Finite groups of local characteristic p An Overview

Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth

Contents

Introduction 2 Notations and Hypothesis 3 1 The Modules 5

1.1 Results 5 1.2 An Example 8

2 The Local Analysis 10

2.1 Pushing Up 12 2.2 Groups with a proper p-Core 14

2.2.1 Strongly p-embedded subgroups 15 2.2.2 CGT 15

2.3 ¬E! 16

2.3.1 The open “¬ E!,b=1”-Problem 20 2.4 E! 14

2.4.1 The Structure Theorem 21 2.4.1.1 The Structure Theorem for Y_{M} ≤ Q 22 2.4.1.2 The Structure Theorem for Y_{M} ≰ Q 25

2.4.2 The P!-Theorem 26 2.4.3 The P̃!-Theorem 29 2.4.4 The Small World Theorem 31

2.4.4.1 The open “gb=2”-Problem 32 2.4.5 Rank 2 32

2.4.5.1 The open “Rank 2,gb=1”-Problem 33 2.4.6 gb=1 33

2.4.6.1 The open “P̃!,gb=1”-Problem 37

3 The Global Analysis 37

Introduction

Let p be a fixed prime and H be a finite group whose order is divisible by p. A p-local subgroup of H is a subgroup of the form N_{H}(U), where U is a non-trivial p-subgroup of H.

H has characteristic p if C_{H}(O_{p}(H)) ≤ O_{p}(H), where O_{p}(H) is the largest normal p-subgroup of H. If all the p-local subgroup of H have characteristic p, we say that H has local characteristic p.

In this paper we describe the current status of a project whose goals are

– to understand the p-local structure of finite simple groups of local characteristic p, and

– to classify the finite simple groups of local characteristic 2.

The generic examples of groups of local characteristic p are the groups of Lie type defined over fields of characteristic p. Also some of the sporadic groups have local characteristic p, for example J_{4}, M_{24} and T h for p = 2, McL for p = 3, Ly for p = 5, and O′N for p = 7.

But also every group with a self-centralizing Sylow p-subgroup of order p, like Alt(p), is of local characteristic p. These latter groups are particular examples of groups with a strongly p-embedded subgroup. Because of such groups we used the word “understand” rather than “classify” in the first item.

We hope to obtain information that allows to understand why, apart from groups with a strongly p-embedded subgroup, p-local subgroups of groups of local characteristic p look like those in the above examples.

For p = 2 Bender’s fundamental classification of groups with a strongly 2-embedded subgroup puts us in a much better situation. In this case the information collected about the 2-local structure actually suffice to classify the finite simple groups of local characteristic 2. This then can be seen as part of a third generation proof of the classification of the finite simple groups.

At this point we also should justify another technical hypothesis we have not mentioned yet. We will assume that the simple sections (i.e., the composition factors of subgroups) of p-local subgroups are “known” simple groups, a property that surely holds in a minimal counterexample to the Classification Theorem of the finite simple groups.

One final word about a possible third generation proof of that classification and its relation to existing proofs. In 1954 R. Brauer [Br] suggested to classify the finite simple groups by the structure of the centralizers of their involutions. In principle the classification went this way, based on the epoch-making Theorem of Feit-Thompson [FeTh] which shows that every non-abelian finite simple group possesses involutions. Of course, a priori, there are as many centralizers as there are finite groups, so one of the main steps in the proof is to give additional information about the possible structure of centralizers of involutions in finite simple groups (this corresponds to the first item of our project).

In a given simple group the centralizers of involutions are particular 2-local subgroups, and there are basically two cases: Either every such centralizer has characteristic 2, in which case the group is of local characteristic 2, or this is not the case.

In the latter case, with a great amount of work, one can prove that there exists a centralizer of an involution that has a certain standard form. There is a well established machinery that then can be used to classify the corresponding groups.

The situation is more complicated if the simple group has local characteristic 2. The actual classification then works with a suitably chosen odd prime p and centralizers of elements of order p rather than involutions. For example, in the groups L_{n}(2^{m}), which are of local characteristic 2, one would choose an element of order p in a standard torus, or an element of order 3 if m = 1. The idea is then to prove that there exists a p-element whose centralizer is again in some standard form. This needs very delicate signalizer functor and uniqueness group arguments, moreover, the classification of quasi-thin groups has to be done separately.

If successful, our classification of groups of local characteristic 2 would give an alternative proof that does not need the above described switch to another prime and also does not need the separate treatment of quasi-thin groups.

