α) such that Z_{α′} ≤ G_{α−1} but Z_{α′} ≰ Q_{αα−1}. In particular R := [Z_{αα−1}, Z_{α′}] ≠ 1, so Z_{αα−1} ≠ Z_{αβ}, and by (3) (α − 1, β) is critical.

The action of G_{αα−1}^{∗} on Z_{α−1α}, see 2.1 (d), gives

R = Z_{αα−1} ∩ Z_{αβ} = [Z_{α−1}, Z_{β}].

Hence, since |[Z_{α}, Z_{α′}]| = q, we get

R = [Z_{α−1}, Z_{β}] = [Z_{α}, Z_{α′}] ≤ Z_{β} ∩ Z_{α′}

and thus (5) Z_{α−1} ≤ G_{α′}.

Again by 2.3 we have Z_{α−1} ≰ Q_{βα′} since (α − 1, β) is critical. As Z_{α}Q_{α′} = Z_{α−1α}Q_{α′} we conclude that |Z_{α−1}Q_{α′}/Q_{α′}| = q^{2} and thus |Z_{α−1} ∩ Q_{α′}| = q. We have shown:

(6) |Z_{α−1}Q_{α′}/Q_{α′}| = q^{2} and Z_{α−1} ∩ Q_{α′} = R ≤ Z_{α′}.

Let A := V_{α′} ∩G_{α}. There are q+1 2-dimensional subspaces of Z_{α} containing R. Since A fixes Z_{αβ} we get that |Z_{αα−1}^{A}| ≤ q, so |A/A ∩ G_{α−1}| ≤ q. By 2.3, applied to \overline{G}_{β}, also |V_{α′}/V_{α′} ∩ G_{α}| ≤ q, so we get

(7) |V_{α′}/V_{α′} ∩ G_{α−1}| ≤ q^{2}.

On the other hand, (5) and (6) imply that [Z_{α−1}, V_{α′} ∩G_{α−1}] ≤ Z_{α−1}∩Q_{α′} ≤ Z_{α′}. Thus (6) yields

|Ṽ_{α′}/CṼ_{α′} (Z_{α−1})| ≤ q^{2} = |Z_{α−1}Q_{α′}/Q_{α′}|,

where Q̃_{α′} := Q_{α′}/Z_{α′}.

Clearly, Φ(Q_{α′}) ≤ D_{α′} = Z_{α′}, so V′_{α′} ≠ 1 shows that Φ(Q_{α′}) = Φ(V_{α′}) = Z_{α′}. Hence, we are able to apply 2.6 with M := G_{α′}, D := V_{α′}, Y := Z_{β}, and V_{1} := Z_{α′}, and get that Ṽ_{α′} is a natural G_{α′}/Q_{α′}-module dual to Z_{α′}. Thus (c) holds.

Note that G_{α} acts transitively on (V_{α}/Z_{α})^{#}. So every element in V_{α} has order p. Since V_{α} is not abelian we conclude that p ≠ 2.

By (3) Q_{α′} = (Q_{α′} ∩ Q_{β})V_{α′} and

[Q_{α′} ∩ Q_{β}, Z_{α}] ≤ Z_{α} ∩ Z_{β} ≤ Z_{β},

so (c) gives [Ṽ_{α′}, Z_{α}] ≤ Z̃_{β}. It follows that |[Q̃_{α′}, Z_{α}]| = |Z̃_{β}| = q.

Let \overline{U} be the parabolic subgroup of \overline{G}_{α′} fixing Z̃_{β}. Then \overline{Z}_{α} ≤ O_{p}(\overline{U}), and since p ≠ 2 there exists \overline{K} ≤ \overline{U} of order q − 1 such that [Z̃_{β}, \overline{K}] = 1 and [\overline{Z}_{α}, \overline{K}] = \overline{Z}_{α}. In addition, since Q̃_{α′}/Ṽ_{α′} is central

Q̃_{α′} = Ṽ_{α′} C̃, where C̃ = CQ̃_{α′} (\overline{K}).

