The P!-Theorem

Ch.W. Parker, University of Birmingham G. Parmeggiani, University of Padova B. Stellmacher, University of Kiel

Let H be a finite group and p be a prime dividing the order of H. 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 the p-local subgroups are ”known” simple groups^{1}.

Every group with a self-centralizing cyclic Sylow p-subgroup, as for example the alternating group A_{p}, is of local characteristic p, and these groups are particular examples of groups with a strongly p-embedded subgroup. Apart from such groups, all groups of Lie type in characteristic p of rank at least 2 and some sporadic groups (for suitably chosen p) have local characteristic p. Therefore it would be a major contribution to a revision of the classification of the finite simple groups to give a classification of all finite groups of local characteristic p that do not have a strongly p-embedded subgroup. This is the goal of a project initiated by U. Meierfrankenfeld. For an overview of this project see [MSS1].

The part of the project our paper deals with uses the following hypothesis:

Q!-Hypothesis. H is a finite K_{p}-group of local characteristic p, S ∈ Syl_{p}(H) and Z := Ω_{1}(Z(S)). There exists a maximal p-local subgroup C̃ of H with N_{H}(Z) ≤ C̃ such that for Q := O_{p}(C̃)

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

In the subdivision given in [MSS1] this hypothesis refers to the E!-case, see [MSS1, Lemma 2.4.2], and we will prove the P!-Theorem, as it was announced in section 2.4.2 of [MSS1]. To state this result we need some further notation.

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

Throughout this paper S ∈ Syl_{p}(H), and Z, C̃ and Q are as in the above hypothesis. Moreover

C := C_{H}(Z), B(T) := Ω_{1}(Z(J(T))) (T a p-subgroup), X^{0} := ⟨Q^{X}⟩ (X a subgroup).

A subgroup P ≤ H is called minimal parabolic (with respect to p), if P is not p-closed and every Sylow p-subgroup of P is contained in a unique maximal subgroup of P. Let X and M be subgroups of H, and let T be a p-subgroup of H:

Loc_{M}(X) := {U ≤ M | X ≤ U and C_{M}(O_{p}(U)) ≤ O_{p}(U)},

M_{M}(X) is the set of maximal elements of Loc_{M}(X).
L_{M}(T) := {U ∈ Loc_{M}(T) | T ∈ Syl_{p}(U)},
P_{M}(T) := {P ∈ L_{M}(T) | P is minimal parabolic},

According to (1.2) below every element U ∈ Loc_{M}(X) contains a unique maximal elementary abelian normal subgroup Y_{U} satisfying O_{p}(U/C_{U}(Y_{U})) = 1.

Let P ∈ P_{H}(S) and B(P) := ⟨B(S)^{P} ⟩. Then P is said to be of type L_{3}, if p is odd, O_{p}(P) = Y_{P} ≤ B(S), B(P)/Y_{P} ≅ SL_{2}(p^{m}), and Y_{P} is a natural SL_{2}(p^{m})-module for B(P)/Y_{P}.

Hypothesis I. The Q!-Hypothesis holds, and there exists P ∈ P_{H}(S) such that P ≰ C̃ and Y_{M} ≤ Q for every M ∈ M_{H}(P).

In this paper we prove:

P!-Theorem. Assume Hypothesis I. Let P^{∗} := P^{0}O_{p}(P) and Z_{0} := Ω_{1}(Z(S ∩ P^{∗})). Then the following hold:

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

(b) Z_{0} is normal in C̃; in particular P ∩ C̃ is the unique maximal subgroup of P containing S.

(c) Then either P is the unique element of P_{H}(S) not in C̃, or every element of P_{H}(S)\P_{C}(S) is of type L_{3}.

The proof of the P!-Theorem uses the Structure Theorem, which was proved in [MSS2]. To state this result we need some further notation. Let

\overline{L}_{H}(S) := \{U ∈ L_{H}(S) | C_{H}(Y_{U}) ≤ U\}.

For U, Ũ ∈ \overline{L}_{H}(S) define

U << Ũ ⟺ U = (U ∩ Ũ)C_{U}(Y_{U}).

Then (1.5) below shows that << is a partial order on \overline{L}_{H}(S). Let

L_{H}^{∗}(S) = \{L ∈ \overline{L}_{H}(S) | L is maximal with respect to <<\}.

Note that M_{H}(S) ⊆ \overline{L}_{H}(S) and L_{H}^{∗}(S) ⊆ M_{H}(S), if H has local characteristic p.

Structure-Theorem. Assume the Q!-Hypothesis. Suppose that there exists M ∈ L_{H}^{∗}(S) \ \{C̃\} such that Y_{M} ≤ Q. Then for M_{0} := M^{0}C_{S}(Y_{M}) and \overline{M} := M/C_{M}(Y_{M}) one of the following holds:

(a) F^{∗}(\overline{M}) = \overline{M}′_{0}, \overline{M}_{0} ≅ SL_{n}(p^{m}), n ≥ 2, Sp_{2n}(p^{m}), n ≥ 2, or Sp_{4}(2)′ (and p = 2), and [Y_{M}, M_{0}] is the corresponding natural module for \overline{M}_{0}. Moreover, either C_{M_{0}}(Y_{M}) = O_{p}(M_{0}) or p = 2 and M_{0}/O_{p}(M_{0}) ≅ 3Sp_{4}(2)′.

(b) P_{1} := M_{0}S ∈ P_{H}(S), Y_{M} = Y_{P_{1}}, and there exists a a normal subgroup P_{1}^{∗} ≤ P_{1} containing C_{P_{1}}(Y_{P_{1}}) but not Q such that

(i) \overline{P} _{1}^{∗} = K_{1} × ⋯ × K_{r}, K_{i} ≅ SL_{2}(p^{m}), Y_{M} = V_{1} × ⋯ × V_{r}, where V_{i} := [Y_{M}, K_{i}] is a natural K_{i}-module,

(ii) Q permutes the components K_{i} of (i) transitively, (iii) O^{p}(P_{1}^{∗}) = O^{p}(M_{0}), and P_{1}^{∗}C_{M}(Y_{M}) is normal in M, (iv) C_{P_{1}}(Y_{P_{1}}) = O_{p}(P_{1}), or r > 1, K_{i} ≅ SL_{2}(2) (and p = 2) and C_{P_{1}}(Y_{P_{1}})/O_{2}(P_{1}) is a 3-group.

We will refer to property (b) (ii) of the Structure Theorem as Q-transitivity. As a corollary of the Structure- and the P!-Theorem we get:

Corollary. Assume Hypothesis I. Then for every L ∈ Loc_{H}(P) the following hold, where \overline{L} := L/C_{L}(Y_{L}) and L_{0} = L^{0}C_{S}(Y_{L}):

(a) F^{∗}(\overline{L}) = \overline{L}′_{0}, \overline{L}_{0} ≅ SL_{n}(p^{m}), Sp_{2n}(p^{m}) or Sp_{4}(2)′ (and p = 2), and [Y_{L}, L_{0}] is the corresponding natural module.

(b) Either C_{L_{0}}(Y_{L}) = O_{p}(L_{0}), or p = 2, L_{0}/O_{p}(L_{0}) ≅ 3Sp_{4}(2)′ and LC_{H}(Y_{L}) ∈ L_{H}^{∗}(S).

Acknowledgement. We would like to thank A. Chermak for pointing out a mistake in a previous version of Lemma (3.7) and also the referee for his helpful suggestions.

1. Elementary Properties.

(1.1) Let X = S_{k} and V be the non-central irreducible constituent of the GF (2)-permutation module for X.

(a) Let k = 2m + 1 and set t_{i} := (2i − 1, 2i) and d_{i} = (2i − 1, 2i, k), i = 1, ..., m. Then X = ⟨t_{i}, d_{i} | i = 1, ..., m⟩.

(b) Let t be a transposition of X and x ∈ X such that [V, t, x] = 0. Then k = 4 or t^{x} = t.

(c) Let k ≠ 4, t_{1}, ..., t_{m} be a maximal set of commuting transpositions and V_{0} = C_{V} (t_{1}, ..., t_{m}). Then C_{X}(V_{0}) = ⟨t_{1}, ..., t_{m}⟩.

Proof. (a): It is well known that Ω := {(k, k + 1) | k = 1, ..., 2m} is a generating set for X. Thus the claim follows from the fact that

t_{m}^{d_{m}} = (2m, 2m + 1) and t_{i}^{d_{i}d_{i+1}} = (2i, 2i + 1), i = 1, ..., m − 1.

(b): Let W = ⟨v_{1}, ..., v_{k}⟩ be the GF (2)-permutation module for X with basis {v_{1}, ..., v_{k}}, where v_{i}x := v_{ix} for x ∈ X. Set

W_{0} := ⟨∑_{i=1}^{k} v_{i}⟩, W_{1} := ⟨v_{i} + v_{j} | i, j ∈ \{1, ..., k\}⟩ and \overline{W}_{1} := (W_{1} + W_{0})/W_{0}.

Then V = \overline{W}_{1}. Let t = (i, j) and t^{x} = (r, s), so

⟨\overline{v}_{i} + \overline{v}_{j}⟩ = [\overline{W}_{1}, t] = [\overline{W}_{1}, t^{x}] = ⟨\overline{v}_{r} + \overline{v}_{s}⟩.

It follows that v_{i} + v_{j} + v_{r} + v_{s} ∈ W_{0}, and either {i, j} = {r, s} and t = t^{x}, or k = 4. (c): This is a direct consequence of (b).

(1.2) Let U be a finite group of characteristic p, T ∈ Syl_{p}(U) and T ≤ Ũ ≤ U. Then the following hold:

(a) There exists a unique maximal elementary abelian normal p-subgroup Y_{U} of U such that O_{p}(U/C_{U}(Y_{U})) = 1.

(b) Y_{U} ≤ Y_{U}.

(c) Ω_{1}(Z(T)) ≤ Y_{U}.

(d) If U = ŨC_{U}(Y_{U}) then Y_{U} = Y_{U}.

(e) If O_{p}(U) = C_{T} (Y_{U}) then Y_{U} = Ω_{1}(Z(O_{p}(U))).

Proof. (a): Let Ω be the set of all elementary abelian normal p-subgroups X of U satisfying O_{p}(U/C_{U}(X)) = 1. For the existence of a unique maximal element in Ω it suffices to show that the product of two elements of Ω is again in Ω.

Let A_{1}, A_{2} ∈ Ω and A = A_{1}A_{2}. Then A ≤ C_{U}(A_{1})∩C_{U}(A_{2}) and thus A is elementary abelian. Let C_{U}(A) ≤ D ≤ U such that D/C_{U}(A) = O_{p}(U/C_{U}(A)). Then DC_{U}(A_{i})/C_{U}(A_{i}) is a p-group since C_{U}(A) ≤ C_{U}(A_{i}). Hence D ≤ C_{U}(A_{1}) ∩ C_{U}(A_{2}) = C_{U}(A).

(b): Set V = ⟨(Y_{U})^{U}⟩. By the definition of Y_{U}, O_{p}(U) ≤ C_{U}(Y_{U}) and so Y_{U} ≤ Ω_{1}(Z(O_{p}(U))) as U is of characteristic p. Hence also V is in Ω_{1}(Z(O_{p}(U))); i.e. V is elementary abelian.

Let C_{U}(V) ≤ D ≤ U such that D/C_{U}(V) = O_{p}(U/C_{U}(V)). Then

D = (D ∩ T)C_{U}(V) ≤ (D ∩ T)C_{U}(Y_{U}).

Hence O_{p}(Ũ/C_{U}(Y_{U})) = 1 gives T ∩ D ≤ C_{U}(Y_{U}) and thus D = C_{U}(V). Since V is elementary abelian we conclude that V ∈ Ω and thus Y_{U} ≤ V ≤ Y_{U}.

(c): This follows from (b) with Ũ := T. (d): According to (b) it suffices to show that Y_{U} ≤ Y_{U}. But this is clear since U/C_{U}(Y_{U}) ≅

Ũ/C_{U}(Y_{U}) and thus O_{p}(Ũ/C_{U}(Y_{U})) = 1.

(e): Let Y := Ω_{1}(Z(O_{p}(U))). Then Y_{U} ≤ Y by the definition of Y_{U}. Let C_{U}(Y) ≤ D ≤ U such that D/C_{U}(Y) = O_{p}(U/C_{U}(Y)). Since C_{U}(Y) ≤ C_{U}(Y_{U}) we get DC_{U}(Y_{U})/C_{U}(Y_{U}) ≤ O_{p}(U/C_{U}(Y_{U})) = 1, and so D ≤ C_{U}(Y_{U}). It follows that D/O_{p}(U) is a p′-group and O_{p}(U/C_{U}(Y)) = 1, so Y ≤ Y_{U}.

(1.3) Let U be a finite group of characteristic p, T ∈ Syl_{p}(U) and P ∈ P_{U}(T). Then the following hold:

(a) U = ⟨P_{U}(T)⟩N_{U}(T).

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

(c) For every normal subgroup T_{0} of T either T_{0} ≤ O_{p}(P) or O^{p}(P) = [O^{p}(P), T_{0}].

(d) Y_{P} = Ω_{1}(Z(O_{p}(P))) or [Ω_{1}(Z(O_{p}(P))), O^{p}(P)] = 1.

Proof. (a): We proceed by induction on |U|. Set U_{0} = ⟨P_{U}(T)⟩N_{U}(T), and note that N_{U}(T) normalizes ⟨P_{U}(T)⟩, so U_{0} is a subgroup of U. By induction all proper subgroups of U containing T are in U_{0}. If U ≠ U_{0}, then U_{0} is the unique maximal subgroup of U containing T. But then U ∈ P_{U}(T) and thus U = U_{0}, a contradiction.

(b): By the Frattini argument P = N_{P} (N ∩ T)N. As T is in a unique maximal subgroup of P at least one of NT and N_{P} (N ∩ T) is not a proper subgroup of P. This gives (b).

(c): Let P_{0} = [O^{p}(P), T_{0}] and P_{1} = [O^{p}(P), T_{0}]T_{0}. Then P_{1} is normal in P. Hence, by (b) either O^{p}(P) ≤ P_{1} and thus P_{0} ≤ O^{p}(P_{1}) = O^{p}(P) ≤ P_{0}, or T_{0} ≤ O_{p}(P).

(d): If C_{T} (Y_{P}) = O_{p}(P), then Y_{P} = Ω_{1}(Z(O_{p}(P)) follows from (1.2)(e). In the other case (c) gives [Ω_{1}(Z(O_{p}(P))), O^{p}(P)] = 1.

Hypothesis and Notation. For the rest of this section the Q!-Hypothesis holds. We use the notation given in the introduction. For L_{1}, L_{2} ∈ L_{H}(S) we define

L_{1} << L_{2} ⟺ L_{1} = (L_{1} ∩ L_{2})C_{L_{1}}(Y_{L_{1}}).
(1.4) Let L, L̃ ∈ L_{H}(S) such that L << L̃. Then L^{0} ≤ L̃^{0}.
Proof. Note that C_{L}(Y_{L}) ≤ C̃. Hence C_{L}(Y_{L}) normalizes Q and Q^{L} = Q^{L∩_{L}}.
(1.5) << is a partial ordering on \overline{L}_{H}(S).

Proof. By (1.2) L_{1} = (L_{1} ∩ L_{2})C_{L_{1}}(Y_{L_{1}}) implies that Y_{L_{1}} = Y_{L_{1}∩L_{2}} ≤ Y_{L_{2}}. This gives the reflexivity and anti-symmetry. Assume now that L_{1} << L_{2} and L_{2} << L_{3}. Then

L_{1} ∩ L_{2} ≤ (L_{2} ∩ L_{3})C_{L_{2}}(Y_{L_{2}}) and Y_{L_{1}} ≤ Y_{L_{2}}.

It follows that C_{L_{2}}(Y_{L_{2}}) ≤ C_{H}(Y_{L_{1}}) = C_{L_{1}}(Y_{L_{1}}) and thus L_{2} = (L_{2} ∩ L_{3})C_{L_{2}}(Y_{L_{1}}). Hence

L_{1} ∩ L_{2} = (L_{1} ∩ L_{2} ∩ L_{3})C_{L_{2}}(Y_{L_{1}}).

This shows L_{1} = (L_{1} ∩ L_{3})C_{L_{1}}(Y_{L_{1}}) and the transitivity of <<.

(1.6) Every p-subgroup of H contains at most one conjugate of Q; in particular Q is the only conjugate in C̃.

Proof. Let g ∈ H and Q^{g} ≤ S. It suffices to show that Q^{g} = Q. As Z ≤ C_{C^{g}}(Q^{g}) = Z(Q^{g}), Q-Uniqueness shows that S ≤ C̃^{g}, so S ≤ C̃ ∩ C̃^{g}. Now Sylow’s Theorem shows that C̃ and C̃^{g} are conjugate by an element of N_{H}(S). As by the definition of C̃, N_{H}(S) ≤ N_{H}(Z) ≤ C̃ we conclude that C̃ = C̃^{g} and thus also Q = Q^{g}.

(1.7) Let P be a subgroup of H with Q ≤ O_{p}(P). Then P ≤ C̃.

Proof. This is a direct consequence of (1.6).

2. Pushing Up

Hypothesis and Notation. In this section the Q!-Hypothesis holds. In addition, P ≤ H is a minimal parabolic subgroup of characteristic p and T ∈ Syl_{p}(P). We set \overline{P} := P/C_{P} (Y_{P}) and

B(T) := C_{T} (Ω_{1}(Z(J(T)))) and Z_{0} := Ω_{1}(Z(J(T))),
U(P) := \{A | A ≤ P, \overline{A} an elem. abelian p-group, and |A/C_{A}(Y_{P})| ≥ |Y_{P} /C_{Y_{P}} (A)|\},
U(P) := ⟨A | A ∈ U(P)⟩ and B(P) := ⟨B(T)^{P} ⟩.

Moreover K(P) denotes the set of all B(T)-invariant subgroups K ≤ P satisfying:

(i) \overline{K} is normal in \overline{U(P)}, (ii) L := KB(T) is minimal parabolic of characteristic p and O_{p}(P) ≤ T ∩ L ∈ Syl_{p}(L), (iii) \overline{K} ≅ SL_{2}(p^{m}) and [Y_{P} , K]/C_{[Y_{P} ,K]}(K) is a natural SL_{2}(p^{m})-module for \overline{K}, or p = 2,

\overline{K} ≅ S_{2^{n}+1} and [Y_{P} , K] is a natural S_{2^{n}+1}-module for \overline{K}.

Note that trivially C_{P} (Y_{P}) ∈ U(P) and so C_{P} (Y_{P}) ≤ U(P). Then recall from (1.3) that either U(P) = C_{P} (Y_{P}) or P = U(P)T and similarly B(P) = B(T) ≤ O_{p}(P) or P = B(P)T.

Let K = SL_{2}(p^{m}) and V be an irreducible GF (p)K-module. Set F := End_{K}(V). By Schur’s Lemma, F is a finite field, so V is an F K-module. We say that V is a natural SL_{2}(p^{m})-module for K if dim_{F} (V) = 2.

(2.1) Suppose that \overline{U(P)} ≠ 1 and A ∈ U(P). Then there exist subgroups U_{1}, ..., U_{r} of U(P) such that the following hold:

(a) \overline{U(P)} = \overline{U}_{1} × ⋯ × \overline{U}_{r}, \overline{U}_{i} ≅ SL_{2}(p^{m}) or S_{2^{n}+1} (and p = 2).

(b) Either [Y_{P} , U_{i}]/C_{[Y_{P} ,U_{i}]}(U(P)) is a natural SL_{2}(p^{m})-module for \overline{U}_{i}, or [Y_{P} , U_{i}] is a natural S_{2^{n}+1}-module for \overline{U}_{i}, i = 1, ..., r.

(c) Y_{P} = C_{Y_{P}} (U(P)) ∏_{i=1}^{r}[Y_{P} , U_{i}] and [Y_{P} , U_{i}, U_{j}] = 1 for i ≠ j.