It follows that

[C̃, \overline{K}, \overline{Z}_{α}] = 1 = [\overline{Z}_{α}, C̃, \overline{K}],

and thus also [\overline{K}, \overline{Z}_{α}, C̃] = [\overline{Z}_{α}, C̃] = 1. As Ṽ_{α′} is a natural module, (4) implies that |Q̃_{α′}/CQ̃_{α′} (Z_{α})| = q. Hence, 2.1 (f) yields Q̃_{α′} = Ṽ_{α′}CQ̃_{α′} (G_{α′}), and 4.5 gives V_{α′} = Q_{α′}. Thus, also (b) holds.

Now let L := G_{α′β}^{∗}, D := Q_{α′β}, V := Z_{α′β} and Y := Z_{β}Z_{α′}. From (b) and (c) we get that Y = Q_{β} ∩ Q_{α′}, |Y | = q^{4} and [Y, L] = V. Moreover, L/D ≅ SL_{2}(q), and D/Y = Q_{β}/Y × Q_{α′}/Y is the direct sum of the natural L/D-modules.

Let λ ∈ {α′, β}. Then C_{D}(Z_{λ}) = Q_{λ} and so C_{D}(Z_{λ})′ = Z_{λ}. Thus D′ = Z_{α′}Z_{β} = Y. Hence, the hypotheses of 2.7 are satisfied, and (a) follows.

□

5.7 Suppose that b = 1. Then G_{δ}/Q_{δ} ≅ Sp_{4}(2) or Sp_{4}(2)′, Q_{δ} = Y_{δ}, and |Y_{δ}| = 2^{4} or 2^{5} for all δ ∈ Γ.

Proof. As α′ ∈ Δ(α) we get from 4.1 together with 2.1 (d) or 2.2 (b) that Q_{αα′}/Q_{α′} is elementary abelian and

[Z_{α}, Z_{α′}] = Z_{α} ∩ Q_{α′} = Z_{αα′}.

In particular Φ(Q_{α}) ≤ Q_{α} ∩Q_{α′} since b = 1, so Φ(Q_{α}) ≤ D_{α} and [D_{α}, Z_{α′}] = 1. Since G_{α} = ⟨Z_{α′}^{G_{α}}⟩Q_{α}, we conclude that

[Q_{α}, O^{p}(G_{α})] ≤ Z_{α} and [D_{α}, O^{p}(G_{α})] = 1.

Now Z(G_{α}) = 1 (4.1 (d)) implies Φ(Q_{α}) = 1 and thus Q_{α} = Y_{α}.

Suppose that \overline{G}_{α} ≅ SL_{3}(q). Then by 4.1 (d) and (e) [Q_{α}, O^{p}(G_{αα′}^{∗})] ≤ Z_{αα′} ≤ Q_{α′} and so by 4.3 [Q_{αα′}/Q_{α′}, O^{p}(G_{αα′}^{∗})] = 1, which contradicts 2.1 (d).

By 4.2 (b) and a symmetric argument Y_{δ} = Q_{δ} and \overline{G}_{δ} ≅ Sp_{4}(q) or Sp_{4}(2)′ for all δ ∈ Γ. Hence Z_{α}Q_{α′}/Q_{α′} is a normal subgroup of order q^{2} in G_{αα′}^{∗}/Q_{α′}, so 2.2 (e) implies q = 2. In particular

(1)  O_{2}(O^{2}(G_{αα′}^{∗})) ≤ Q_{α}Q_{α′}.

It remains to prove that |Q_{δ}| = 2^{4} or 2^{5}.

From 2.2 (d) we get that C_{Z_{α}}(G_{αα′}^{∗}) = 1. In particular, by 4.1 (a), C_{Ω_{1}(Z(T))}(G_{αα′}^{∗}) = 1 for T ∈ Syl_{2}(G_{αα′}^{∗}). It follows that also

(2)  C_{Q_{α}}(O^{2}(G_{αα′}^{∗})) = 1.