(d) \overline{T} acts transitively on \{\overline{U}_{1}, ..., \overline{U}_{r}\}.

(e) [Y_{P} , A, A] = 1 and |\overline{A}| = |Y_{P} /C_{Y_{P}} (A)|. In particular |E| ≤ |Y_{P} /C_{Y_{P}} (E)| for every elementary abelian p-group E ≤ \overline{P}.

(f) \overline{A} = \overline{A ∩ U_{1}} × ⋯ × \overline{A ∩ U_{r}} and A ∩ U_{i}C_{P} (Y_{P}) ∈ U(P), i = 1, ..., r.

(g) \overline{A ∩ U_{i}} ∈ Syl_{p}(\overline{U}_{i}) if \overline{U}_{i} ≅ SL_{2}(p^{m}) and \overline{A ∩ U_{i}} ≠ 1.

(h) \overline{A ∩ U_{i}} is generated by a set of commuting transpositions if \overline{U}_{i} ≅ S_{2^{n}+1}.

Proof. See [Cher].

(2.2) A(T) ⊆ U(P) and \overline{J(T)} = \overline{B(T)} ≤ \overline{U(P)}.

Proof. Assume that J(T) ≤ C_{P} (Y_{P}). Then clearly A(T) ⊆ U(P) and Y_{P} ≤ Z_{0}; in particular B(T) ≤ C_{P} (Y_{P}) and 1 = \overline{J(T)} = \overline{B(T)} ≤ \overline{U(P)}.

Assume now that J(T) ≰ C_{P} (Y_{P}). Let A ∈ A(T) such that \overline{A} ≠ 1. The maximality of A gives C_{Y_{P}} (A) = A ∩ Y_{P}. Hence

|C_{A}(Y_{P})||Y_{P} ||C_{Y_{P}} (A)|^{−1} = |C_{A}(Y_{P})||Y_{P} ||A ∩ Y_{P} |^{−1} = |C_{A}(Y_{P})Y_{P} | ≤ |A|

and A ∈ U(P); in particular \overline{J(T)} ≤ \overline{U(P)} ≠ 1.

We now use the notation given in (2.1). In addition we set Y_{i} := [Y_{P} , U_{i}] and Ỹ_{P} := Y_{P} /C_{Y_{P}} (U(P)). Then (2.1)(c) implies

(∗) Ỹ_{P} = Ỹ_{1} × ⋯ × Ỹ_{r} and [Y_{i}, U_{j}] = 1 for i ≠ j.

Assume first that \overline{U}_{i} ≅ SL_{2}(p^{m}). Then (2.1)(f) and (g) show that \overline{J(T)} ∈ Syl_{p}(\overline{U(P)}), and (2.1)(b),

(e) and (f) that [Y_{i}, J(T)] ≤ Y_{i} ∩ Z_{0} and |Y_{i}/Y_{i} ∩ Z_{0}| = p^{m}; in particular Y_{i} ∩ Z_{0} ≰ C_{Y_{i}}(U_{i}). As B(T) centralizes Y_{i} ∩ Z_{0}, we get from (∗) that \overline{B(T)} ≤ N\overline{P} (\overline{U}_{i}).

Let F := End_{\overline{U}_{i}}(Ỹ_{i}). Then the elements of N\overline{P} (\overline{U}_{i}) induce field automorphisms on F and semi-linear transformations on Ỹ_{i}. As Ỹ_{i} ∩ Z_{0} is a 1-dimensional F -subspace centralized by B(T), we conclude that the elements of B(T) act F -linear on Ỹ_{i}, so \overline{B(T)} ≤ (\overline{J(T)}∩\overline{U}_{i})C\overline{P} (\overline{U}_{i}) by (2.1)(g). It follows that \overline{B(T)} ≤ \overline{J(T)} since C\overline{P} (\overline{U(P)}) ≤ \overline{U(P)}, whence \overline{B(T)} = \overline{J(T)}.

Assume now that \overline{U}_{i} ≅ S_{2^{n}+1}. Recall that any two transpositions of S_{m} commute if they generate a 2-group. Hence, by (2.1)(h) \overline{J(T)} ∩ \overline{U}_{i} is generated by a maximal set of commuting transpositions, and as above, by (2.1)(e) and (f) [Y_{i}, J(T)] ≤ Y_{i} ∩ Z_{0} and \overline{B(T)} ≤ N\overline{P} (\overline{U}_{i}). Now (1.1)(c) shows that \overline{B(T)} ≤ (\overline{J(T)} ∩ \overline{U}_{i})C\overline{P} (\overline{U}_{i}) and, again as above, \overline{B(T)} = \overline{J(T)}.

(2.3) Suppose that \overline{U(P)} ≠ 1. Then K(P) ≠ ∅, and for every K ∈ K(P) and L := KB(T):

(a) U(L)/C_{L}(Y_{L}) ≠ 1; i.e. L satisfies the hypothesis of (2.1).
(b) Y_{L} ≤ Y_{P} and [Y_{L}, K] = [Y_{P} , K].
(c) B(T) ≤ O_{p}(P) or L = [K, B(T)](T ∩ L).

(d) There exists U_{i} as in (2.1) such that \overline{K} = \overline{U}_{i}.

Proof. We first show that K(P) ≠ ∅. Let U_{1}, ..., U_{r} be as in (2.1) and fix U ∈ {U_{1}, ..., U_{r}}. By (2.1) and (2.2) \overline{J(T)} = \overline{B(T)} ≤ N\overline{P} (\overline{U}) and B(T) ≤ N_{P} (UC_{P} (Y_{P})). Among all subgroups K_{0} ≤ UC_{P} (Y_{P}), which are B(T)-invariant and satisfy

(∗) \overline{K}_{0} = \overline{U} and O_{p}(P) ≤ T ∩ K_{0}B(T) ∈ Syl_{p}(K_{0}B(T)), we choose K minimal and set L = KB(T). According to (2.1)(a) there exists C_{L}(Y_{P})(T ∩L) ≤ L_{0} ≤ L such that \overline{L}_{0} is the unique maximal subgroup of \overline{L} containing \overline{T ∩ L}. Hence, the minimality of K implies that L_{0} is the unique maximal subgroup of L containing T ∩ L, so L is minimal parabolic. Moreover, L is of characteristic p since O_{p}(P) ≤ O_{p}(L). This shows that K ∈ K(P).

Now let K ∈ K(P). Then (d) follows from (2.1)(a). Let L = KB(T). From (1.3)(d) we get Ω_{1}(Z(O_{p}(L))) = Y_{L} ≤ Ω_{1}(Z(O_{p}(P))) = Y_{P}, so Y_{L} = C_{Y_{P}} (O_{p}(L)). Since [\overline{K}, \overline{O_{p}(L)}] = 1 the P × Q-Lemma gives [Y_{L}, K] ≠ 1 and thus by (2.1)(b) [Y_{L}, K] = [Y_{P} , K]. This is (b).

From (1.3)(c) we get either L = [K, B(T)](T ∩ L) or B(T) ≤ O_{p}(L). In the latter case [\overline{K}, \overline{B(T)}] = 1, and (2.1)(d) implies B(T) ≤ C_{T} (Y_{P}). This shows (c) since C_{T} (Y_{P}) = O_{p}(P) by (1.3)(c).

According to (2.1)(d) and (f) there exists A ∈ U(P) such that \overline{A} ≠ 1 and \overline{A} ≤ \overline{T ∩ K}. Since C_{T} (Y_{P}) = O_{p}(P) ≤ L and \overline{A} is a p-group we may assume that A ≤ T ∩ L. Set A_{0} = C_{A}(Y_{L}). By (2.1)(e)

|\overline{A}_{0}| ≤ |Y_{P} /C_{Y_{P}} (A_{0})| ≤ |Y_{P} /Y_{L}C_{Y_{P}} (A)| = |Y_{P} /C_{Y_{P}} (A)||Y_{L}/C_{Y_{L}}(A)|^{−1} = |\overline{A}||Y_{L}/C_{Y_{L}}(A)|^{−1}

and |Y_{L}/C_{Y_{L}}(A)| ≤ |A/A_{0}|. It follows that U(L) ≠ C_{L}(Y_{L}), and (a) holds.

(2.4) Suppose that \overline{U(P)} ≠ 1. Let A ∈ U(P) and A_{1} ≤ P such that [Y_{P} , A, A_{1}] = 1. Then

[Y_{P} , A_{1}] ≤ [Y_{P} , A][C_{Y_{P}} (A), A_{1}].

Proof. We apply (2.1) and choose the subgroups U_{1}, ..., U_{r} as in (2.1). Let V_{i} := [Y_{P} , U_{i}]. By (2.1)(c)

[Y_{P} , A_{1}] = [C_{Y_{P}} (A), A_{1}] ∏_{i=1}^{r}[V_{i}, A_{1}].

Hence, it suffices to show that (∗) [V_{i}, A_{1}] ≤ [V_{i}, A][C_{Y_{P}} (A), A_{1}]. If \overline{A ∩ U_{i}} = 1, then by (2.1)(c) and (f) V_{i} ≤ C_{Y_{P}} (A), and (∗) is obvious. Hence, we may assume

that \overline{A ∩ U_{i}} ≠ 1. Then [V_{i}, A, A_{1}] = 1 shows that A_{1} normalizes U_{i} and V_{i}.

Assume first that \overline{U}_{i} ≅ SL_{2}(p^{m}). By (2.1)(g) \overline{A ∩ U_{i}} ∈ Syl_{p}(\overline{U}_{i}), so [V_{i}, A, A_{1}] = 1 implies A_{1} ≤ AC_{P} (V_{i}), and (∗) follows.

Assume now that \overline{U}_{i} ≅ S_{2^{n}+1}. By (2.1)(h) \overline{A ∩ U_{i}} = ⟨t_{1}, ..., t_{s}⟩, t_{1}, ..., t_{s} commuting transpositions of S_{2^{n}+1}; in particular

C_{\overline{U}_{i}}(\overline{A}) = C_{\overline{U}_{i}}(\overline{A ∩ U_{i}}) = ⟨t_{1}, ..., t_{s}⟩ × X, X ≅ S_{2^{n}+1−2s} and [V_{i}, X] = [C_{V_{i}}(A), X].

Since [V_{i}, t_{j}, A_{1}] = 1 for j = 1, ..., s we get \overline{A}_{1} ≤ C_{\overline{U}_{i}}(\overline{A})C\overline{P} (V_{i}). Hence,

[V_{i}, A_{1}] ≤ [V_{i}, A][C_{V_{i}}(A), A_{1}] ≤ [V_{i}, A][C_{Y_{P}} (A), A_{1}],

and again (∗) follows.

(2.5) Suppose that T = S, \overline{U(P)} ≠ 1 and P ≰ C̃. Let K ∈ K(P). Then the following hold:

(a) Z(P) = Z(U(P)) = 1.

(b) Y_{P} = ×_{\{\overline{K}|K∈K(P)\}}[Y_{P} , \overline{K}], and [Y_{P} , \overline{K}] is an natural \overline{K}-module.

(c) Q acts transitively on \{\overline{K} | K ∈ K(P)\}.

(d) \overline{K} ≅ SL_{2}(p^{m}) or p = 2 and \overline{K} = \overline{U(P)} ≅ S_{5}.

(e) If \overline{K} ≅ SL_{2}(p^{m}) and A ≤ P with [Y_{P} , A, A] = 1, then [Y_{P} , K, A] = [Y_{P} , K, a] for all a ∈ A \ C_{P} ([Y_{P} , K]). Moreover, either |A/C_{A}([Y_{P} , K])| = 2 (= p) or \overline{A} ≤ \overline{K}C\overline{A}(\overline{K}).

Proof. (a): It suffices to show that C_{Y_{P}} (U(P)) = 1 since Ω_{1}(Z(P)) ≤ Y_{P}. If C_{Y_{P}} (U(P)) ≠ 1, then there exists 1 ≠ x ∈ C_{Y_{P}} (U(P)) ∩ Z(Q), and by Q-Uniqueness U(P) ≤ C_{H}(x) ≤ C̃. Since also S ≤ C̃ we get that P = U(P)S ≤ C̃, a contradiction.

(b): This follows from (a) and (2.1)(c). (c): By (b) and (2.1)(c),(d) together with (2.3)(d)

Y_{P} = [Y_{P} , K_{1}] × ⋯ × [Y_{P} , K_{r}],

where K_{i} ∈ K(P) and Ω := \{\overline{K} | K ∈ K(P)\} = \{\overline{K}_{1}, ..., \overline{K}_{r}\}. Assume that Q is not transitive on Ω. Then there exist 1 ≠ x ∈ Z(Q) ∩ Y_{P} and K_{i} ∈ {K_{1}, ..., K_{r}} such that [K_{i}, x] = 1. Again by Q-Uniqueness K_{i} ≤ C̃ and thus P = ⟨K_{i}, S⟩ ≤ C̃, a contradiction.

(d): We use (2.1) and (2.3)(d). Assume that \overline{K} ≅ S_{2^{n}+1}, n ≥ 2 (and p = 2). The action of U(P) on Y_{P} shows that there exists 1 ≠ x ∈ Z(Q) ∩ Y_{P} such that \overline{C_{K}(x)} ≅ S_{2^{n}}. On the other hand by Q-Uniqueness C_{H}(x) ≤ C̃ and thus [C_{K}(x), Q] ≤ Q. Since S_{2^{n}} is not a 2-group we get \overline{K}^{Q} = \overline{K}, and \overline{P} ≅ S_{2^{n}+1} follows with (c). Moreover \overline{Q} is a normal 2-subgroup of \overline{C_{K}(x)}.

If n = 2, then (d) follows. In the other cases \overline{Q} = 1 and thus Q ≤ C_{S}(Y_{P}) = O_{2}(P). But this contradicts (1.7).

(e): By (b) V := [Y_{P} , K] is a natural SL_{2}(p^{m})-module for \overline{K}. Assume first that V^{A} = V. Then again (b) implies that \overline{K}\overline{A} = \overline{K}. Since V is a faithful irreducible \overline{K}-module we conclude that C\overline{A}(\overline{K}) = C\overline{A}(V).

Let V_{0} := [V, A] and F := End\overline{K}(V). Recall that the elements of \overline{A} induce semi-linear transformations on the F -vector space V. Thus, if V_{0} contains a 1-dimensional F -subspace, then \overline{A} ≤ \overline{K}C\overline{P} (\overline{K}). In the other case no element of \overline{A}^{♯} induces an F -linear transformation on V. As ΓL(V)/GL(V) has cyclic Sylow p-subgroups, we get in this case that |A/C_{A}(V)| = p. Moreover, the quadratic action of A on V shows that the elements of A^{♯} induce field automorphisms of order 2 in F, so p = 2.

Assume now that V^{A} ≠ V. Then the quadratic action of A gives

⟨V^{A}⟩ = V × V^{a} for a ∈ A \ N_{A}(V);

in particular |A/N_{A}(K)| = p (= 2). Since

[V, N_{A}(K)] ≤ C_{V} (A) ≤ C_{V} (a) = 1

we get N_{A}(K) ≤ C_{A}(V) and |A/C_{A}(V)| = p. Now again (e) is obvious.

(2.6) Suppose that neither Ω_{1}(Z(T)) nor B(T) is normal in P. Then \overline{B(P)} = \overline{U(P)} ≠ 1 and \overline{B(T)} = \overline{J(T)} ≠ 1.

Proof. According to (1.3) C_{T} (Y_{P}) = O_{p}(P) since Ω_{1}(Z(T)) is not normal in P. Hence B(T) ≰ C_{P} (Y_{P}) since also B(T) is not normal in P. It follows with (2.2) that \overline{B(T)} = \overline{J(T)} ≤ \overline{U(P)} ≠ 1, and (2.1) gives \overline{B(P)} = \overline{U(P)}.

(2.7) Suppose that neither Ω_{1}(Z(T)) nor B(T) is normal in P. Then Z_{0} ≤ Ω_{1}(Z(J(O_{p}(P)))) and

[Ω_{1}(Z(J(O_{p}(P)))), J(T)] ≤ Z_{0} ∩ Y_{P};

in particular [Ω_{1}(Z(J(O_{p}(P)))), O^{2}(P)] ≤ Y_{P}. Moreover, if in addition \overline{K} ≅ SL_{2}(p^{m}) for K ∈ K(P), then B(T) ∈ Syl_{p}(O^{p}(K)B(T)).

Proof. By (2.6) \overline{U(P)} ≠ 1 and \overline{J(T)} = \overline{B(T)} ≠ 1. Let A ∈ A(T) such that \overline{A} ≠ 1 and Z_{1} := Ω_{1}(Z(J(O_{p}(P)))). Then by (2.1) [Y_{P} , A] ≤ C_{Y_{P}} (J(T)) ≤ Z_{0}, and (2.1)(e) gives Y_{P} C_{A}(Y_{P}) ∈ A(T). This shows that

Y_{P} C_{A}(Y_{P}) ∈ A(O_{p}(P)) ⊆ A(T).

Hence Z_{1} ≤ Y_{P} C_{A}(Y_{P}) and Z_{0} ≤ Z_{1}. It follows that [Z_{1}, A] ≤ Y_{P} ∩Z_{0} and thus [Z_{1}, J(T)] ≤ Y_{P} ∩Z_{0}. Since O^{p}(P) ≤ ⟨J(T)^{P} ⟩ by (1.3) we get [Z_{1}, O^{p}(P)] ≤ Y_{P}.

Assume now that \overline{K} ≅ SL_{2}(p^{m}), where K ∈ K(P). By (2.2) and (2.1)(d), (g) we can choose A such that \overline{A} ∩ \overline{K} ∈ Syl_{p}(\overline{K}); in particular

⟨\overline{A} ∩ \overline{K}, (\overline{A} ∩ \overline{K})\overline{g}⟩ = \overline{K} for some g ∈ K.

Set L = KB(T), W = [Y_{L}, K], Z_{0}^{∗} := Z_{0} ∩ Z_{0}^{g} and L_{0} = C_{L}(Z_{0}^{∗}). Then B(T) ≤ L_{0} and L = L_{0}C_{L}(Y_{P}). Since L is minimal parabolic and by (1.3) C_{T} (Y_{P}) = O_{p}(P) we get

(1) L = L_{0}O_{p}(P), and L_{0} is normal in L.

By (2.3) L satisfies the hypothesis of (2.1), and W = [Y_{P} , K]. As [Z_{0}, K] = [Z_{0}, K, K] ≤ W, Z_{0}W is normal in L, and (2.1)(b),(g), applied to L, gives Z_{0}W = Z_{0}Z_{0}^{g}, C_{W} (T ∩ L) = W ∩ Z_{0} and |W Z_{0}/Z_{0}| = p^{m}; in particular Z_{0}^{∗} ∩ W = C_{W} (L). It follows that

|Z_{0}^{∗}W/Z_{0}^{∗}| = |W/W ∩ Z_{0}^{∗}| = p^{2m} and |Z_{0}W/Z_{0}^{∗}| = |Z_{0}Z_{0}^{g}/Z_{0}^{∗}| ≤ p^{2m}.

This shows that Z_{0}^{∗}W = Z_{0}W and Z_{0} = Z_{0}^{∗}C_{W} (T ∩ L); in particular (2) B(T) = C_{T ∩L}(Z_{0}) = C_{T ∩L}(Z_{0}^{∗}). By (1) and (2) B(T) ∈ Syl_{p}(L_{0}) and O^{p}(K) ≤ O^{p}(L) ≤ L_{0}, so B(T) ∈ Syl_{p}(O^{p}(K)B(T)).

(2.8) Suppose that neither B(T) nor Ω_{1}(Z(T)) is normal in P and Z(P) = 1. Then O_{p}(P) ≤ B(T).

Proof. By (2.7) Z_{0}Y_{P} is normal in P. Hence, R := [Z_{0}Y_{P} , O_{p}(P)] is a normal subgroup of P in Z_{0}. But then by (2.6) and (1.3) O^{p}(P) centralizes R, and Z(P) = 1 implies R = 1. This gives O_{p}(P) ≤ B(T).

(2.9) Suppose that neither B(T) nor Ω_{1}(Z(T)) is normal in P. Then there exist subgroups L_{1}, ..., L_{k} ≤ P such that for i = 1, ..., k and L̂_{i} = L_{i}/C_{L_{i}}(Y_{L_{i}}):

(a) L_{i} is minimal parabolic of characteristic p and O_{p}(P)B(T) ∈ Syl_{p}(L_{i}).

(b) L̂_{i} ≅ SL_{2}(p^{m}), and Y_{L_{i}}/C_{Y_{L_{i}}} (L_{i}) is a natural SL_{2}(p^{m})-module for L̂_{i}.
(c) [Y_{L_{i}}, O^{p}(L_{i})] = [Y_{P} , O^{p}(L_{i})].
(d) L_{1}, ..., L_{k} are conjugate under T, ⟨L_{1}, ..., L_{k}⟩T = P, and ∩_{i=1}^{k}O_{p}(L_{i}) = O_{p}(P).
(e) [Y_{P} , B(P)] ∩ Z_{0} = ∏_{i=1}^{k}[Y_{L_{i}}, B(T)] and [Y_{L_{i}}, B(T), L_{j}] = 1 for i ≠ j.

Proof. By (2.6) \overline{U(P)} ≠ 1, and we are allowed to apply (2.1) and (2.3) to P. Let K ∈ K(P), and set L = KB(T) and L̂ = L/C_{L}(Y_{L}). Then (2.3) shows that L satisfies (2.1) and [Y_{L}, O^{p}(L)] = [Y_{P} , O^{p}(L)].

Assume first that \overline{K} ≅ SL_{2}(p^{m}). Then (2.1)(f),(g) gives

\overline{L} = \overline{K} × C\overline{B(T)}(\overline{K}) and \overline{B(T)} ∩ \overline{K} ∈ Syl_{p}(\overline{K});

in particular O_{p}(P)B(T) ∈ Syl_{p}(L) and [O_{p}(L), O^{p}(L)] ≤ O_{p}(P). Now (a) – (d) follow for k = 1, and (e) is a consequence of (2.1)(b).

Assume now that \overline{K} ≅ S_{2^{n}+1} (and p = 2). Then \overline{K ∩ B(T)} is generated by a maximal set \{\overline{t}_{1}, ..., \overline{t}_{2^{n−1}}\} of transpositions, where t_{1}, ..., t_{2^{n−1}} ∈ K. For every t_{i} there exists d_{i} ∈ K such that \overline{d}_{i} has order 3 and

⟨\overline{d}_{i}, \overline{K ∩ B(T)}⟩ = ⟨\overline{d}_{i}, \overline{t}_{i}⟩ × ⟨\overline{t}_{j} | i ≠ j⟩ and ⟨\overline{d}_{i}, \overline{t}_{i}⟩ ≅ SL_{2}(2).

Note that the subgroups ⟨\overline{d}_{i}, \overline{t}_{i}⟩, i = 1, ..., 2^{n−1}, are conjugate under \overline{T ∩ K} and that by (1.1)

⟨\overline{d}_{i}, \overline{t}_{i} | i = 1, ..., 2^{n−1}⟩ = \overline{K}.

Note further that by (2.1)(b)

(∗) [Y_{P} , K] ∩ Z_{0} = [Y_{P} , ⟨t_{1}, ..., t_{2^{n−1}}⟩] and [Y_{P} , t_{i}, d_{j}] = 1 for i ≠ j.

We now choose L_{1} ≤ ⟨d_{1}, B(T)⟩ minimal with respect to

O_{2}(P)B(T) ≤ T_{1} := T ∩ L_{1} ∈ Syl_{2}(L_{1}) and \overline{L}_{1} = ⟨\overline{d}_{1}, \overline{B(T)}⟩.

Then L_{1} is a minimal parabolic subgroup of characteristic 2. Moreover O_{2}(P)B(T) = T_{1} and [O_{2}(L_{1}), O^{2}(L_{2})] ≤ O_{2}(P), and (a) follows for L_{1}. Since Y_{L_{1}} ≤ Ω_{1}(Z(O_{2}(L_{1}))) ≤ Ω_{1}(Z(O_{2}(P))) we get from (1.3)(d) that Y_{L_{1}} ≤ Y_{P}. It follows that |[Y_{L_{1}}, L_{1}]| = 4, and (b) and (c) hold for L_{1} since O^{2}(\overline{L}_{1}) ≅ C_{3}.

Finally, for every i ∈ {1, ..., 2^{n−1}} there exists a T -conjugate L_{i} of L_{1} with d_{i} ∈ L_{i}, and ⟨\overline{L}_{1}, ..., \overline{L}_{2^{n−1}}⟩\overline{B(T)} = \overline{L}. Since L is minimal parabolic we get ⟨L_{1}, ..., L_{2^{n−1}}⟩B(T) = L. Similarly, since P is minimal parabolic (2.1)(d) and (2.3)(d) imply (d); and (e) follows from (d) and (∗).

Notation. Let

P_{0} := P_{H}(S) \ (P_{N_{H}(B(S))}(S) ∪ P_{C}(S)) and P_{0}^{∗} := {P^{g} | P ∈ P_{0}, g ∈ N_{H}(B(S))},

and let P be the set of all subgroups X ≤ H satisfying:

(i) X is minimal parabolic of characteristic p and B(S) ∈ Syl_{p}(X),

(ii) ⟨X, S⟩ = P for some P ∈ P_{0},

(iii) X/C_{X}(Y_{X}) ≅ SL_{2}(p^{m}) and Y_{X}/C_{Y_{X}} (X) is a natural SL_{2}(p^{m})-module for X/C_{X}(Y_{X}). Let P^{∗} := {X^{g} | X ∈ P, g ∈ N_{H}(B(S))}, G := ⟨X | X ∈ P^{∗}⟩ and L := GN_{H}(B(S)).

Theorem 1. One of the following holds:

(a) L ∈ L_{H}(S) and P_{H}(S) = P_{L}(S) ∪ P_{C}(S).

(b) P_{H}(S) = P_{N_{H}(B(S))}(S) ∪ P_{C}(S).

(c) O_{p}(P) = Y_{P} and Z(P) = 1 for every P ∈ P^{∗}.

Proof. We may assume that neither (a) nor (b) holds. Then P_{0} ≠ ∅ ≠ P_{0}^{∗}. Let P^{∗} ∈ P_{0}^{∗} and set Z_{0} := Ω_{1}(Z(B(S))) .

(1) P^{∗} satisfies the hypotheses of (2.1), (2.8) and (2.9), and, after a suitable conjugation, also that of (2.5).

By the definition of P_{0}^{∗} there is P_{0} ∈ P_{0} and g ∈ N_{H}(B(S)) such that P_{0}^{g} = P^{∗}. Hence, it suffices to show the claim for P_{0}.

From the choice of P_{0} and the definition of C̃ follows that neither B(S) nor Z is normal in P_{0}. Hence, P_{0} satisfies the hypotheses of (2.6) and (2.9), and by (2.6) also those of (2.1) and (2.5). Finally, by (2.5) P_{0} satisfies the hypothesis of (2.8).

(2) Z(P^{∗}) = 1 and O_{p}(P^{∗}) ≤ B(S). This follows from (1), (2.5) and (2.8). Let P_{0} ∈ P_{0}. According to (2.9) and (2) there exists a subset

Ω(P_{0}) := {L_{1}, ..., L_{k}} ⊆ P

such that the subgroups L_{1}, ..., L_{k} satisfy (2.9)(a) – (e) (with respect to P_{0} and S). We fix this notation. From (2), (2.1)(c) and (2.9)(e) we get

(3) Z_{0} = ∏_{i=1}^{k}[Y_{L_{i}}, B(S)].

Next we prove:

(4) L = ⟨N_{H}(B(S)), P_{0} | P_{0} ∈ P_{0}⟩.

Let L̃ := ⟨N_{H}(B(S)), P_{0} | P_{0} ∈ P_{0}⟩. By the definition of P^{∗} we have L ≤ L̃. On the other hand, for P_{0} ∈ P_{0} by (2.9)(d) P_{0} ≤ GS and so also L̃ ≤ L.

(5) O_{p}(G) = 1 = O_{p}(L).

From (4) we get

P_{H}(S) = P_{L}(S) ∪ P_{C}(S).

Hence, O_{p}(L) = 1 since (a) does not hold. As G is normal in L we also have O_{p}(G) = 1.

In the following let

Δ^{∗} := ∪_{P_{0}∈P_{0}}Ω(P_{0}).

We now apply the amalgam method to G with respect to the subgroups in P^{∗} and use the standard notation, see for example [DS] or [KS]. For the convenience of the reader we repeat some of the notation:

Γ = {P x | x ∈ G, P ∈ P^{∗}} is the set of vertices, and two vertices are adjacent, if they are different and have non-empty intersection. P^{∗} is a (maximal) set of pairwise adjacent vertices (where the elements of P^{∗} are understood as cosets), and every pair of adjacent vertices is conjugate (under G) to a pair of vertices from P^{∗}. For a vertex δ ∈ Γ the stabilizer of δ in G is denoted by G_{δ}. Moreover

Q_{δ} = O_{p}(G_{δ}) and Z_{δ} = ⟨Ω_{1}(Z(X)) | X ∈ Syl_{p}(G_{δ})⟩.

A critical pair (δ, δ′) of vertices satisfies Z_{δ} ≰ Q_{δ} with the distance d(δ, δ′) being minimal. This distance is denoted by b.

Note that by (2.9)(b) Z_{δ} = Y_{G_{δ}} for every δ ∈ Γ. Since by (1.3)(b) C_{B(S)}(Y_{P}) = O_{p}(P) for every P ∈ P^{∗} we get from (2.1)(g):

(6) Z_{α}Q_{α′} ∈ Syl_{p}(G_{α′} ∩ G_{α′−1}) and Z_{α′}Q_{α} ∈ Syl_{p}(G_{α} ∩ G_{α+1}) for every critical pair (α, α′). Let (α, α′) be a critical pair with G_{α} ∈ P^{∗}. Then there exists T_{1} ∈ Syl_{p}(G_{α}) such that

G_{α} = ⟨T_{1}, Z_{α′}⟩. Thus, possibly after conjugation in G_{α}, we may assume

(∗) (α, α′) is a critical pair such that G_{α} ∈ P^{∗} and G_{α} = ⟨B(S), Z_{α′}⟩.

In the steps (7), (8) and (9) below (α, α′) is a critical pair satisfying (∗). Further we set R_{ρ} := [Z_{ρ}, Q_{α}] for every ρ ∈ P^{∗}. Note that by (2.1)(e) and (g) R_{ρ} ≤ Z(B(S)). We first show:

(7) Let ρ ∈ P^{∗} and b > 1 or Z_{ρ} ≤ Q_{α′−1}. Then R_{ρ} ≤ Z(G_{α}).

Assume first that Z_{ρ} ≤ Q_{α′−1}. Then by (6) Z_{ρ} ≤ Z_{α}Q_{α′} and
[Z_{ρ}, Z_{α′}] ≤ [Z_{α}, Z_{α′}] ≤ Z_{α}.

Hence, Z_{ρ}Z_{α} is normal in ⟨B(S), Z_{α′}⟩ = G_{α}; so also [Z_{ρ}, Q_{α}] = R_{ρ} is normal in G_{α}. Since R_{ρ} ≤ Z(B(S)) we get R_{ρ} ≤ Z(G_{α}).

Assume now that Z_{ρ} ≰ Q_{α′−1}. Then (ρ, α′ − 1) is a critical pair, and (6) gives [Z_{ρ}, Z_{α′−1}] = [Z_{ρ}, Q_{α}] = R_{ρ}. If b > 1, then R_{ρ} is centralized by ⟨B(S), Z_{α′}⟩ = G_{α}.

Next we show:

(8) Let ρ ∈ P^{∗}. Suppose that b > 1 or Z_{ρ} ≤ Q_{α′−1}. Then either Q_{α} = Q_{ρ} or Q_{α}Q_{ρ} = B(S). Let T := Q_{α}Q_{ρ}. Assume that Q_{α} ≤ Q_{ρ} but Q_{α} ≠ Q_{ρ}. Then the action of G_{α} on Z_{α} shows

that

Z_{ρ} ≤ C_{Z_{α}}(T) = Z_{0},

so B(S) ≤ Q_{ρ}, a contradiction. Hence, we may assume now that Q_{ρ} < T < B(S).

There exists x ∈ G_{α} such that (α + 1)^{x} ∈ P^{∗} and (α, α′^{x}) is a critical pair; so by (6) B(S) = Z_{α′}^{x}Q_{α}. If (ρ, α′^{x}) is not a critical pair, we get Z_{α′}^{x} ≤ Q_{ρ} and thus T = B(S), a contradiction. Hence, also (ρ, α′^{x}) is a critical pair, and by (6) B(S) = Z_{α′}^{x}Q_{ρ} and T = Q_{ρ}(Z_{α′}^{x} ∩ T).

Let t ∈ Z_{α′}^{x} such that t ∈ T \ Q_{ρ}. Then there exists y ∈ Z_{ρ} such that [t, y] ≠ 1, and by (7) [t, y] ∈ Z(G_{α}). On the other hand, according to (6) (applied to (ρ, α′^{x}) and (α, α′^{x})) there exists y′ ∈ Z_{α} such that [t, y] = [t, y′]. The action of Z_{α′}^{x} on Z_{α} gives [t, y′] ∉ Z(G_{α}), a contradiction.

We now let N_{H}(B(S)) act on Γ in the following way: Let g ∈ N_{H}(B(S)) and δ ∈ Γ, so δ = P y for some P ∈ P^{∗} and y ∈ G. Then

g : δ ↦ δ^{g} := P^{g}y^{g}.

(9) For every P ∈ Δ^{∗} there exists a critical pair (δ, δ′) satisfying (∗) such that G_{δ} = P. There exists P_{0} ∈ P_{0} such that P ∈ Ω(P_{0}) ⊆ Δ^{∗}. Hence, there exist δ_{1}, ..., δ_{k} ∈ P^{∗} such that

Ω(P_{0}) = {G_{δ_{1}}, ..., G_{δ_{k}}}.

Note that by (2.9)(d) the subgroups in Ω(P_{0}) are conjugate under S. We will show that there exists a critical pair (δ_{i}, δ′_{i}) for some i ∈ {1, ..., k}. The (∗)-property then can be achieved by a suitable conjugation in G_{δ_{i}} and the claim for the other δ_{j} by the action of S.

Hence, we may assume that Z_{δ_{i}} ≤ Q_{α′−1} for all i = 1, ..., k. If there exists j ∈ {1, ..., k} such that Q_{δ_{j}} = Q_{α}, then (δ_{j}, α′^{x}) is a critical pair, where x ∈ G_{α} such that B(S)^{x^{−1}} ≤ G_{α+1}. Thus, we may also assume that Q_{α} ≠ Q_{δ_{i}} for all i = 1, ..., k. Now (7) and (8) give

R_{δ_{i}} = [Z_{δ_{i}}, B(S)] ≤ Z(G_{α}), i = 1, ..., k,

and by (3)

Z_{0} = ∏_{i=1}^{k}[Z_{δ_{i}}, B(S)] = ∏_{i=1}^{k} R_{δ_{i}} ≤ Z(G_{α}),

a contradiction.

(10) There exists ρ ∈ P^{∗} and P ∈ Δ^{∗} such that Q_{ρ}^{q} ≠ O_{p}(P) for all q ∈ Q.

Assume that (10) does not hold. Let P_{0} ∈ P_{0} and Ω(P_{0}) = {L_{1}, ..., L_{k}}. By (2.9)(d)
∩_{i=1}^{k}O_{p}(L_{i}) = O_{p}(P_{0}).

Now let ρ ∈ P^{∗} and L_{i} ∈ Ω(P_{0}). Then there exists q ∈ Q such that Q_{ρ}^{q} = O_{p}(L_{i}); in particular O_{p}(P_{0}) ≤ Q_{ρ}^{q}. Since O_{p}(P_{0}) is Q-invariant we get

O_{p}(P_{0}) ≤ Q_{ρ} for all ρ ∈ P^{∗} and all P_{0} ∈ P_{0}.

Note that P^{∗} is invariant under N_{H}(B(S)). Hence also

O_{p}(P^{∗}) ≤ Q_{ρ} for all ρ ∈ P^{∗} and all P^{∗} ∈ P_{0}^{∗}.

It follows that

O_{p}(P^{∗}) ≤ ∩_{L_{i}∈Ω(P_{0})}O_{p}(L_{i}) = O_{p}(P_{0}) for all P_{0} ∈ P_{0} and all P^{∗} ∈ P_{0}^{∗}.

This shows that O_{p}(P^{∗}) = O_{p}(P_{0}) for all P^{∗} ∈ P_{0}^{∗} and all P_{0} ∈ P_{0}, and by (4) O_{p}(P_{0}) is normal in L, a contradiction to (5).

By (10) there exists ρ ∈ P^{∗} and P ∈ Δ^{∗} such that Q_{ρ}^{q} ≠ O_{p}(P) for all q ∈ Q, and by (9) there exists a critical pair (α, α′) satisfying (∗) such that G_{α} = P. We fix this notation with the additional property that P_{0} := ⟨P, S⟩ ∈ P_{0} and P ∈ Ω(P_{0}).

(11) There exists q ∈ Q such that (ρ^{q}, α) is a critical pair; in particular b = 1.

Suppose that b > 1 or Z_{ρ^{q}} ≤ Q_{α′−1} for all q ∈ Q. Then (8) shows that B(S) = Q_{α}Q_{ρ}^{q} for all q ∈ Q. Hence [Z_{ρ}^{q}, Q_{α}] = [Z_{ρ}^{q}, B(S)] and by (7)

R := ∏_{q∈Q}[Z_{ρ}^{q}, B(S)] ≤ Z(G_{α});

in particular R is a Q-invariant and non-trivial subgroup of Z(G_{α}). Hence, Q-Uniqueness gives G_{α} = P ≤ C̃. But then also P_{0} ≤ C̃, which contradicts P_{0} ∈ P_{0}. This shows that b = 1 and there exists q ∈ Q such that (ρ^{q}, α) is a critical pair.

(12) Let γ ∈ P^{∗} such that G_{γ} ≤ P_{0}. Then Y_{G_{γ}} ≤ Y_{P_{0}}; in particular Z_{α} ≤ Y_{P_{0}} and no Q-conjugate of G_{ρ} is contained in P_{0}.

Since by (2) O_{p}(P_{0}) ≤ B(S), we have Ω_{1}(Z(Q_{γ})) ≤ Ω_{1}(Z(O_{p}(P_{0}))). Hence (1.3)(d) and (2) yield Y_{G_{γ}} ≤ Y_{P_{0}}. This gives, together with (11), that there exists q ∈ Q such that G_{ρ}^{q} is not contained in P_{0}, and, since Q ≤ S ≤ P_{0}, no Q-conjugate of G_{ρ} is contained in P_{0}.

Let μ := ρ^{q} be as in (11). Then (6) and b = 1 give

B(S) = Z_{μ}Z_{α}(Q_{α} ∩ Q_{μ});

in particular

Φ(Q_{α}) = Φ(Q_{α} ∩ Q_{μ}) = Φ(Q_{μ}).

This gives [Q_{α}, Z_{μ}] = [Z_{α}, Z_{μ}] ≤ Z_{α}. Hence (2), (1.3)(b) and (12) yield

[O_{p}(P_{0}), O^{p}(G_{α})] ≤ [Q_{α}, O^{p}(G_{α})] ≤ [Q_{α}, ⟨Z_{μ}^{G_{α}}⟩] ≤ Z_{α} ≤ Y_{P_{0}}.