Let D := Q_{α} ∩ Q_{α′}. Then Q_{α}Q_{α′} centralizes D and

[D, O^{2}(G_{αα′}^{∗})] ≤ Z_{α} ∩ Q_{α′} ≤ Z_{α} ∩ Z_{α′} = Z_{αα′}.

Moreover, by (1) O^{2}(G_{αα′}^{∗}) acts as a cyclic group of order 3 on D. Thus

D = Z_{αα′}C_{D}(O^{2}(G_{αα′}^{∗})) =^{(2)} Z_{αα′},

and |Q_{δ}/Z_{αα′}| ≤ 2^{3} for δ = α, α′. As |Z_{αα′}| = 4, the edge-transitivity of G on Γ gives |Q_{δ}| = 2^{4} or 2^{5} for every δ ∈ Γ. □

6 The Proof of Theorem 1 and the P̃!-Theorem

Theorem 1 follows from 5.7 and 5.6.

The proof of the P̃!-Theorem: Suppose that there exists P̃_{1} ∈ P_{H}(S) such that P̃_{1} ≰ N_{H}(P^{∘}) and ⟨P̃_{1}, P ⟩ ∈ L_{H}(P). Set M_{1} := ⟨P̃_{1}, P ⟩^{∘}C_{S}(Y_{P}). Then we are allowed to apply the Local P̃-Theorem 3.5 with M_{1} in place of U^{∗}. Observe (with the notation from 3.5) that \overline{U}^{∘} = \overline{U}_{0}, and also \overline{U}^{∘} = \overline{U}^{∗} or \overline{U}^{∘} ≅ Sp_{4}(2)′ and \overline{U}^{∗} ≅ Sp_{4}(2). Hence 3.5 gives (a_{1}) and (a_{2}) for M_{1}.

Suppose now, in addition, that there exists P̃_{2} ∈ P(S) such that P̃_{1} ≠ P̃_{2}, P̃_{2} ≰ N_{H}(P^{∘}) and ⟨P̃_{2}, P ⟩ ∈ L_{H}(P). Set

M_{i} := ⟨P̃_{i}, P ⟩^{∘}C_{S}(Y_{P}) and V_{i} := [Y_{M_{i}}, O^{p}(M_{i})], i = 1, 2

According to 3.6 G = ⟨M_{1}, M_{2}⟩ satisfies the hypothesis of Theorem 1 with respect to M_{1} and M_{2}.

Assume first that we are in case (a) of Theorem 1. Then p = 2 and Y_{M_{2}} ≰ Y_{M_{1}}, and P ∩ M_{1} stabilizes a 2-dimensional singular subspace of V_{1}. Moreover, by 2.4 Y_{M_{2}} does not centralize a hyperplane in V_{1}. It follows that

|V_{2}/V_{1} ∩ V_{2}| = |V_{1} ∩ V_{2}| = 4.

Pick 1 ≠ x ∈ Z(S) ∩ V_{1} ∩ V_{2}. Then 2.2 (g) implies that V_{2} ≰ O_{2}(C_{M_{i}}(x)). On the other hand, Q-uniqueness gives C_{H}(x) ≤ C̃. Hence, also V_{2} ≰ Q, but this contradicts the hypothesis of the P̃!-Theorem.

Assume now that case (b) of Theorem 1 holds. Let L := P^{∘}C_{P} (Y_{P}) and observe that C_{H}(L/O_{p}(L))/O_{p}(L) is a p′-group since P contains a Sylow p-subgroup of H. Then L satisfies the hypothesis of 2.7, and H, L, M_{1} and M_{2} the hypothesis of 2.9. Hence, q = 3 or 5 follows.

References

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[Go] D. Gorenstein, Finite groups, Harper & Row, New York (1968).

[MSS] U. Meierfrankenfeld, B. Stellmacher, G. Stroth, Finite groups of local characteristic p, an overview, in: Proc. Durham Conference, 2001.

[PPS] Ch.W. Parker, G.Parmeggiani, B. Stellmacher, The P!-Theorem, J. Alg. 263 (2003), 17 – 58.

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