From G_{α} ∈ Ω(P_{0}) and (2.9)(d) we get [O_{p}(P_{0}), O^{p}(P_{0})] ≤ Y_{P_{0}}. Now Z(P_{0}) = 1 yields Y_{P_{0}} = O_{p}(P_{0}), and (2.1) and (2.9) applied to P_{0} give B(S) = Y_{P_{0}}⟨Z_{μ}^{S}⟩. From (2.1) and (3) it follows that Φ(B(S)) = Z_{0}; in particular

Φ(Q_{α}) = Φ(Q_{μ}) ≤ Z(G_{α}) ∩ Z(G_{μ}).

Assume that Ω(P_{0}) = {P }. Then Z(G_{α}) = 1 and Z_{α} = Q_{α} is a natural G_{α}/Q_{α}-module. In particular

B(S) = Z_{α}Z_{μ} and Z_{α} ∩ Z_{μ} = Z_{0}.

Thus, also Q_{μ} = Z_{μ}, and the action of Z_{α} on Z_{μ} also shows that Z(G_{μ}) = 1.

Let λ ∈ P^{∗}. If Q_{λ}^{q} ≠ Q_{α} for all q ∈ Q, then, as for ρ and μ, Q_{λ} = Z_{λ} and Z(G_{λ}) = 1. If Q_{λ}^{q} = Q_{α} for some q ∈ Q, then Z_{α} = Z_{λ}^{q} = Q_{λ}^{q}, and the action of Z_{μ} shows that Z(G_{λ}^{q}) = Z(G_{λ}) = 1. Hence, (c) holds in the case Ω(P_{0}) = {P }.

Assume now that Ω(P_{0}) ≠ {P } and choose L_{i} ∈ Ω(P_{0})\{P }; i.e. L_{i} = G_{ν} for some α ≠ ν ∈ P^{∗}. Since [Z_{μ}, Q_{α}] = [Y_{P} , B(S)] and by (2.9)(e) [Y_{L_{i}}, B(S)] ≠ [Y_{P} , B(S)] we get from b = 1 and (6) that Z_{ν} ≤ Q_{μ} ∩ Q_{α}. Hence,

R_{0} := [Z_{ν}, B(S)] = [Z_{ν}, Q_{α} ∩ Q_{μ}] ≤ Z(G_{α}) ∩ Z(G_{μ}).
Let U = N_{H}(R_{0}). Then U is of characteristic p and ⟨G_{α}, G_{μ}⟩ ≤ C_{H}(R_{0}). Thus
O_{p}(U) ∩ Q_{μ} = O_{p}(U) ∩ B(S) = O_{p}(U) ∩ Q_{α},

so O_{p}(U)∩B(S) is normal in G_{α} and [O_{p}(U)∩B(S), Z_{μ}] = 1. Note that [O_{p}(U), Z_{μ}] ≤ O_{p}(U)∩B(S). Since O^{p}(G_{α}) ≤ ⟨Z_{μ}^{G_{α}}⟩ we get that [O_{p}(U), O^{p}(G_{α}), O^{p}(G_{α})] = 1. This contradicts the fact that U is of characteristic p.

Corollary 1. Suppose that the cases (a) and (b) of Theorem 1 do not hold. Let P ∈ P_{H}(S) \ P_{C}(S) such that Ω_{1}(Z(B(S))) is not normal in P. Then \overline{B(P)} ≅ SL_{2}(p^{m}), and O_{p}(P) is a natural SL_{2}(p^{m})-module for \overline{B(P)}. Moreover, either N_{H}(B(S)) ≤ N_{H}(O_{p}(P)), or P is of type L_{3}.

Proof. By the choice of P and the definition of C̃, P satisfies the hypothesis of (2.6). Hence \overline{U(P)} ≠ 1 and by (2.5)(a) Z(P) = 1. Thus (2.8) gives O_{p}(P) ≤ B(S). Applying (2.9) and Theorem 1 (c) we get that \overline{B(P)} ≅ SL_{2}(p^{m}) and that O_{p}(P) = Y_{P} is a natural SL_{2}(p^{m})-module for \overline{B(P)}. Hence either P is of type L_{3} or p = 2.

Assume that p = 2. Suppose that N_{H}(B(S)) is not contained in N_{H}(Y_{P}) and pick x ∈ N_{H}(B(S)) \ N_{H}(Y_{P}). Then B(S) = Y_{P} Y_{P}^{x} and A(S) = {Y_{P} , Y_{P}^{x}}. Since N_{H}(B(S)) acts on A(S) we get O^{2}(N_{H}(B(S))) ≤ N_{H}(Y_{P}) and thus also N_{H}(B(S)) ≤ N_{H}(Y_{P}), a contradiction.

3. P-Uniqueness

Throughout this section we assume Hypothesis I. In particular, the Structure Theorem applies to all M ∈ L_{H}^{∗}(S) with P ≤ M. In addition, among all P satisfying Hypothesis I we choose P maximal (with respect to inclusion).

Local P!-Theorem. Let P^{∗} = U(P) and P ≤ M ∈ L_{H}^{∗}(S). Then one of the following holds:

(a) Case (a) of the Structure Theorem holds for M, P^{∗} = P ∩ M_{0} and

(i) P^{∗}/O_{p}(P) ≅ SL_{2}(p^{m}) and Y_{P} is a natural SL_{2}(p^{m})-module,

(ii) P_{M}(S) = {P } ∪ P_{M∩_{C}}(S),

(iii) M ∩ C̃ = N_{M}(Ω_{1}(Z(S ∩ P^{∗}))).

(b) Case (b) of the Structure Theorem holds for M, and

(i) P_{M}(S) = P_{P} (S) ∪ P_{M∩_{C}}(S), in particular P = O^{p}(M_{0})S, (ii) M ∩ C̃ ≤ N_{M}(Ω_{1}(Z(S ∩ P^{∗}))), (iii) M_{H}(P) = {M}.

Proof. We discuss the two cases of the Structure Theorem separately. Assume first that case (a) of the Structure Theorem holds for M. Let \overline{M} := M/C_{M}(Y_{M}), S_{0} := S ∩M_{0} and Z_{0} := Ω_{1}(Z(S_{0})). The p-local structure of M_{0}/O_{p}(M_{0}) shows:

(+) There exists a unique U ∈ P_{M_{0}}(S_{0}) such that [Z_{0}, U] ≠ 1; in particular P_{M_{0}}(S_{0}) =

{U} ∪ P_{M_{0}∩_{C}}(S_{0}).

(++) U/O_{p}(U) ≅ SL_{2}(p^{m}), and Y := C_{Y_{M}} (O_{p}(U)) is a natural SL_{2}(p^{m})-module for U/O_{p}(U). Since Q ≤ S_{0} from (1.7) it follows N_{H}(S_{0}) ≤ C̃, hence (+) gives N_{H}(S_{0}) ≤ N_{H}(U), in

particular S normalizes U.

Let P_{1} ∈ P_{M}(S) such that P_{1} ≰ C̃. By (1.7) Q ≰ O_{p}(P_{1}), and so by (1.3)(b) P_{1} = (P_{1})^{0}S and (P_{1})^{0}S_{0} ≤ M_{0}. Since O_{p}(M) ≤ O_{p}((P_{1})^{0}S_{0}), (P_{1})^{0}S_{0} has characteristic p, whence (1.3)(a) and the uniqueness of U give

(P_{1})^{0}S_{0} = ⟨U, (P_{1})^{0}S_{0} ∩ C̃⟩.

Since P_{1} is a minimal parabolic subgroup not contained in C̃ we get that P_{1} = US; in particular P = US, and (a)(ii) follows.

From O_{p}(U) ≤ O_{p}(P) and (1.2)(b) we get Y_{P} ≤ Y_{M}, thus Y_{P} ≤ Y and (++) yields Y_{P} = Y. Now (2.1) gives P^{∗} = UO_{p}(P) ≤ M_{0}, whence (a)(i) and P ∩ M_{0} = P^{∗} follow.

Note that M_{0}C_{M}(Y_{M}) is a normal subgroup of M. It follows that

M ∩ C̃ = C_{M}(Y_{M})(M_{0} ∩ C̃)N_{M∩_{C}}(S_{0}) ≤ (M_{0} ∩ C̃)N_{M}(Z_{0}),

so (+) and (1.3)(a) yield M ∩ C̃ ≤ N_{M}(Z_{0}). On the other hand by Q-Uniqueness C_{M}(Z_{0}) ≤ C̃, so by (1.6) Q is the unique conjugate of Q in C_{M}(Z_{0}). Hence N_{M}(Z_{0}) ≤ N_{M}(Q) = M ∩ C̃.

By the Structure Theorem C_{S}(Y_{M}) = O_{p}(M_{0}) ∈ Syl_{p}(C_{M_{0}}(Y_{M})), whence by (1.2)(e) Y_{M_{0}} = Ω_{1}(Z(C_{S}(Y_{M}))) ≤ Y_{M}. This gives Z_{0} ≤ Y_{M} and thus Z_{0} = C_{Y_{M}} (S_{0}). From (++) it follows that Z_{0} ≤ Y = Y ∩ Z(O_{p}(P)), therefore S ∩ P^{∗} = O_{p}(P)S_{0} yields Z_{0} = Ω_{1}(Z(S ∩ P^{∗})). This shows (a)(iii).

Assume now that case (b) of the Structure Theorem holds. Let P_{1} and P_{1}^{∗} be as given there and set S_{0} := P_{1}^{∗} ∩ S and Z_{0} := Ω_{1}(Z(S_{0})). Then P_{1} = M^{0}S and by (2.1) P_{1}^{∗} = U(P_{1}); moreover, by (1.3)(c) and (1.7) P_{M}(S) = P_{P_{1}}(S) ∪ P_{M∩_{C}}(S). The maximality of P gives P = P_{1} and P^{∗} = P_{1}^{∗}, and (b)(i)holds.

Since P^{∗}C_{M}(Y_{M}) is normal in M we get as above

M ∩ C̃ = C_{M}(Y_{M})(P^{∗} ∩ C̃)N_{M∩_{C}}(S_{0}).

As P is a minimal parabolic subgroup, the structure of P^{∗} and its action on Y_{P} show that N_{P} (Z_{0}) is the unique maximal subgroup containing S. It follows that P^{∗} ∩ C̃ ≤ N_{P} (Z_{0}) and thus M ∩ C̃ ≤ N_{M}(Z_{0}). This is (b)(ii).

Let P ≤ L ∈ M_{H}(S) and L << L̃ ∈ L_{H}^{∗}(S). Then L = (L ∩ L̃)C_{L}(Y_{L}) and thus

P^{0} ≤ L^{0} = (L ∩ L̃)^{0} ≤ L̃^{0}.

It follows that P = P^{0}S ≤ L̃, and we are allowed to apply the Structure Theorem to L̃.

If case (a) of the Structure Theorem holds for L̃, then by case (a) of the Local P! Theorem P ∩ L̃_{0} = P^{∗} = U(P). But then Q ≤ P^{∗}, a contradiction.

If case (b) of the Structure Theorem holds for L̃, then the maximality of P gives Y_{P} = Y_{L} and thus Y_{L} = Y_{M}; in particular M = L̃. This shows (b)(iii).

Notation. We fix M, P and P^{∗} as in the Local P!-Theorem. (Observe that in case (b) of the Local P!-Theorem the definition of P^{∗} differs from that given in the P!-Theorem. But it will be shown in section 4 that this case does not occur.) Furthermore, we set \overline{P} := P/C_{P} (Y_{P}), S_{0} := S ∩ P^{∗} and Z_{0} := Ω_{1}(Z(S_{0})). Recall that P satisfies the hypotheses of (2.1) – (2.5) and if B(S) ≰ O_{p}(P) also those of (2.6) – (2.9). Later in the course of the amalgam method we will apply these Lemmata not only to P but also to conjugates of P.

(3.1) P admits the decompositions

(D_{1}) \overline{P}^{∗} = K_{1} × ⋯ × K_{r}, K_{i} ≅ SL_{2}(p^{m}), and
(D_{2}) Y_{P} = V_{1} × ⋯ × V_{r}, V_{i} a natural SL_{2}(p^{m})-module for K_{i}.

Moreover, [Y_{P} , Q ∩ S_{0}] = Z_{0} and either S_{0} = B(S) or B(S) ≤ O_{p}(P).

Proof. The decompositions D_{1} and D_{2} are from the Local P!-Theorem. Assume that B(S) ≰ O_{p}(P). Since P^{∗} = U(P) (2.6), (2.1) and (2.8) show that S_{0} = B(S).

Remark. The next result, Theorem 2, establishes part (a) and (c) of the P!-Theorem if case (a) of the Local P!-Theorem holds. We then embark on the proof of the main result of this section, Theorem 3, where we show that Z_{0} is normal in C̃. This establishes part (b) of the P!-Theorem in all cases. It then remains to treat case (b) of the Local P!-Theorem. This is done in the next section, where the F!-Theorem eliminates this case.

Theorem 2. Assume Hypothesis I. Then either P_{H}(S) = P_{P} (S) ∪ P_{C}(S), or the following hold:

(a) Z_{0} is normal in C̃.

(b) Q = B(S) = S_{0}.

(c) P̃ is of type L_{3} for every P̃ ∈ P_{H}(S) \ P_{C}(S).

Proof. Assume first that P is of type L_{3}. Then by (2.2) Y_{P} ∈ A(S), B(S) = S_{0}, and for every A ∈ A(S) either

S_{0} = AY_{P} or A = Y_{P} .

Moreover, Y_{P} ≤ Q by (1.2)(b) and Hypothesis I. It follows that also J(S) = S_{0} ≤ Q since Y_{P} is not normal in C̃. But then J(S) = J(Q) and Z_{0} = Ω_{1}(Z(S_{0})) = Ω_{1}(Z(J(S))) is normal in C̃. On the other hand, N_{P} (S_{0}) is transitive on Z_{0} and by (1.7) contained in C̃, so Z_{0} ≤ Z(Q) and Q ≤ S_{0}. We conclude that Q = S_{0}; in particular N_{H}(B(S)) = C̃.

Let P̃ ∈ P_{H}(S) \ P_{C}(S). Then P̃ ≰ N_{H}(Ω_{1}(Z(B(S)))), and Corollary 1 shows that also P̃ is of type L_{3}. Hence, Theorem 2 holds if P is of type L_{3}.

We may assume now:

(1) P is not of type L_{3} and P_{H}(S) ≠ P_{P} (S) ∪ P_{C}(S).

By (1) there exists P̃ ∈ P_{H}(S) such that

(2) P̃ ≰ P and P̃ ≰ C̃.

Assume that O_{p}(⟨P, P̃⟩) ≠ 1. Then there exists L ∈ \overline{L}_{H}(S) such that ⟨P, P̃⟩ := R ≤ L. Since P = P^{0}S and P̃ = P̃^{0}S, we also get R ≤ L^{0}S. Now (1.4) shows that there exists M̃ ∈ L_{H}^{∗}(S) such that R ≤ M̃. The Local P!-Theorem applied to M̃, together with the maximal choice of P, gives P̃ ≤ P, which contradicts (2). We have shown:

(3) O_{p}(⟨P, P̃⟩) = 1.

We now apply Theorem 1. Then (3) shows that the cases (a) and (b) of Theorem 1 do not hold. Assume that B(S) is not normal in P, so by (1.3) also Ω_{1}(Z(B(S))) is not normal in P. Hence by Corollary 1 O_{p}(P) = Y_{P} and P^{∗}/Y_{P} ≅ SL_{2}(p^{m}), and Corollary 1 and (1) show that N_{H}(B(S)) ≤ N_{H}(Y_{P}). On the other hand, as above, Y_{P} ≤ Q implies B(S) = S_{0} = Q since Y_{P} is not normal in C̃. Hence C̃ = N_{H}(B(S)) ≤ N_{H}(Y_{P}), and Y_{P} is normal in C̃, a contradiction. We have shown:

(4) P ≤ N_{H}(B(S)).

By (3) and (4) Ω_{1}(Z(B(S))) is not normal in P̃. Hence again (3) and Corollary 1 show that P̃ is of type L_{3}. In particular p ≠ 2, and there exists an involution t ∈ N_{P}(S) such that [S, t] = Y_{P}. Since Y_{P} ≤ B(S) and Y_{P} = O_{p}(P̃) we get Y_{P} ≤ Ω_{1}(Z(B(S)) ≤ Y_{P}. Hence Y_{P} = [Y_{P} , t], and t inverts Y_{P}. This shows that [t, P] ≤ C_{H}(Y_{P}) ∩ N_{H}(O_{p}(P)) =: X, and P^{0} normalizes ⟨t⟩X. Since

[⟨t⟩X, Q] ≤ Q ∩ ⟨t⟩X ≤ C_{S}(Y_{P}) = O_{p}(P)

we conclude that [t, P^{0}] ≤ O_{p}(P) and thus also [t, P] ≤ O_{p}(P). Hence, P normalizes ⟨t⟩O_{p}(P) and thus also O^{p}(⟨t⟩O_{p}(P)) = ⟨t⟩Y_{P}. It follows that P normalizes Y_{P}, which contradicts (3). This completes the proof of Theorem 2.

(3.2) Suppose that O^{p}(\overline{P}) ≤ ⟨\overline{x}, \overline{A}⟩, where x is a p-element in P and A a normal subgroup of S in Q. Then O^{p}(P) ≤ ⟨x, A⟩.

Proof. Let P_{0} = ⟨x, A⟩ and P_{1} = O^{p}(P). Note that P_{1} ≰ C_{P} (Y_{P}) by our choice of P, so P_{1} ≤ ⟨A^{P} ⟩ by (1.3)(b). Note further that [C_{P} (Y_{P}), A] ≤ O_{p}(P) since A ≤ Q and that P_{1} ≤ P_{0}C_{P} (Y_{P}). It follows that

P_{1} ≤ ⟨A^{P} ⟩ = ⟨A^{P_{1}}⟩ ≤ ⟨A^{P_{0}}⟩O_{p}(P).

Since ⟨A^{P_{0}}⟩ is normal in P_{0}O_{p}(P) we get that

P_{1} = O^{p}(⟨A^{P} ⟩) = O^{p}(P_{0}O_{p}(P)) = O^{p}(P_{0}).

Hypothesis II. Assume Hypothesis I and P_{H}(S) = P_{P} (S) ∪ P_{C}(S). Further assume that there exists P̃ ∈ P_{C}(S) such that (P, P̃) is an amalgam and N_{P}(Z_{0}) is a maximal subgroup of P̃.

Our goal, which we will achieve in (3.9), is to prove that no group H satisfies Hypothesis II.

(3.3) Assume Hypothesis II. Let x ∈ P̃ and O_{p}(P) ≤ N_{P}(Z_{0}^{x}). Then x ∈ N_{P}(Z_{0}).

Proof. Assume first that J(S) ≤ O_{p}(P̃). Then J(S) is normal in P̃ and thus not normal in P since (P, P̃) is an amalgam. Hence, by (3.1) S_{0} = B(S) and Z_{0} = Ω_{1}(Z(J(S))). But then Z_{0} is normal in P̃, a contradiction. Thus, J(S) is not normal in P̃. Since P̃ is minimal parabolic we get that N_{P}(J(S)) ≤ N_{P}(Z_{0}) and that N_{P}(Z_{0}) is self-normalizing.

Assume now that x ∉ N_{P}(Z_{0}) but O_{p}(P) ≤ N_{P}(Z_{0}^{x}), so N_{P}(Z_{0}) ≠ N_{P}(Z_{0}^{x}). We choose x in addition such that |T | is maximal, where

O_{p}(P) ≤ T ∈ Syl_{p}(N_{P}(Z_{0}) ∩ N_{P}(Z_{0}^{x})).

Note that O_{p}(P̃) ≤ T ∩ S.

After conjugation in N_{P}(O_{p}(P)) we may assume that T_{1} := N_{T} (O_{p}(P)) ≤ S, so T_{1} = T ∩ S. Note that T ∉ Syl_{p}(P̃) since P̃ is minimal parabolic; in particular T is not a Sylow p-subgroup of N_{P}(Z_{0}^{x}). Hence, the maximality of T yields

(1) N_{P}(T) ≰ N_{P}(Z_{0}).

From (1) and N_{P}(J(S)) ≤ N_{P}(Z_{0}) we get:

(2) J(S) ≠ J(T) and J(S) ≰ T. In particular J(S) ≰ O_{p}(P), and (3.1) and (2.7) yield

(3) S_{0} = B(S) = J(S)O_{p}(P) and [Ω_{1}(Z(J_{0})), J(S)] = Z_{0} = Ω_{1}(Z(J(S))), where J_{0} := J(O_{p}(P)).

Assume that J(T_{1}) ≠ J_{0}. Since Q ≤ T_{1} the Q-transitivity and (2.1) imply

S_{0} = J(T_{1})O_{p}(P) ≤ T.

This contradicts (2) and (3). We have shown:

(4) J_{0} = J(T_{1}). Since (P, P̃) is an amalgam and O_{p}(P̃) ≤ T_{1} we get from (4) J_{0} ≰ O_{p}(P̃) and thus N_{P}(J_{0}) ≤

N_{P}(Z_{0}).

Set T_{2} := N_{T} (J_{0}) and note that J_{0} ≠ J(T_{2}) by (1). There exists y ∈ N_{P}(J_{0}) such that T_{2} ≤ S^{y}. From (3) we get

[Ω_{1}(Z(J_{0})), J(S)^{y}] = Z_{0},

in particular J(T_{2}) ≤ N_{H}(Y_{P}) since Y_{P} ≤ Ω_{1}(Z(J_{0})). Hence also T_{3} := ⟨O_{p}(P), J(T_{2})⟩ ≤ N_{H}(Y_{P}), and O_{p}(P) = C_{T_{3}}(Y_{P}) is normal in T_{3} since O_{p}(P) ∈ Syl_{p}(N_{H}(Y_{P})). It follows that T_{3} ≤ T_{1} and thus by (4) J_{0} = J(T_{2}), a contradiction.

(3.4) Assume Hypothesis II. Let V = ⟨Y _{P} ; P⟩. Then V is abelian.

Proof. Set V_{0} = ⟨Z_{P} ; 0⟩. By Hypothesis I and (1.2)(b) Y_{P} ≤ Q and thus V ≤ O_{p}(P̃) ≤ S. Assume that V is not abelian. Then there exists x ∈ P̃ such that A := Y_{P}^{x} ≰ O_{p}(P). Then (2.1) and the Q-invariance of A show that [V, Y_{P}] = [A, Y_{P}] = Z_{0} and AO_{p}(P) = V O_{p}(P) = S_{0}. Moreover V_{0} ≤ Z(V) ≤ O_{p}(P), and O^{p′}(C_{P}(V_{0})) ≤ O_{p}(P̃) since Z_{0} is not normal in P̃.

There exists y ∈ P such that ⟨V, V^{y}⟩C_{P^{∗}}(Y_{P}) = P^{∗}. Since V is contained in Q and normal in S (3.2) implies O^{p}(P) ≤ ⟨V, V^{y}⟩. Hence Z(P) = 1 gives Z(⟨V, V^{y}⟩) = 1.

Note that V_{0} ≤ O_{p}(P) ≤ S^{y} and thus

[V_{0}, V_{0}^{y}] ≤ V_{0} ∩ V_{0}^{y} ≤ Z(⟨V, V^{y}⟩) = 1.

Let z′, z ∈ P̃ such that for A_{1} = Y_{P}^{z} and A_{2} = Y_{P}^{z′}
[A_{1}, A_{2}] = Z_{0}^{z} ≠ Z_{0}.

It follows that U := ⟨A_{1}, A_{2}⟩ ≤ O_{p}(P). In addition V_{0}^{y} ≤ O^{p′}(C_{P}(V_{0})) ≤ C_{O_{p}(_{P})}(Z_{0}^{z}) and thus [V_{0}^{y}, U] ≤ V_{0}^{y} ∩ V_{0} = 1. Hence U ≤ C_{O_{p}(_{P^{y}})}(V_{0}^{y}) and thus [A_{1}, A_{2}, V^{y}] = 1. It follows that Z_{0}^{z} centralizes V^{y} and

Z_{0}^{z} ≤ Z(⟨V, V^{y}⟩) = 1,

a contradiction.

Notation. From now through (3.9) we will apply the amalgam method to the amalgam (P, P̃). With one exception we will use the standard terminology (see [DS], [KS] and the proof of Theorem 1). In particular we choose α, β, α′ ∈ Γ so that (α, α′) is a critical pair and so that {G_{α}, G_{β}} = {P, P̃}. The exception to standard notation is the definition of Z_{δ}. For δ ∈ Γ we define

Z_{δ} := Y_{G_{δ}}.

In addition, we define for g ∈ G, δ = α^{g} and λ = β^{g}

\begin{matrix}Z_{λ}^{∗} = C_{Z_{δ}}(O^{p}(G_{λ})), Q̃_{λ} = Q^{g}, Z(δ, λ) = Z_{0}^{g}, C̃_{λ} = C̃^{g}, \\ V_{λ}^{∗} = ⟨x^{h} | h ∈ G_{λ}, x ∈ Z_{δ} and [x, S^{g}] ≤ Z_{λ}^{∗}⟩. \end{matrix}

Note that Z_{λ}^{∗} is normal in G_{λ} and thus [V_{λ}^{∗}, Q_{λ}] ≤ Z_{λ}^{∗}. Note further that

V_{λ}^{∗} = ⟨(Z_{δ} ∩ V_{λ}^{∗})^{G_{λ}}⟩.

(3.5) Assume Hypothesis II. Then Z = Y_{P} and P̃ = G_{β}.

Proof. Clearly Z = Y_{P} implies P̃ = G_{β}. Thus, we may assume that Z ≠ Y_{P}. Then by (1.3)(b)

C_{S}(Y_{P}) = O_{p}(P̃) and [Z_{α}, Z_{α′}] ≠ 1. Let 1 ≠ x ∈ [Z_{α}, Z_{α′}].

Assume that G_{α} = P̃. Then Z_{α} ≤ Y_{C} ≤ Z(Q) by (1.2)(b) and C_{H}(x) ≤ C̃ by Q-Uniqueness. Since Z_{α} ≰ Q_{α′} we get G_{α′} ≰ C̃. It follows that G_{α′} is conjugate to P.

Hence, after switching to another critical pair we may assume that G_{α} = P. (3.4) shows that b > 2. Let α − 1 ∈ Δ(α) such that ⟨G_{α−1} ∩ G_{α}, Z_{α′}⟩ = G_{α} and set A := Z_{α′−1}(Z_{α′} ∩ Q_{α}). Since b > 2 we have

(∗) [Z_{α−1}, A, Z_{α′}] = 1.

Assume first that [Z_{α−1}, A] =: R ≠ 1. As above C_{H}(R) ≤ C̃_{α−1} since G_{α−1} is conjugate to P̃. Hence (∗) gives

⟨G_{α−1}, Z_{α′}⟩ = ⟨G_{α−1}, G_{α}⟩ ≤ C̃_{α−1},

a contradiction.

Assume now that [Z_{α−1}, A] = 1. Then Z_{α−1} ≤ G_{α′} and

Z_{α′} ∩ Q_{α} = C_{Z_{α′}} (Z_{α}) ≤ C_{Z_{α′}} (Z_{α−1}),

while (2.1) gives

|Z_{α}/C_{Z_{α}}(Z_{α′})| = |Z_{α′}/C_{Z_{α′}} (Z_{α})|.

It follows that

(∗∗) |Z_{α′}/C_{Z_{α′}} (Z_{α}Z_{α−1})| = |Z_{α′}/C_{Z_{α′}} (Z_{α})| = |Z_{α}/C_{Z_{α}}(Z_{α′})| ≤ |Z_{α}Z_{α−1}/C_{Z_{α}Z_{α−1}}(Z_{α′})|.

According to (2.1)(e), this time applied to G_{α′}, equality holds in (∗∗), so Z_{α−1} ≤ Z_{α}Q_{α′} and [Z_{α−1}, Z_{α′}] ≤ [Z_{α}, Z_{α′}] ≤ Z_{α}. Hence, Z_{α−1}Z_{α} and thus also [Z_{α−1}, Q_{α}] is normal in G_{α}. Now the irreducibility of Z_{α} and (1.2)(e) yield Z_{α−1} ≤ Z_{α}. But then Q_{α} ≤ Q_{α−1} and thus also Q_{α} ≤ Q_{β}. Since Z_{α′} ≤ Q_{β} (2.1) and (3.1) give S_{0} ≤ Q_{β} and S_{0} = B(S). Hence, Z_{0} is normal in P̃, which contradicts Hypothesis II.

(3.6) Assume Hypothesis II. Then [Z_{α}, Z_{α′}] = 1.

Proof. Asssume that [Z_{α}, Z_{α′}] ≠ 1. From (3.5) we get that P̃ = G_{β} and Z_{β} = Z. In particular b is even, and G_{α′} is conjugate to G_{α}. Moreover, (3.4) gives:

(1) V_{β} is an elementary abelian subgroup of Q_{β}, and b ≥ 4.

Pick α′ + 1 ∈ Δ(α′) such that Z(α′, α′ + 1) ≠ Z(α′, α′ − 1). The Q̃_{α′+1}-transitivity shows that O^{p}(G_{α′}) ≤ ⟨Z_{α}, Q̃_{α′+1}⟩C_{G_{α′}} (Z_{α′}). So (3.2) yields O^{p}(G_{α′}) ≤ ⟨Z_{α}, Q̃_{α′+1}⟩.

(2) Z_{α} ∩ V_{β}^{∗} ≤ Z(α, β).

Note that S_{0} = Q_{α}⟨Z_{Q_{β}} ; α′⟩ by (2.1) and Q-transitivity since Z_{α′} ∈ U(P), so [Z_{β}^{∗}, S_{0}] = 1. Hence Z_{β}^{∗} ≤ Z(α, β). Moreover

D := [Z_{α} ∩ V_{β}^{∗}, S_{0}] ≤ [Z_{α} ∩ V_{β}^{∗}, Q_{α}Q_{β}] ≤ [V_{β}^{∗}, Q_{β}] ≤ Z_{β}^{∗}.

Note that D is Q-invariant. Hence, the action of S_{0} on Z_{α} and the Q-transitivity either give D = 1, or D = Z(α, β). The first case implies (2). In the second case Z(α, β) = Z_{β}^{∗} is normal in G_{β}, which contradicts Hypothesis II.

(3) V_{α′+1}^{∗} ≤ Q_{α+2}.

This follows from (2) since Z_{α+2} centralizes Z(μ, α′ + 1) for all μ ∈ Δ(α′ + 1).

(4) Let A ≤ V_{α′+1}^{∗} such that O^{p}(G_{α′}) ≤ N_{G}(AZ_{α′}). Then A ≤ Z(α′, α′ + 1).

Since ⟨A^{G_{α′} ∩G_{α′+1}}⟩ satisfies the hypothesis of (4) we may assume that A is (G_{α′} ∩ G_{α′+1})- invariant; i.e. AZ_{α′} is normal in G_{α′}. Then also Y := [AZ_{α′}, Q_{α′}] is normal in G_{α′} and Y ≤ V_{α′+1}^{∗}.

If Y = 1, then (1.2)(e) shows that A ≤ Ω_{1}(Z(Q_{α′})) = Z_{α′} since G_{α′} is conjugate to P. Now (2) yields A ≤ Z(α′, α′ + 1). If Y ≠ 1, then the irreducibility of Z_{α′} gives Z_{α′} ≤ Y, which contradicts (2).

(5) V_{α′+1}^{∗} ≰ G_{α}.
Assume that V_{α′+1}^{∗} ≤ G_{α}. As b > 2 and thus V_{α′+1}^{∗} ≤ Q_{α′}, (2.4) gives
[Z_{α}, V_{α′+1}^{∗}] ≤ [Z_{α}, Z_{α′}][Z_{α} ∩ Q_{α′}, V_{α′+1}^{∗}] ≤ Z_{α′}V_{α′+1}^{∗},

so Z_{α′}V_{α′+1}^{∗} is normal in ⟨Z_{α}, G_{α′} ∩ G_{α′+1}⟩ = G_{α′}. Now (4) shows that V_{α′+1}^{∗} = Z(α′, α′ + 1), which contradicts Hypothesis II.

By (1.3)(b) and Hypothesis II Q_{β} is the unique Sylow p-subgroup of ∩_{ρ∈Δ(β)}N_{G_{β}}(Z(ρ, β)). Hence, by (5) there exists ρ ∈ Δ(β) such that V_{α′+1}^{∗} ≰ N_{G_{β}}(Z(ρ, β)). Note that by (3) and (3.3) also ⟨Q_{ρ}^{V_{α′+1}^{∗}}⟩ ≰ N_{G_{β}}(Z(ρ, β)).

(6) Z_{ρ} ≤ Q_{α′}.

Assume that Z_{ρ} ≰ Q_{α′}. Then (ρ, α′) is a critical pair, and Z(ρ, β) = ⟨[Z_{ρ}, Z_{α′}]_{Q_{β}}⟩ centralizes ⟨Q_{ρ}^{V_{α′+1}^{∗}}⟩, a contradiction.

(7) Set R := [Z_{ρ}, V_{α′+1}^{∗}]. Then |R| < |Z(ρ, β)|.

Note that by (3) and (6) R ≤ V_{α′+1}^{∗} ∩ V_{β} and by (1) [R, Z_{α}] = 1. Since [V_{α′+1}^{∗}, Q_{α′+1}] ≤ Z_{α′+1}^{∗} ≤ Z_{α′} we get that RZ_{α′} is normalized by ⟨Z_{α}, Q_{α′+1}⟩ anf thus by O^{p}(G_{α′}). Now (4) shows that R ≤ Z(α′, α′ + 1); and equality does not hold since Z_{α} centralizes R but not Z(α′, α′ + 1).

We now derive a final contradiction. Let t ∈ V_{α′+1}^{∗}\N_{G_{β}}(Z(ρ, β)), U = ⟨Q_{ρ}, t⟩ and Y_{0} = C_{Z_{ρ}}(t). Note that

|Z_{ρ}/Y_{0}| ≤ |[Z_{ρ}, t]| ≤ |[Z_{ρ}, V_{α′+1}^{∗}]|,

so by (7) |Z_{ρ}/Y_{0}| < |Z(ρ, β)|. On the other hand, by (3.1) |Z_{ρ}| = |Z(ρ, β)|^{2} and so |Y_{0}| > |Z(ρ, β)|.

Set U_{0} := ⟨Q_{ρ}^{U}⟩ and Y_{1} = C_{Z_{ρ}}(U_{0}). By (3.3) U_{0} ≰ N_{G_{β}}(Z(ρ, β)). Since Y_{0} ≤ Y_{1} we also have |Y_{1}| > |Z(ρ, β)|. Moreover, Y_{1} and U_{0} are Q_{β}-invariant.

Let x ∈ G_{β} such that α^{x} = ρ. As seen above S_{0}^{x} ≤ Q_{β}Q_{ρ}, so Y_{1} is S_{0}^{x}-invariant. Moreover, since |Y_{1}| > |Z(ρ, β)| we also have [Y_{1}, S_{0}^{x}] ≠ 1. Now (3.1), applied to P^{x} (= G_{ρ}), and the Q-transitivity yield

Z(ρ, β) = ⟨[Y_{1}, S_{0}^{x}]^{Q}⟩ ≤ Y_{1}.

This contradicts U_{0} ≰ N_{G_{β}}(Z(ρ, β)).

(3.7) Assume Hypothesis II. Let A ≤ P̃ and Y_{0} := [Y_{P} , A ∩ P]. Suppose that A ≰ N_{P}(Z_{0}) and [Y_{0}, A] = 1. Then either Y_{0} = 1, or the following hold:

(a) p = 2 and \overline{P} ≅ S_{3} wr C_{2} or S_{5}.

(b) |A ∩ P/A ∩ O_{2}(P)| = 2, |Y_{0}| = |Z_{0}| = 4 and C_{P^{∗}}(Y_{0}) = O_{2}(P).

Proof. Set A_{0} := A ∩ P, U := ⟨O_{p}(P), A⟩, U_{0} := ⟨O_{p}(P)^{U}⟩ and Y_{1} := C_{Y_{P}} (U_{0}). Note that (1) Y_{0} ≤ Y_{1}, and U_{0} is Q-invariant. Hence Y_{1} is the largest Q-invariant subgroup of Y_{P} centralized by U_{0}. By (3.3) U_{0} ≰ N_{P}(Z_{0})

and thus (2) Z_{0} ≰ Y_{1}. From now on we assume that Y_{0} ≠ 1 and use the notation of (3.1); in addition we set q := p^{m}

and R_{i} := [V_{i}, A_{0}], i = 1, ..., r. It is convenient to treat the following two cases separately: (∗) There exists i ∈ {1, ..., r} such that 1 ≠ R_{i} ≤ V_{i}. (∗∗) R_{i} ≰ V_{i} for all i ∈ {1, ..., r} with R_{i} ≠ 1. Case (∗): We have A_{0} ≤ N_{H}(V_{i}) and thus \overline{A}_{0} ≤ N\overline{P} (K_{i}). If \overline{A}_{0} ≤ K_{i}C\overline{P} (V_{i}), then R_{i} =

Z_{0} ∩ V_{i} ≤ Y_{0}, and (1) and the Q-transitivity give ⟨R_{i}^{Q}⟩ = Z_{0} ≤ Y_{1}, which contradicts (2). Hence, by (2.5)(e) |A_{0}/C_{A_{0}}(V_{i})| = 2 = p.

Assume that r > 1. Then there exists x ∈ Q such that K\overline{x} ; i = K_{j} ≠ K_{i} and

[K_{i} ∩ \overline{S}, \overline{x}]C\overline{P} (V_{i}) = (K_{i} ∩ \overline{S})C\overline{P} (V_{i}).

It follows that

[R_{i}, K_{i} ∩ \overline{S}] = [R_{i}, [K_{i} ∩ \overline{S}, \overline{x}]] ≤ [R_{i}, \overline{Q}],

so by (1) [R_{i}, K_{i} ∩ \overline{S}] = Z_{0} ∩ V_{i} ≤ Y_{1}. Now as above the Q-transitivity yields Z_{0} ≤ Y_{1}, which contradicts (2). Hence r = 1. Thus |A_{0}/A_{0} ∩ O_{2}(P)| = 2; moreover |Y_{P} /C_{Y_{P}} (A_{0})| = q and C_{Y_{P}} (A_{0}) = Y_{0} since \overline{A}_{0} acts as a field automorphism on \overline{P}^{∗}.

We have proved:

(3) In case (∗) r = 1, p = 2, C_{P^{∗}}(Y_{0}) = O_{2}(P), |A_{0}/A_{0} ∩ O_{2}(P)| = 2 and |Y_{P} /Y_{0}| = q.

Case (∗∗): Fix i ∈ {1, .., r} such that R_{i} ≠ 1. Then A_{0} ≰ N_{P} (V_{i}) since R_{i} ≰ V_{i}, and from (2.5)(e) we get that |A_{0}/C_{A_{0}}(V_{i})| = 2(= p) and there exists j ≠ i such that ⟨V_{i}^{A_{0}}⟩ = V_{i} × V_{j}. Note that

V_{i}V_{j} = V_{i}(Y_{1} ∩ V_{i}V_{j}) = V_{j}(Y_{1} ∩ V_{i}V_{j}).

Assume that r > 2. Then by the Q-transitivity there exists x ∈ Q such that V_{i}^{x} ∉ {V_{i}, V_{j}}. In particular, there exists \overline{b} ∈ (K_{i} × K\overline{x} ; i) ∩ \overline{Q} such that

V_{i} ∩ Z_{0} = [V_{i}, b] ≤ [V_{i}V_{j}, b] = [V_{j}Y_{1}, b] = [Y_{1}, b].

As above, (1) and the Q-transitivity give Z_{0} ≤ Y_{1}, which contradicts (2). We have shown that r = 2, so N_{A_{0}}(V_{i}) = C_{A_{0}}(V_{i}) implies |A_{0}/A_{0} ∩ O_{2}(P)| = 2.

For every c ∈ P^{∗} \ O_{2}(P) we have [Y_{0}, c] ≠ 1 since Y_{P} = Y_{0}V_{i} for i = 1, 2. It follows that C_{P^{∗}}(Y_{0}) = O_{2}(P). Moreover V_{i} ∩ Y_{0} = 1 implies |Y_{0}| = |V_{i}| = |Y_{P} /Y_{0}| = q^{2}. We have shown:

(4) In case (∗∗) r = 2 = p, C_{P^{∗}}(Y_{0}) = O_{2}(P), |A_{0}/A_{0} ∩ O_{2}(P)| = 2 and |Y_{P} /Y_{0}| = q^{2}.

Assume that case (a) of the Local P!-Theorem holds for P. Then r = 1, QO_{2}(P) = S_{0} and [y, Q] = Z_{0} for every y ∈ Y_{P} \ Z_{0}. As Y_{0} ≰ Z_{0} by (3), this gives Z_{0} ≤ Y_{1}, which contradicts (2). We have shown:

(5) Case (b) of the Local P!-Theorem holds for P; in particular M_{H}(P) = {M}.

As a trivial consequence of (5) we get:

(6) N_{H}(J(O_{2}(P))) ≤ M.

Let O_{2}(P) ≤ T ∈ Syl_{2}(U_{0}) and T_{0} = N_{T} (J(O_{2}(P))). Note that T_{0} ≤ M by (6). By (3.1) J(S) ≤ S_{0} and by (2.1)(e)

A(O_{2}(P)) ⊆ A(S),

so J(T_{0}) ≤ S_{0}^{x} for some x ∈ M. According to (5) P^{∗}C_{M}(Y_{P}) is normal in M, hence J(T_{0}) ≤ P^{∗}C_{M}(Y_{P}). Now by (1), (3) and (4) imply

J(T_{0}) ≤ C_{M}(Y_{0}) ∩ P^{∗}C_{M}(Y_{P}) = C_{P^{∗}}(Y_{0})C_{M}(Y_{P}) = O_{2}(P)C_{M}(Y_{P}) = C_{M}(Y_{P}).

Since O_{2}(P) is a Sylow 2-subgroup of C_{M}(Y_{P}) we conclude that J(T_{0}) = J(O_{2}(P)) and thus also J(T) = J(O_{2}(P)); in particular T = N_{T} (J(O_{2}(P))) = T_{0} ≤ M. In addition, (3.3) implies T ≤ N_{P}(Z_{0}) and (5) implies Y_{P} = Y_{M}. We have shown:

(7) J(T) = J(O_{2}(P)), and T normalizes Y_{P} and Z_{0}.

According to (5), (6), (7) and (b)(ii) of the Local P!-Theorem N_{U_{0}}(T) ≤ M ∩ C̃ ≤ N_{M}(Z_{0}). Since U_{0} ≰ N_{P}(Z_{0}) there exists F ∈ P_{U_{0}}(T) such that F ≰ N_{H}(Z_{0}); see (1.3)(a). As O_{2}(P̃) ≤ N_{H}(U_{0}) we get [U_{0}, O_{2}(P̃)] ≤ O_{2}(U_{0}); in particular, F is O_{2}(P̃)-invariant and [F, O_{2}(P̃)] ≤ O_{2}(F). In addition, (3.3) and (7) show O_{2}(P) ≰ O_{2}(F) and thus by (1.3)(c)

(8) O^{2}(F) = [O^{2}(F), O_{2}(P)] ≤ ⟨O_{2}(P)^{F} ⟩.

Set W = ⟨Y_{P}^{F} ⟩. Clearly [W, O^{2}(F)] ≠ 1 since by (7) O^{2}(F) ≰ N_{H}(Z_{0}). Moreover, (3.4) shows that W is elementary abelian. Assume that O_{2}(P) ∩ O_{2}(F) is normal in F. Then by (8)

[O^{2}(F), O_{2}(P̃)] ≤ [⟨O_{2}(P)^{F} ⟩, O_{2}(P̃)] ≤ O_{2}(P) ∩ O_{2}(F)

and W = ⟨Y_{P}^{O^{2}(F)}⟩ ≤ Z(O_{2}(P) ∩ O_{2}(F)) since Y_{P} ≤ Z(O_{2}(P) ∩ O_{2}(P̃)) by Hypothesis I and (1.2)(b). The P × Q-Lemma implies that [C_{W} (O_{2}(P̃)), O^{2}(F)] ≠ 1; in particular [Y_{P}, O^{2}(P̃)] ≠ 1, which contradicts (3.5). We have shown:

(9) O_{2}(P) ∩ O_{2}(F) is not normal in F.

Note that F ≰ M since M ∩ C̃ ≤ N_{M}(Z_{0}), so by (6) and (7) J(O_{2}(P)) = J(T) ≰ O_{2}(F). Assume that there exists only one non-central F -chief factor (in a given F -chief series) of W. As by (9)

[O^{2}(F), O_{2}(F)] ≰ O_{2}(F) ∩ O_{2}(P) and C_{O_{2}(F)}(W) ≤ O_{2}(F) ∩ O_{2}(P),

we get [O^{2}(F), O_{2}(F), W] ≠ 1. Thus by [Ste2, 3.3] there exists B ≤ O_{2}(F) such that

[Y_{P} , B, B] = 1 ≠ [Y_{P} , B] and |[Y_{P} , B]| ≤ |B/C_{B}(Y_{P})|.

The structure of P given in (3.1) shows that B ≤ P^{∗}. But then (1), (3) and (4) imply B ≤ C_{P^{∗}}(Y_{0}) = O_{2}(P) = C_{P^{∗}}(Y_{P}), a contradiction.

We have shown that there are at least two non-central F -chief factors in W. Let B_{1} ∈ A(O_{2}(P)) with B_{1} ≰ O_{2}(F). From (2.1) we get that

|B_{1}/C_{B_{1}}(W^{∗})| ≤ |W^{∗}/C_{W^{∗}}(B_{1})|

for all non-central F -chief factors W^{∗} of W.

We now apply the qrc-Lemma [Ste2, 3.1(c)] to F and B_{1} and get (q − 1)(rc − 1) ≤ 1 (where q, r and c are the parameters defined in [Ste]). Since by [Cher] r ≥ 1 it follows that q ≤ 2. Hence, there exists B ≤ O_{2}(F) such that

(+) |B/C_{B}(Y_{P})|^{2} ≥ |Y_{P} /C_{Y_{P}} (B)|.

Again by (3) and (4) C_{P^{∗}}(Y_{0}) = O_{2}(P) and thus B ∩ P^{∗} ≤ O_{2}(P).

As above, we now treat the two cases (∗) and (∗∗) separately. It remains to prove the isomorphism type of \overline{P}.

Assume case (∗). Then \overline{B} induces a field automorphism of order 2 on \overline{P}^{∗}. Hence (+) gives |Y_{P} | = 4^{2} and \overline{P} ≅ S_{5}.

Assume case (∗∗). Then Y_{P} = Y_{0}V_{i}, i = 1, 2, and again |\overline{B}| = 2 and |Y_{P} | = 4^{2}, so \overline{P} ≅ S_{3} wr S_{2}.

L-Lemma. Let X ∈ P_{H}(S) and A ≤ S such that A ≰ O_{p}(X), and let M be the unique maximal subgroup of X containing S. Then there exists a subgroup O_{p}(X) ≤ L ≤ X with A ≤ L satisfying:

(i) AO_{p}(L) is contained in a unique maximal subgroup L_{0} of L, and L_{0} = L ∩ M^{g} for some g ∈ X.

(ii) L = ⟨A, A^{x}⟩O_{p}(L) for every x ∈ L \ L_{0}.

(iii) L is not contained in any X-conjugate of M.

Proof. For U ≤ X set

U^{∗} := ⟨A^{g} | g ∈ X, A^{g} ≤ U⟩.

Note that N_{X}(U) ≤ N_{X}(U^{∗}); in particular N_{X}(S^{∗}) ≤ M. Choose Y among all X-conjugates of M such that Y ≠ M and for T ∈ Syl_{p}(Y ∩ M)

|T^{∗}| is maximal.

Without loss of generality we may assume that T ≤ S. Let h ∈ X such that T ≤ S^{h} ≤ Y and set N := N_{X}(T^{∗}) and S_{1} := S ∩ N. Then T ≠ S^{h} since Y ≠ M, so also T < N_{S^{h}}(T) ≤ N ∩ S^{h}. As T ∈ Syl_{p}(Y ∩ M) this gives N ≰ M. Since N_{X}(S^{∗}) ≤ M this implies that T^{∗} ≠ S^{∗} and thus also T^{∗} ≠ S_{1}^{∗}. Hence, there exists a conjugate B = A^{g}, g ∈ X, such that B ≤ S_{1} and B ≰ T. Choose z ∈ N \ M such that L_{1} := ⟨B, z⟩T^{∗} is minimal, and set L := L_{1}^{g^{−1}}O_{p}(X).

Since T^{∗} ≠ BT^{∗} = (BT^{∗})^{∗} the maximality of T^{∗} shows that M is the unique conjugate containing BT^{∗}. In particular, (iii) holds since L_{1} ≰ M. Moreover, the minimality of L_{1} gives (i). Let x ∈ L_{1} \ M. Then M^{x} is the unique conjugate of M containing B^{x}T^{∗} and M ≠ M^{x}, so B^{x} ≰ M and ⟨B, B^{x}⟩T^{∗} = L_{1}. This gives (ii).

(3.8) Assume Hypothesis II. Let A ≤ S such that [V_{β}, A, A] = 1 and A ≰ Q_{β}. Then there exist τ ∈ Δ(β), T ∈ Syl_{p}(G_{β} ∩ G_{τ}) and L ≤ G_{β} such that for L(τ) := N_{L}(Z(τ, β)), W := ⟨Z_{τ}^{L}⟩ and W^{∗} := ⟨v^{h} | v ∈ Z_{τ} , h ∈ L, [v, T] ≤ Z_{β}^{∗}⟩ the following hold:

(a) Q_{β} ≤ AO_{p}(L) ≤ T ∩ L ∈ Syl_{p}(L(τ)), and L(τ) is a maximal subgroup of L.

(b) L = ⟨y, A^{x}⟩O_{p}(L) for every x ∈ L and every y ∈ L \ L(τ)^{x}.
(c) [W^{∗}, O^{p}(L)] ≠ 1 and [W, O^{p}(L)] ≰ W^{∗}.

(d) Let U be a non-central L-chief factor of W. Then C_{U}(A) = C_{U}(a) for every a ∈ A \ O_{p}(L), and |U/C_{U}(A)| ≥ |A/A ∩ O_{p}(L)|.

Proof. According to (3.1), (3.4), (3.5) and (3.6) b ≥ 3 and α′ ∈ β^{G}; in particular Q_{τ} ≰ Q_{β} for all τ ∈ Δ(β) since Z_{α} ≤ Q_{α′−1} and Z_{α} ≰ Q_{α′}. We apply the L-Lemma with G_{β} in place of X. Then there exists Q_{β} ≤ L ≤ G_{β} and τ ∈ Δ(β) such that

(i) L(τ) is the unique maximal subgroup of L containing AO_{p}(L), and AO_{p}(L) ≤ T ∩ L ∈ Syl_{p}(L(τ)) for some T ∈ Syl_{p}(G_{β} ∩ G_{τ}).

(ii) L = ⟨A, A^{x}⟩O_{p}(L) for every x ∈ L \ L(τ). (iii) ⟨L, T_{0}⟩ = G_{β} for every T_{0} ∈ Syl_{p}(G_{β}). Claim (a) follows directly from (i). Let y and x be as in (b). Then y′ := y^{x^{−1}} ∈ L \ L(τ) and by (ii)

L = ⟨A, A^{y′}⟩O_{p}(L) = ⟨A, y′⟩O_{p}(L).

This implies (b).

For the proof of (c) assume first that [W^{∗}, O^{p}(L)] = 1. Then W^{∗} ≤ Z_{τ} and [W^{∗}, T] ≤ Z_{β}^{∗} ≤ W^{∗} since L = O^{p}(L)(T ∩L). By (iii) W^{∗} is normal in ⟨L, T ⟩ = G_{β}. But this implies that W^{∗} = Z_{β}^{∗} = Z_{τ}, a contradiction.

Assume now that [W, O^{p}(L)] ≤ W^{∗}. Then W = W^{∗}Z_{τ} and
Z_{β}^{∗}[W, Q̃_{β}] = Z_{β}^{∗}[Z_{τ} , Q̃_{β}] ≤ Z_{τ} .

Hence Z_{β}^{∗}[Z_{τ} , Q̃_{β}] is normal in ⟨T, L⟩ = G_{β}. On the other hand Q_{τ} ≰ Q_{β} and thus by (1.3)(b) [Z_{τ} , Q̃_{β}] ≤ Z_{β}^{∗}. Let g ∈ G_{β} such that τ = α^{g}. Then the action of P^{g} on Z_{τ}, as described in (3.1), shows that

[Z_{τ} , Q̃_{β} ∩ S_{0}^{g}] = Z(τ, β) ≤ Z_{β}^{∗},

which contradicts Hypothesis II. Hence, (c) is proved.

Note that L is minimal parabolic (with respect to T ∩ L and L(τ)). Hence by (1.3)(b) C_{T ∩L}(U) = O_{p}(L) for every non-central L-chief factor U in W. (2.1)(e) shows that

|U/C_{U}(A)| ≥ |A/A ∩ O_{p}(L)|.

Let a ∈ A \ O_{p}(L). Then by (1.3)(b) there exists x ∈ L such that a ∉ L(τ)^{x}. By (b) L = ⟨a, A^{x}⟩O_{p}(L) and thus, together with the quadratic action of A on U,

U = [U, a] × [U, A^{x}] = C_{U}(a) × C_{U}(A^{x});

in particular C_{U}(a) = [U, a] ≤ C_{U}(A) and equality holds. This is (d).

(3.9) No group satisfies Hypothesis II.

Proof. Assume Hypothesis II. By (3.1), (3.4), (3.5) and (3.6) [Z_{α}, Z_{α′}] = 1 and b ≥ 3. In particular, α′ ∈ β^{G} and V_{β} acts quadratically on V_{α′}, and vice versa. We apply (3.8) with (G_{α′}, V_{β}) in place of (G_{β}, A) and choose the notation τ, L, T, W, W^{∗} as there.

(1) Z_{μ} ≰ G_{ρ} for every ρ ∈ Δ(β) and μ ∈ τ^{L} such that Z_{ρ} ≰ L(μ).

Assume that there exist ρ ∈ Δ(β) and μ ∈ τ^{L} such that Z_{ρ} ≰ L(μ) but Z_{μ} ≤ G_{ρ}. Let x ∈ L such that μ = τ^{x}. Then, with the notation of (3.1) applied to G_{ρ}, there exists a submodule V_{i} ≤ Z_{ρ} such that V_{i} ≰ L(μ). By (3.8)(b) ⟨V_{i}, V_{β}^{x}⟩O_{p}(L) = L. On the other hand Z_{μ} ≤ G_{ρ}, and (3.1) together with the quadratic action of Z_{μ} on Z_{ρ} gives either

[V_{i}, Z_{μ} ∩ W^{∗}] = 1 or [V_{i}, Z_{μ}] = [V_{i}, Z_{μ} ∩ W^{∗}].

In the first case Z_{μ} ∩ W^{∗} is normal in L. Hence W^{∗} = Z_{μ} ∩ W^{∗}, and by (1.3)(b) [W^{∗}, O^{p}(L)] = 1 since V_{i} ≰ O_{p}(L). In the second case [W, O^{p}(L)] ≤ W^{∗} since O^{p}(L) ≤ ⟨V_{i}^{L}⟩, so both cases contradict (3.8)(c), and (1) is proved.

In particular, (1) together with V_{β} ≰ O_{p}(L) gives W ≰ Q_{β}. Hence, we are allowed to apply (3.8) to (G_{β}, W) in place of (G_{β}, A). Again we use the notation of (3.8), but this time indicated by to distinguish from the above notation, so τ̃, L̃, T̃, W̃, W̃^{∗} are given as there. With the same argument as above we get

(2) Z_{μ} ≰ G_{ρ} for every ρ̃ ∈ Δ(α′) and μ̃ ∈ τ_{L} such that Z_{ρ} ≰ L̃(μ̃).

As above, (2) implies W̃ ≰ O_{p}(L). We now choose μ ∈ τ^{L} and μ̃ ∈ τ_{L} such that W̃ ≰ L(μ) and W ≰ L̃(μ̃). From (1) and (2) we get that Z_{μ} ≰ O_{p}(L) and Z_{μ} ≰ O_{p}(L̃). Moreover, we may assume that |W/W ∩ O_{p}(L̃)| ≤ |W̃/W̃ ∩ O_{p}(L)|, since the other case follows by the same argument with the roles of W and W̃ reversed.

From (3.8)(c) we get that there exist two non-central L- chief factors U_{1} and U_{2} in W. As Z_{μ} ≰ O_{p}(L) (3.8)(d) implies that C_{U_{i}}(V_{β}) = C_{U_{i}}(Z_{μ}), so, again by (3.8)(d),

|W̃/W̃ ∩ O_{p}(L)| ≤ |V_{β}/V_{β} ∩ O_{p}(L)| ≤ |U_{i}/C_{U_{i}}(V_{β})| = |U_{i}/C_{U_{i}}(Z_{μ})|.

Hence

|W̃/W̃ ∩ O_{p}(L)|^{2} ≤ |U_{1}/C_{U_{1}}(Z_{μ})||U_{2}/C_{U_{2}}(Z_{μ})| ≤ |W/C_{W} (Z_{μ})|

≤ |W/W ∩ Q_{μ}| ≤ |W/W ∩ O_{p}(L̃)||W ∩ G_{μ}/W ∩ Q_{μ}|

≤ |W̃/W̃ ∩ O_{p}(L)||W ∩ G_{μ}/W ∩ Q_{μ}|.

On the other hand by (3.7), applied to G_{μ} with A = W, we get |W ∩ G_{μ}/W ∩ Q_{μ}| ≤ 2. It follows that

(3) |W/W ∩ O_{p}(L̃)| = |W̃/W̃ ∩ O_{p}(L)| = 2 = p and |Z_{μ}| = |Z_{μ}| = 16.

(4) |W/C_{W} (Z_{μ})| = |W̃/C_{W} (Z_{μ})| = 4.

As a consequence we get from (3)

(5) Z_{μ} ≰ L(μ) and Z_{μ} ≰ L̃(μ̃).

Next we prove:

(6) L/C_{L}(W) ≅ L̃/C_{L}(W̃) ≅ S_{3}.

Let t ∈ Z_{μ} \ O_{2}(L) and x ∈ L such that μ = τ^{x}. Then by (3) and (5) L = ⟨t, t^{x}⟩O_{2}(L) and thus O^{2}(L) ≤ ⟨t^{L}⟩. Hence, (3.8)(c) gives W^{∗} ≰ C_{W} (t) and W^{∗}C_{W} (t) ≠ W, and (6) follows for L from |W/C_{W} (Z_{μ})| = 4. A similar argument gives the claim for L̃.

Set W_{0} := W and W_{i} := [W_{i−1}, Q̃_{α′}] for i ≥ 1, and note that W_{i} = ⟨(W_{i} ∩ Z_{μ})^{L}⟩.

(7) Assume that (W_{i} ∩ Z_{μ})W_{i+1} = W_{i}. Then W_{i} ≤ Z_{μ}.

Note that W_{i+1} = [W_{i}, Q̃_{α′}] ≤ [Z_{μ}W_{i+1}, Q̃_{α′}] ≤ Z_{μ}W_{i+2}. It follows that W_{i} = (W_{i} ∩ Z_{μ})W_{k} for all k ≥ i + 1 and thus W_{i} ≤ Z_{μ}.

(8) [Z_{μ}, Z_{μ} ∩ O_{2}(L̃)] ≠ 1.

Let A_{1} := Z_{μ} ∩ O_{2}(L̃),and assume that [Z_{μ}, A_{1}] = 1. By (6) L(μ) = (L(μ) ∩ G_{μ})C_{L}(W). Suppose that Z_{μ} = A_{1}(Z_{μ} ∩ W_{1}). Then W = Z_{μ}W_{1} and by (7) W = Z_{μ}. But then Z_{μ} is normal in ⟨L, G_{α′} ∩ G_{μ}⟩ = G_{α′}, a contradiction. We have shown that Z_{μ} ∩ W_{1} ≤ A_{1}. It follows that Z_{μ} ∩ W_{1} is centralized by Z_{μ} and thus normalized by L, so W_{1} ≤ A_{1} and [W_{1}, O^{2}(L)] = 1. In particular [Z_{μ}, Q̃_{α′}] is normalized by L and centralized by O^{2}(L). Hence, by the L-Lemma (iii) it is also normalized by G_{α′} and centralized by O^{2}(G_{α′}). Since Z(μ, α′) ≤ [Z_{μ}, Q_{α′}] we get that Z(μ, α′) is normal in G_{α′}, a contradiction to Hypothesis II.

(9) R := [Z_{μ} ∩ O_{2}(L), Z_{μ} ∩ O_{2}(L̃)] ≠ 1, and R is centralized by a Sylow 2-subgroup of G_{μ} and G_{μ}.

Let A := Z_{μ} and A_{0} := A ∩ G_{μ}. By (8) Y_{0} := [Z_{μ}, A_{0}] ≠ 1, and by (5) A and G_{μ} satisfy the hypothesis of (3.7). Then (3.7) shows that |Y_{0}| = 4 and |A_{0}/A_{0} ∩ Q_{μ}| = 2; in particular A_{0} = A∩O_{2}(L̃). Moreover, (3.7) gives |Z_{μ}/C_{Z_{μ}}(A_{0})| = 4 and thus R ≠ 1 since |Z_{μ}/Z_{μ}∩O_{2}(L)| = 2.

The action of G_{μ} on Z_{μ} also shows that all elements of Y_{0} are centralized by a Syolw 2-subgroup of G_{μ}. This and the symmetric argument in G_{μ} yields the additional claim of (9).

We now derive a final contradiction. According to (9) there exist y ∈ G_{μ} and z ∈ G_{μ} such that R = Z_{β}^{y} = Z_{α′}^{z}. Then by (1.6) C̃_{β}^{y} = C̃_{α′}^{z} and thus Q̃_{β}^{y} = Q̃_{α′}^{z}. On the other hand, Hypothesis I and (1.2)(b) yield Z_{μ} ≤ Q̃_{β}^{y}, so Z_{μ} ≤ Q̃_{α′}^{z} ≤ G_{μ}, which contradicts (2) and (5).

Theorem 3. Assume Hypothesis I. Then Z_{0} is normal in C̃.

Proof. Assume that Z_{0} is not normal in C̃. By the definition of C̃ N_{H}(S) ≤ C̃. Hence, N_{H}(S) acts on P_{H}(S) \ P_{C}(S), and Theorem 2 implies that N_{H}(S) ≤ N_{H}(P) and thus also N_{H}(S) ≤ N_{H}(P^{∗}) since P^{∗} = U(P). It follows that N_{H}(S) ≤ N_{H}(S_{0}) ≤ N_{H}(Z_{0}).

According to (1.3)(a) there exists P̃ ∈ P_{C}(S) such that Z_{0} is not normal in P̃. We choose |P̃| minimal with this property. If (P, P̃) is an amalgam, then (P, P̃) satisfies Hypothesis II, which is impossible by (3.9).

Thus, (P, P̃) is not an amalgam, and there exists L ∈ L_{H}(S) such that ⟨P, P̃⟩ ≤ L. Let L << M̃ ∈ L_{H}^{∗}(S). Then by (1.2) Y_{L} ≤ Y_{M} and by (1.4) P^{0} ≤ L^{0} ≤ M̃^{0} ≤ M̃.

We now apply the Local P!-Theorem to M̃. Assume that also P̃ ≤ M̃. Then P̃ ≤ M̃ ∩ C̃ ≤ N_{M}(Z_{0}), a contradiction. Thus, we have P̃ ≰ M̃.

Assume first that case (a) of the Local P!-Theorem holds. Then Q ≤ S_{0}, so Z_{0} ≤ Z(Q) and thus also W := ⟨Z_{P} ; 0⟩ ≤ Z(Q). Note that

Z_{0} ≤ Y_{P} = [Y_{P} , P^{0}] ≤ [Y_{L}, L^{0}] and W ≤ [Y_{L}, L^{0}]

by (1.2). It follows that W ≤ [Y_{M}, M̃_{0}] since Y_{L} ≤ Y_{M} and L^{0} ≤ M̃^{0}. In case (a) [Y_{M}, M̃_{0}] is a natural SL_{n}(p^{m})- or Sp_{2n}(p^{m})′-module. In particular, C_{[Y_{M},_{M_{0}}]}(Q) = Z_{0} and so Z_{0} = W and P̃ ≤ N_{H}(Z_{0}), a contradiction.

Assume finally that case (b) of the Local P!-Theorem holds for M̃. Then P^{0} = L^{0} = M̃^{0} and P̃ ≤ N_{H}(M̃^{0}) = M̃, which contradicts P̃ ≰ M̃.

Corollary 2. Assume Hypothesis I and p = 2. Then P_{H}(S) = {P } ∪ P_{C}(S).

Proof. We apply Theorem 2. Then P_{H}(S) = P_{P} (S)∪P_{C}(S), and the structure of P, see (3.1), implies P_{P} (S) = {P } ∪ P_{N_{P} (Z_{0})}(S). Now Theorem 3 yields the assertion.

Corollary 3. Assume Hypothesis I. Suppose that case (b) of the Local P!-Theorem holds for P ≤ M ∈ L_{H}^{∗}(S). Then the following holds:

(a) p = 2 and M_{H}(P) = {M},

(b) \overline{P}^{∗} = K_{1} × ⋯ × K_{r}, K_{i} ≅ SL_{2}(2),

(c) Y_{P} = V_{1} × ⋯ × V_{r}, where V_{i} = [Y_{P} , K_{i}] is a natural SL_{2}(2)-module for K_{i},

(d) r ≥ 4.

(e) Q is transitive on K_{1}, ..., K_{r}.

Proof. We are in case (b) of the Structure Theorem. According to Theorem 3 Z_{0} is normal in C̃. Hence

(∗) [N_{P} (Z_{0}), Q] ≤ O_{p}(N_{P} (Z_{0})).

We apply (3.1). Then either \overline{P}^{∗} ≅ SL_{2}(p^{m}), or the Q-transitivity and (∗) show that N_{K_{i}}(Z_{0}) is a p-group and r ≥ 2.

In the first case Y_{P} is a natural SL_{2}(p^{m})-module for P^{∗}. Thus, Y_{P} is an F -vector space for F := End\overline{P}^{∗}(Y_{P}), and P induces semi-linear transformations on Y_{P}. As N_{P^{∗}}(Z_{0}) is irreducible on Z_{0}, we get from (∗) that [Z_{0}, Q] = 1, so Q centralizes a 1-dimensional F -subspace of Y_{P}. Hence, Q induces F -linear transformations on Y_{P}, and Q ≤ P^{∗}. But this contradicts case (b) of the Structure Theorem.

In the second case (a) – (c) and (e) are clear. For the proof of (d) note that Q-transitivity yields r = 2 or (d). Assume r = 2, so P/C_{P} (Y_{P}) ≅ O_{4}^{+}(2) and |Z_{0}| = 4. Hence, Theorem 3 shows that C̃/C_{C}(Z_{0}) is a subgroup of S_{3}. If all involutions in Z_{0} are conjugate in C̃, then Q-Uniqueness implies that P ≤ C̃, which is not the case. It follows that C̃ = C_{C}(Z_{0})S, in particular C_{C}(Z_{0}) ≰ M. We conclude that C_{H}(x) ≰ M for all 1 ≠ x ∈ Y_{P}. Now Theorem 3 of [MSS2] shows that Y_{M} ≰ Q, a contradiction.

4. F-Uniqueness

In this section we treat the exceptional case described in Corollary 3, so in this section we assume:

Hypothesis III. Hypothesis I and case (b) of the Local P!-Theorem holds for P ≤ M ∈ L_{H}^{∗}(S); in particular M_{H}(P) = {M}.

Notation. We use the notation given in Corollary 3 (and (3.1)). Set

F := C_{C}(Z_{0}) and Ω := {K_{1}, ..., K_{r}}.

Recall that by Theorem 3 F is normal in C̃, and by Corollary 3 (∗) p = 2, K_{i} ≅ SL_{2}(2), r ≥ 4, and Q is transitive on Ω. We will use these facts without further reference.

(4.1) P^{∗} ∩ C̃ = S_{0}C_{P^{∗}}(Y_{P}) and C̃ = C.

Proof. Assume that U := C_{P^{∗}}(Y_{P})S_{0} < P^{∗} ∩ C̃. Then by Corollary 3 (b) K_{i} ≤ [\overline{S}_{0}, \overline{P^{∗} ∩ C̃}]\overline{S}_{0} for some i, and the Q-transitivity yields P^{∗} ≤ C̃, which is not the case.

Let Z^{∗} = ⟨Z_{C}⟩. By Theorem 3 Z^{∗} ≤ Z_{0} ∩ Z(Q), and by Q-uniqueness C_{P^{∗}}(z) ≤ P^{∗} ∩ C̃ = S_{0}C_{P^{∗}}(Y_{P}) for all 1 ≠ z ∈ Z^{∗}. Now Corollary 3 (c) yields |Z^{∗}| = 2, so C = C̃.

(4.2) N_{H}(B(S)) ≤ M.

Proof. It suffices to show that P and N_{H}(B(S)) are contained in a 2-local subgroup of H since M_{H}(P) = {M}. Assume that this is not the case; i.e. O_{2}(⟨P, N_{H}(B(S))⟩ = 1. Then B(S) is not normal in P and by (3.1) B(S) = S_{0}. Hence, N_{H}(B(S)) = N_{H}(S_{0}) ≤ N_{H}(Z_{0}) = C̃. For every i = 1, ..., r we choose X_{i} ≤ P^{∗} minimal with respect to

B(S) ≤ X_{i} and \overline{X}_{i} = K_{i}\overline{B(S)}.

Then X_{i} ∈ P_{H}(B(S)) and ⟨X_{i}, S⟩ = P. Moreover V_{i} = [Y_{X_{i}}, O^{2}(X_{i})] = [Y_{P} , O^{2}(X_{i})] since Y_{X_{i}} ≤ Ω_{1}(Z(O_{2}(P))) = Y_{P}.

Suppose that there is a non-trivial characteristic subgroup A of B(S), which is normal in X_{1}. Then ⟨S, X_{1}, N_{H}(B(S))⟩ = ⟨P, N_{H}(B(S))⟩ ≤ N_{H}(A), which contradicts O_{2}(⟨P, N_{H}(B(S))⟩ = 1.

Hence, no non-trivial characteristic subgroup of B(S) is normal in X_{1}. Now [Ste1] gives [O_{2}(X_{1}), O^{2}(X_{1})] = V_{1} ≤ Y_{P}. Hence also [O_{2}(P), O^{2}(P)] ≤ Y_{P}, and Z(P) = 1 yields

Y_{P} = O_{2}(P) = V_{1} × ⋯ × V_{r}.

Since Q is transitive on {V_{1}, ..., V_{r}} and N_{H}(B(S)) does not normalize Y_{P} there exists t ∈ N_{H}(B(S)) such that R := [V_{1}, V_{1}^{t}] ≠ 1. It follows that also [V_{1}^{t}, V_{1}^{t^{2}}] ≠ 1, so

R^{t} = [V_{1}^{t}, V_{1}^{t^{2}}] = [V_{1}^{t}, V_{1}] = R

since ⟨V_{1}, V_{1}^{t^{2}}⟩ ≤ B(S) ≤ N_{P} (V_{1}^{t}). As t ∈ C̃ and Y_{P} is normal in Q the Q-transitivity gives

(∗) S_{0} = Y_{P} Y_{P}^{t} and Y_{P} ∩ Y_{P}^{t} = Z_{0}.

Let U = N_{H}(R) and W = O_{2}(U). Then ⟨t, X_{2}, ..., X_{r}⟩ ≤ U, and V_{i} ∩ W is X_{i}-invariant for every i ≥ 2. It follows that either there exists an i ≥ 2 such that V_{i} ≤ W, or V_{i} ∩ W = 1 for every i ≥ 2. The first case gives V_{i}^{t} ≤ W and so V_{i}^{t} ≤ O_{2}(X_{2} ⋯ X_{r}). On the other hand, by (∗) [Y_{P} , V_{i}^{t}] ≠ 1, so we get that [V_{i}^{t}, V_{1}] = R. But this implies that R ≤ V_{i}^{t} and R = R^{t} ≤ V_{i}, which is impossible since V_{1} ∩ V_{i} = 1 for i > 1.

We have shown that V_{i} ∩ W = 1 for i > 1. It follows that [S_{0} ∩ W, O^{2}(X_{2})] = 1. Since S_{0} ∩ W is normalized by X_{2} and W we get [(S_{0} ∩ X_{2})^{x}, W] ≤ S_{0} ∩ W for every x ∈ X_{2}. Hence [W, O^{2}(X_{2})] ≤ S_{0} ∩ W and [W, O^{2}(X_{2}), O^{2}(X_{2})] = 1. But then U is not of characteristic 2, a contradiction.

(4.3) Let S_{0} ≤ T, T a 2-subgroup of H. Then S_{0} is normal in N_{H}(T) and N_{H}(T) ≤ M ∩ C̃.

Proof. Note that N_{H}(T) ≤ N_{H}(B(S)) ≤ M by (3.1) and (4.2). Moreover, by the Structure Theorem, case (b), Y_{P} = Y_{M} and P^{∗}C_{M}(Y_{M}) is normal in M, so T ∩ P^{∗}C_{M}(Y_{M}) = S_{0}. Hence Theorem 3 gives N_{M}(T) ≤ N_{M}(S_{0}) ≤ M ∩ C̃.

(4.4) Let L̃ ∈ L_{H}(S). Then either L̃ ≤ C̃, or P ≤ L̃ ≤ M and F ≰ L̃.

Proof. Assume that L̃ ≰ C̃. Then (1.3)(a) and the Corollaries 2 and 3 show that P ≤ L̃ ≤ M. If in addition F ≤ M, then the Frattini argument and (4.3) imply that C̃ = F N_{H}(S_{0}) ≤ M, a contradiction.

(4.5) Suppose that S_{0} ≤ T ≤ S such that |S/T | = 2 and S = T Q. Let T ≤ L ≤ H and O_{2}(L) ≠ 1. Then one of the following holds:

(a) L ≤ M.

(b) L ≤ C̃.
(c) L ∈ L_{H}(T).

Proof. Let U = N_{H}(O_{2}(L)) and T ≤ T_{0} ∈ Syl_{2}(U). By (4.3) T_{0} ≤ M ∩ C̃ and thus either T = T_{0} or T_{0} ∈ Syl_{2}(C̃) and Q ≤ T_{0}. In the second case T_{0} = T Q = S, and (4.4) yields L ≤ U ≤ M or L ≤ U ≤ C̃. In the first case U ∈ L_{H}(T) and thus also L ∈ L_{H}(T).

Notation. From now on we fix a maximal subgroup T of S containing N_{S}(K_{1}). Recall that B(S) ≤ S_{0} ≤ T. Let Q_{0} := T ∩ Q and

L_{0}(T) := {U ∈ L_{H}(T) | U ≰ C̃ and U ∩ C̃ ≰ M}.

By L_{0}(T)_{∗} we denote the set of minimal elements of L_{0}(T).

(4.6) Let \overline{P}^{∗} := P^{∗}/C_{P^{∗}}(Y_{P}) and 1 ≠ K ≤ O^{2}(\overline{P}^{∗}). Suppose that K is Q_{0}-invariant. Then K = O^{2}(\overline{P}^{∗}) or K = ×_{X∈Ω_{i}}X′ for some T -orbit Ω_{i} of Ω; in particular [K, \overline{Q}_{0}] ≠ 1.

Proof. Since K ≠ 1 there exist K_{i} ∈ Ω and t ∈ S_{0} ∩ K_{i} such that [K, \overline{t}] = K′_{i}. Let q ∈ Q such that K_{i}^{q} ≠ K_{i}, and let q_{0} := [t, q] and R := [K, \overline{q}_{0}]. Then q_{0} ∈ S_{0} ∩ Q ≤ Q_{0} and R ≤ (K_{i} × K_{i}^{q}) ∩ K with [R, \overline{t}] = K′_{i}.

Since r > 2 there exists x ∈ Q such that K_{i}^{x} ∉ {K_{i}, K_{i}^{q}}. Let x_{0} = [t, x]. Then as above x_{0} ∈ Q_{0} ∩ S_{0}, while \overline{x}_{0}C_{\overline{S}_{0}}(K_{i} × K_{i}^{q}) = \overline{t}C_{\overline{S}_{0}}(K_{i} × K_{i}^{q}). It follows that [R, \overline{x}_{0}] = K′_{i} ≤ K.

We have shown that K′_{i} ≤ K for every K_{i} ∈ Ω such that [K, K_{i}] ≠ 1. Now the action of Q_{0} on K and Ω gives the desired structure of K. Moreover, r > 2 implies that [K, \overline{Q}_{0}] ≠ 1.

(4.7) |S/T | = 2, S = T Q, and T has two orbits Ω_{1} and Ω_{2} on Ω such that for Z_{i} := C_{Ω_{1}(Z(T))}(Ω_{i}) the following hold:

(a) |Ω_{i}| = \frac{r}{2} and |Z_{i}| = 2, i = 1, 2, and

(b) Ω_{1}(Z(T)) = Z_{1} × Z_{2}.

Proof. This is a direct consequence of the choice of T.

(4.8) L_{0}(T) ≠ ∅.

Proof. Let L := C_{H}(Z_{1}), Z_{1} as in (4.7). Then L ≰ C̃, and by (4.4) L∩C̃ ≰ M since F ≤ L∩C̃. Now (4.5) shows that L ∈ L_{0}(T).

(4.9) Let L ∈ L_{0}(T). Then O_{2}(⟨O^{2}(P^{∗}), L ∩ C̃⟩) = 1.

Proof. Let L_{0} := ⟨O^{2}(P^{∗}), L ∩ C̃⟩ and assume that O_{2}(L_{0}) ≠ 1. Let t ∈ Q \ T. Then T ⟨t⟩ = S since T has index 2 in S. Moreover, [t, L ∩ C̃] ≤ Q_{0} ≤ O_{2}(L ∩ C̃). It follows that t normalizes L_{0}. Hence S ≤ L_{0}⟨t⟩ and 1 ≠ O_{2}(L_{0}) ≤ O_{2}(L_{0}⟨t⟩). This contradicts (4.4) since L_{0} ≰ M as L ∩ C̃ ≰ M and L_{0} ≰ C̃ as O^{2}(P^{∗}) ≰ C̃.

Theorem 4. Suppose that L ∈ L_{0}(T). Then

P_{L}(T) = P_{L∩M}(T) ∪ P_{L∩_{C}}(T).

Proof. Assume that there exists X ∈ P_{L}(T) such that X ≰ M and X ≰ C̃. By (4.2) and (1.3)(b) neither B(S) nor Ω_{1}(Z(T)) is normal in X. Hence, (2.9) implies that there exists a minimal parabolic subgroup X_{0} of characteristic 2 in X such that X_{0} satisfies (2.9)(a) – (e) (in place of L_{i}); in particular X = ⟨T, X_{0}⟩, O_{2}(X)B(S) ∈ Syl_{2}(X_{0}) and X_{0}/C_{X_{0}}(Y_{X_{0}}) ≅ SL_{2}(2^{k}). We choose X^{∗} ≤ X_{0} minimal with respect to

B(S) ≤ X^{∗} and X_{0} = X^{∗}C_{X_{0}}(Y_{X_{0}}).

Then X^{∗} is a minimal parabolic subgroup and X = ⟨X^{∗}, T ⟩. Moreover B(S) ∈ Syl_{2}(X^{∗}) by (2.7) applied to X^{∗}.

Assume that there exists a non-trivial characteristic subgroup A of B(S) which is normal in X^{∗}. As A is also characteristic in S we get

(∗) X = ⟨T, X^{∗}⟩ ≤ N_{H}(A) and S ≤ N_{H}(A).

Hence by (4.4) N_{H}(A) ≤ C̃ or N_{H}(A) ≤ M, which contradicts X ≤ N_{H}(A).

Thus, no non-trivial characteristic subgroup of B(S) is normal in X^{∗}. As X^{∗} is a minimal parabolic subgroup the hypothesis of [Ste1] is satisfied. We get [O^{2}(X^{∗}), O_{2}(X^{∗})] = [Y_{X^{∗}}, O^{2}(X^{∗})] and Y_{X^{∗}}/C_{Y_{X^{∗}}} (X^{∗}) is a natural SL_{2}(2^{k})-module for X^{∗}/C_{X^{∗}}(Y_{X^{∗}}), so [O^{2}(X^{∗}), O_{2}(X^{∗})] ≤ Y_{X}. Since [O_{2}(X), B(S)] ≤ B(S) ∩ O_{2}(X) ≤ O_{2}(X^{∗}) we also get

[O_{2}(X), O^{2}(X^{∗})] ≤ Y_{X} and [O_{2}(X), O^{2}(X)] ≤ Y_{X}.

As in the proof of (4.9) pick t ∈ Q \ T. Then

(∗∗) [L ∩ C̃, t] ≤ Q ∩ T ≤ O_{2}(L ∩ C̃).

Assume first that Y_{X}^{t} ≤ O_{2}(X). Then

S ≤ ⟨X, t⟩ ≤ N_{H}(Y_{X}Y_{X}^{t}) ∈ L_{H}(S),

and by (4.4) N_{H}(Y_{X}Y_{X}^{t}) ≤ M or N_{H}(Y_{X}Y_{X}^{t}) ≤ C̃. But this contradicts X ≤ N_{H}(Y_{X}Y_{X}^{t}).

We have shown that Y_{X}^{t} ≰ O_{2}(X). As |Y_{X}/C_{Y_{X}} (Y_{X}^{t})| = |Y_{X}^{t}/C_{Y_{X}^{t}} (Y_{X})| we get Y_{X}^{t} ∈ U(X) (for the definition see section 2). Since Y_{X}^{t} is normal in T we conclude with (2.1) that Y_{X}^{t}O_{2}(X) = B(S)O_{2}(X). In addition, (2.1) shows that B(S)C_{X}(Y_{X})/C_{X}(Y_{X}) is self-centralizing in X/C_{X}(Y_{X}). It follows that O_{2}(X^{t}) ≤ Y_{X}^{t}O_{2}(X). Hence, for D := O_{2}(X) ∩ O_{2}(X^{t}) we get O_{2}(X^{t}) = Y_{X}^{t}D and similarly O_{2}(X) = Y_{X}D. This gives

Φ(O_{2}(X^{t})) = Φ(D) = Φ(O_{2}(X));

in particular ⟨X, S⟩ ≤ N_{H}(Φ(D)). Now as above (4.4) implies that Φ(D) = 1, so O_{2}(X) = Y_{X} and B(S) = Y_{X}Y_{X}^{t}.

The action of T on B(S) shows that Y_{X} and Y_{X}^{t} are the only maximal T -invariant elementary abelian normal subgroups of B(S); in particular Y_{X} = Y_{L}, and by (∗∗) L ∩ C̃ normalizes B(S). Now (4.2) yields L ∩ C̃ ≤ M, which contradicts L ∈ L_{0}(T).

(4.10) Let L ∈ L_{0}(T)_{∗} and N be a normal subgroup of L that is minimal with respect to N ≰ C̃. Then N = [N, Q_{0}] = O^{2}(L).

Proof. As N(L ∩ C̃) ∈ L_{H}(T) the minimality of L yields L = N(L ∩ C̃). Hence N_{0} := [N, Q_{0}] is a normal subgroup of L. Assume that N ≠ N_{0}. The the minimal choice of N gives N_{0} ≤ C̃, so N_{0}Q_{0} is a normal subgroup of L in C̃. It follows that Q_{0} ≤ O_{2}(N_{0}Q_{0}) ≤ O_{2}(L). But then [Q, O_{2}(L)] ≤ Q_{0} ≤ O_{2}(L) and S = T Q ≤ N_{H}(O_{2}(L)), so (4.4) implies that L ≤ C̃ or L ≤ M. This contradicts the definition of L_{0}(T).

We have shown that N = N_{0}. The minimality of N also gives that N = O^{2}(N). Thus, it remains to prove that L = NT. Assume now that L ≠ NT. By Theorem 4

P_{NT} (T) ⊆ P_{M}(T) ∪ P_{C}(T).

Since NT ≰ C̃ the minimality of L shows that NT ∩ C̃ ≤ M. Thus P_{NT} (T) ⊆ P_{M}(T). As by (4.3) also N_{L}(T) ≤ M we conclude from (1.3)(a) that NT ≤ M.

Now N = [N, Q_{0}] ≤ P, and N = O^{2}(N) implies N ≤ O^{2}(P^{∗}). Since N is also S_{0}-invariant we get from (4.1) that [Z, N] is normal in P^{∗}. On the other hand by (4.6) [Z, N] = [Z, L], so [Z, L] is normalized by L and P^{∗}. But this contradicts (4.9).

Corollary 4. Let L ∈ L_{0}(T)_{∗}. There exists a unique P_{1} ∈ P_{L}(T) such that P_{1} ≰ C̃. Moreover, the following hold:

(a) Q_{0} ≰ O_{2}(P_{1}),

(b) O^{2}(P_{1}) ≤ O^{2}(P^{∗}), and
(c) O^{2}(P_{1})C_{P^{∗}}(Y_{P})/C_{P^{∗}}(Y_{P}) = K′_{1} × ⋯ × K′_{s}, where {K_{1}, ..., K_{s}} is a T -orbit of Ω.

Proof. By (4.3) N_{L}(T) ≤ C̃, so by (1.3)(a) there exists P_{1} ∈ P_{L}(T) such that P_{1} ≰ C̃. Now Theorem 4 gives P_{1} ≤ M and again by (4.3) S_{0} ≰ O_{2}(P_{1}). Since P^{∗}C_{M}(Y_{P}) is normal in M we get from (1.3)(c) that O^{2}(P_{1}) = [O^{2}(P_{1}), S_{0}] ≤ P^{∗}C_{M}(Y_{P}).

Let \overline{M} := M/C_{M}(Y_{P}). Note that O^{2}(\overline{P}_{1}) ≠ 1 and by (4.1) (a) and (c) hold. By (a) and (1.3)(c) O^{2}(P_{1}) = [O^{2}(P_{1}), Q_{0}] ≤ [M, Q] ≤ M^{0} ≤ P, so also (b) holds.

Let P_{0} be another minimal parabolic in P_{L}(T), which is not in C̃. Then (a) – (c) hold for P_{0} in place of P_{1}. By (4.6) either

O^{2}(P_{0})O^{2}(P_{1})C_{P^{∗}}(Y_{P}) = O^{2}(P^{∗})C_{P^{∗}}(Y_{P}) or O^{2}(P_{0})C_{P^{∗}}(Y_{P}) = O^{2}(P_{1})C_{P^{∗}}(Y_{P}).

Note that [C_{P^{∗}}(Y_{P}), Q_{0}] ≤ O_{2}(P) ≤ T and Q_{0} is normal in S. Hence, in the first case (1.3)(c) implies that O^{2}(P^{∗}) = [O^{2}(P^{∗}), Q_{0}] ≤ O^{2}(P_{0})O^{2}(P_{1})O_{2}(P^{∗}) ≤ L, which contradicts (4.9). In the second case we conclude that O^{2}(P_{0})O_{2}(P) = O^{2}(P_{1})O_{2}(P) and thus O^{2}(O^{2}(P_{0})O_{2}(P)) = O^{2}(P_{0}) = O^{2}(P_{1}). Hence P_{0} = P_{1}.

(4.11) Let X be a finite group and V a faithful GF (2)X-module, and let S ∈ Syl_{2}(X) and V_{0} = C_{V} (S). Suppose that F^{∗}(X) is simple, V = ⟨V_{0}^{X}⟩ ≠ V_{0}, and

(∗) there exists an elementary abelian subgroup 1 ≠ A ≤ S such that |V/C_{V} (A)| ≤ |A|. Then there exists a minimal parabolic subgroup P_{1} containing S such that P_{1} ≰ C_{X}(V_{0}) and (P_{1} ∩ F^{∗}(X))/O_{2}(P_{1} ∩ F^{∗}(X)) ≅ SL_{2}(2^{k}) or S_{ℓ}.

Proof. A theorem of Gaschütz (see for example [Hu, I.17.4]), applied to the semidirect product of V with X, shows that V = C_{V} (X)[V, X]. Hence, there exists a X-submodule W such that \overline{V} := V/W is a faithful irreducible X-module. Moreover, property (∗) implies that|\overline{V} /C\overline{V} (A)| ≤ |A|. Thus, the F-Module Theorem for K-groups, see [GM1] and [GM2], gives the conclusion.

F!-Theorem. No group satisfies the hypothesis of this section.

Proof. We will derive a contradiction using the previous results of this chapter. According to (4.8) there exists L ∈ L_{0}(T)_{∗}. We fix the following additional notation:

C_{L} = L ∩ C̃, V = ⟨Z^{L}⟩, \overline{L} = L/C_{L}(V).

As in Corollary 4 let P_{1} be the unique element of P_{L}(T) with P_{1} ≰ C̃. Then

(1) O^{2}(P_{1}) ≤ O^{2}(P^{∗}) and O^{2}(P_{1})C_{P^{∗}}(Y_{P})/C_{P^{∗}}(Y_{P}) = K′_{1} × ⋯ × K′_{s}, where Ω_{1} := {K_{1}, ..., K_{s}} is one of the two T -orbits of Ω. From (1.3)(b) and (1) we get

O^{2}(P_{1}) ∩ C_{P^{∗}}(Y_{P}) = O_{2}(O^{2}(P_{1})) ≥ O^{2}(P_{1}) ∩ C_{L}(V),

in particular (∗) O^{2}(\overline{P}_{1})/O_{2}(O^{2}(\overline{P}_{1})) = K′_{1} × ⋯ × K′_{s}. As in (4.10) let N be a normal subgroup of L that is minimal with respect to N ≰ C_{L}. Then

by (4.10)

(2) N = [N, Q_{0}] = O^{2}(L). Moreover, since by (4.1) every normal subgroup of L in C_{L} centralizes V we get

(3) \overline{N} is a minimal normal subgroup of \overline{L}, and O_{2}(\overline{L}) = 1. Next we show:

(4) C_{L}(V) ≤ M, in particular L ≠ (L ∩ M)C_{L}(V). Assume that C_{L}(V) ≰ M. Then the minimality of L yields L = C_{L}(V)P_{1}. It follows from (2)

that

N = N ∩ (O^{2}(P_{1})C_{L}(V)) = O^{2}(P_{1})(N ∩ C_{L}(V))

and

L = NT = [N, Q_{0}]T = O^{2}(P_{1})T = P_{1} ≤ M,

which contradicts the choice of L in L_{0}(T).

(5) \overline{N} ∩ \overline{T} ≠ 1; in particular \overline{N} is not abelian. Assume that \overline{N} ∩ \overline{T} = 1. For every prime q the Frattini argument gives a \overline{Y}_{q} ∈ Syl_{q}(\overline{N}) such

that \overline{T} ≤ N\overline{U}(\overline{Y}_{q}) and \overline{N} = ⟨\overline{Y}_{q} | q ∈ π(\overline{N})⟩.

Let Y_{q} be the inverse image of \overline{Y}_{q} in L. From (1), (∗) and (1.3)(a) we get that Y_{q} ≤ C_{L} for every q ≠ 3. Hence \overline{N} = \overline{Y}_{3}C\overline{N}(\overline{Q}_{0}), so by (2)

\overline{N} = [\overline{N}, \overline{Q}_{0}] = [\overline{Y}_{3}C\overline{N}(\overline{Q}_{0}), \overline{Q}_{0}] = [\overline{Y}_{3}, \overline{Q}_{0}] ≤ \overline{Y}_{3}.

Now (3) shows that \overline{N} is elementary abelian, moreover \overline{N} = O^{2}(\overline{P}_{1}). Thus (4) gives L ≤ M, a contradiction. Hence, (5) is proved.

Let Ω_{2} be the T -orbit of Ω different from Ω_{1} = {K_{1}, ..., K_{s}}. Then by (4.7)

Ω_{1}(Z(T)) = Z_{1} × Z_{2}, Z_{i} := C_{Y_{P}} (Ω_{i}),

and P_{1} ≤ L_{1} := C_{L}(Z_{2}).

Assume that L_{1} ∩C̃ ≤ M. Then L∩F ≤ L_{1} ∩C̃ ≤ M since Ω_{1}(Z(T)) ≤ Z_{0}. Now (4.3) and the Frattini argument imply C_{L} ≤ N_{C_{L}}(S_{0})(L ∩ F) ≤ M, which contradicts the choice of L ∈ L_{0}(T). Thus L_{1} ∩ C̃ ≰ M, and the minimality of L yields:

(6) Z_{2} ≤ Z(L), in particular O^{2}(P^{∗}) ≰ L. Next we show:

(7) \overline{N} is simple.

According to (3) and (5) there exist subgroups C_{L}(V) ≤ N_{i} ≤ NC_{L}(V), i = 1, ..., k such that \overline{N} = \overline{N}_{1} × ⋯ × \overline{N}_{k}, and \overline{N}_{1}, ..., \overline{N}_{k} are simple groups conjugate under \overline{T}.

Assume first that \overline{N}_{i} ∩ \overline{C}_{L} ≤ \overline{T}, i = 1, ..., k. The projection \overline{C}_{i} of \overline{N} ∩ \overline{C}_{L} in \overline{N}_{i} is a subgroup of \overline{N}_{i} that normalizes \overline{N}_{i} ∩ \overline{T}. Hence by (5) \overline{C}_{L}(\overline{C}_{1} × ⋯ × \overline{C}_{k}) is a proper subgroup of \overline{L}, and the minimality of L implies that \overline{C}_{i} ≤ \overline{C}_{L} ∩ \overline{N}_{i}, so \overline{N} ∩ \overline{T} = \overline{N} ∩ \overline{C}_{L}. Now (4) yields C_{L} ≤ M, which contradicts the choice of L ∈ L_{0}(T).

Assume now that there exists a component \overline{N}_{1} such that \overline{N}_{1} ∩ \overline{C}_{L} is not a 2-group. Then O^{2}(\overline{N}_{1} ∩ \overline{C}_{L}) = O^{2}((\overline{N}_{1} ∩ \overline{C}_{L})O_{2}(\overline{N} ∩ \overline{C}_{L})) ≠ 1 and

[\overline{N}_{1} ∩ \overline{C}_{L}, \overline{Q}_{0}] ≤ O_{2}(\overline{C}_{L}) ∩ \overline{N} ≤ O_{2}(\overline{N} ∩ \overline{C}_{L}),

so \overline{Q}_{0} normalizes O^{2}(\overline{N} ∩ \overline{C}_{L}) and thus also \overline{N}_{1}. It follows:

(∗∗) \overline{Q}_{0} normalizes every component of \overline{N}. Among all T -invariant subgroups U ≤ N satisfying

(i) \overline{U} = \overline{U}_{1} × ⋯ × \overline{U}_{k}, U_{i} ≤ N_{i}, and

(ii) O^{2}(P_{1}) ≤ U

we choose U to be minimal. Then \overline{U ∩ N_{i}} is the projection of O^{2}(\overline{P}_{1}) into \overline{N}_{i}. From (∗) and (3) we conclude that UT ≠ L. The minimality of L implies that UT ∩ C̃ ≤ M and thus by (1.3)(a) and (4.3) UT ≤ M. On the other hand the minimality of U yields U = [U, Q_{0}] = O^{2}(U). It follows that U is a Q_{0}-invariant subgroup of O^{2}(P^{∗}). Now (4.6) and (6) show that

\overline{U} = [\overline{U}, \overline{Q}_{0}] = O^{2}(\overline{P}_{1}) = \overline{U}_{i} × ⋯ × \overline{U}_{k}.

By (∗∗) \overline{U}_{1} is Q_{0}-invariant. Hence, another application of (4.6) shows that O^{2}(\overline{P}_{1}) ≤ \overline{N}_{1}. As O^{2}(\overline{P}_{1}) is \overline{T} -invariant, also \overline{N}_{1} is. Since the groups N_{1}, ..., N_{k} are conjugate under T we conclude that k = 1.

(6) J(S) ≰ C_{L}(V).

Assume that J(S) ≤ C_{L}(V). Then V ≤ Ω_{1}(Z(J(S))) and thus also B(S) ≤ C_{L}(V). Now the Frattini argument and (4.2) yield L = N_{L}(B(S))C_{L}(V) = (L ∩ M)C_{L}(V), which contradicts (4).

We now derive a final contradiction. According to (8) there exists A ∈ A(S) such that \overline{A} ≠ 1. Hence, the maximality of A implies that |V/C_{V} (\overline{A})| ≤ |\overline{A}|, so by (7) we can apply (4.11) to \overline{L}. Thus, there exists C_{L}(V)T ≤ P_{0} ≤ L such that \overline{P}_{0} is a minimal parabolic subgroup of \overline{L}, \overline{P}_{0} ≰ C\overline{L}(V_{0}), where V_{0} := C_{V} (T) = Ω_{1}(Z(T)), and

(∗ ∗ ∗) (\overline{P}_{0} ∩ \overline{N})/O_{2}(\overline{P}_{0} ∩ \overline{N}) ≅ SL_{2}(2^{k}) or S_{ℓ}.

Since by (6) V_{0} = Z_{2} × Z ≤ Z(L)Z we get C_{L} = C_{L}(V_{0}), P_{0} ≰ C_{L} and P_{1} ≤ P_{0}. Now (∗) and (∗ ∗ ∗) show that s = 1 and r = 2, which contradicts r ≥ 4.

The proof of the P!-Theorem and the Corollary. Let P ≤ M ∈ L_{H}^{∗}(S). Then the F!- Theorem and Corollary 3 show that case (a) of the Local P!-Theorem and case (a) of the Structure Theorem hold for M. The P!-Theorem now follows from Theorem 2 and Theorem 3.

For the proof of the Corollary let L ∈ Loc_{H}(P). We may assume that C_{H}(Y_{L}) ≤ L. By (1.5) there exists M ∈ L_{H}^{∗}(S) such that

P = P^{0}S ≤ L^{0}S ≤ M.

Hence, M satisfies case (a) of the Structure Theorem. In particular, we get from the structure of M/C_{M}(Y_{M}) and its action on Y_{M}:

(i) (L∩M_{0})/C_{L∩M_{0}}(Y_{L}) ≅ SL_{k}(p^{m}) or Sp_{2k}(p^{m}), and [Y_{L}, L∩M_{0}] is the corresponding natural module.

(ii) L_{0} = (L ∩ M_{0})C_{S}(Y_{L}) and C_{L_{0}}(Y_{L}) = C_{S}(Y_{L})C_{L_{0}}(Y_{M}). This gives claim (a) of the Corollary.

Assume that C_{L_{0}}(Y_{L}) ≠ O_{p}(L_{0}). Then C_{L_{0}}(Y_{M}) ≠ O_{p}(M_{0}), and we get M_{0}/O_{2}(M_{0}) ≅ Sp_{4}(2)′ (and p = 2). But then L_{0} = M_{0} since otherwise L^{0}/O_{2}(L^{0}) ≅ SL_{2}(2) and C_{L_{0}}(Y_{L}) = O_{2}(L_{0}).

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