In fact, in view of the part of the classification that deals with groups that are not of local characteristic 2, it might be desirable to classify groups of parabolic characteristic 2 rather than of local characteristic 2. Here a parabolic subgroup of H is a subgroup of H which contains a Sylow p-subgroup of H. And H has parabolic characteristic p if all p-local, parabolic subgroups of H have characteristic p. The remaining simple groups would then have a 2-central involution whose centralizer is not of characteristic 2, a condition which seems to be fairly strong. We hope that our methods also work in the more general situation of groups of parabolic characteristic p, but have not spent much time on it.

Notation and Hypothesis

Let p be a fixed prime and H be a finite group whose order is divisible by p. The largest normal p-subgroup of H, O_{p}(H), is called the p-radical of H. H is p-minimal if every Sylow p-subgroup S of H is contained in a unique

maximal subgroup of H and S ≠ O_{p}(H). The p-minimal parabolic subgroups of H are called minimal parabolic subgroups.

If every simple section of H is a known finite simple group, then H is a K-group. If every p-local subgroup of H is a K-group, then H is a K_{p}-group.

A proper subgroup M of H is called strongly p-embedded if p divides |H|, but does not divide |H ∩ H^{g}| for any g ∈ G \ H.

F_{p}^{∗}(H) is defined by F_{p}^{∗}(H)/O_{p}(H) = F^{∗}(H/O_{p}(H)).

For any set T of subgroups of H and U ≤ H we set

T_{U} := {T ∈ T | T ≤ U} and T (U) := {T ∈ T | U ≤ T }.

We further set

L := {L ≤ H | C_{H}(O_{p}(L)) ≤ O_{p}(L)} and P := {P ∈ L | P is p-minimal},

and denote the set of maximal elements of L by M. Observe that in the case when H has local characteristic p and S ∈ Syl_{p}(H)

L contains every p-local subgroup of H,

M is the set of maximal p-local subgroups of H,

L(S) is the set of parabolic subgroups containing S with a non-trivial p-radical,

P(S) is the set of p-minimal parabolic subgroups containing S with a non-trivial p-radical.

Let 1 = D_{0} < D_{2} < … D_{n−1} < D_{n} = H be a chief-series for H and put V_{i} = D_{i}/D_{i−1}. The shape of H is define to be the ordered tuple (H/C_{H}(V_{i}), V_{i})_{1≤i≤n}. Isomorphisms between the shapes of two groups are defined in the canonical way. Note that by the Jordan Hölder Theorem the shape of H is unique up to isomorphism. Abusing language we will say that two groups have the same shape if they have isomorphic shapes.

From now on we assume

Main Hypothesis G is a finite K_{p}-group of local characteristic p with trivial p-radical.

In the following we will discuss the principal steps and subdivisions in the investigation of G. It splits into three major parts:

• Modules

• Local Analysis

• Global Analysis.

In the first part we collect information about pairs (H, V), where H is a finite K-group and V is a faithful 𝔽_{p}H-module fulfilling certain assumptions, like quadratic action or 2F. The results of this part serve as an invaluable background for the local analysis.

The local analysis generates information about the structure of the p-local subgroups of G, and in the global analysis this information is used to identify G up to isomorphism.

1 The Modules

In this part we collect some theorems about finite groups and their 𝔽_{p}-modules that are needed in the local analysis of groups of local characteristic p. Some of these theorems are known, others are not. Proofs for the theorems in this section will appear in [BBSM].

Let H be a finite group, V an 𝔽_{p}H module and A ≤ H. We say that A acts quadratically on V if [V, A, A] = 0. Let i be a positive real number. We say that A is an iF -offender provided that |V/C_{V} (A)| ≤ |A/C_{A}(V)|^{i}. A is an offender if A is an 1F -offender. If in addition [V, A] ≠ 0, A is called a non-trivial iF -offender. If there exists a non-trivial iF -offender in H then V is called an iF -module. An F F -module is a 1F -module.

We say that H is a CK-group if every composition-factor of H is isomorphic to one of the known finite simple groups.

1.1 Results

Theorem 1.1.1 (Quadratic Module Theorem) Let H be a finite CK-group with F^{∗}(H) quasi-simple, V be a faithful irreducible 𝔽_{p}H-module and A ≤ G such that

(i) [V, A, A] = 0.

(ii) H = ⟨A^{H}⟩.

(iii) |A| > 2.

Then one the following holds:

1. p = 2, H ≅ Alt(n) or Sym(n) and V is the natural module.

2. p = 2, H ≅ Alt(n) and V is the spin-module.

3. p = 2, H ≅ 3.Alt(6), Alt(7), 3.U_{4}(3), M_{12}, Aut(M_{12}), Aut(M_{22}), 3.M_{22}, M_{24}, J_{2}, Co_{1}, Co_{2} or 3.Sz and V is known.

4. p = 2, H ≅ O_{2n}^{±}(2) and V is the natural module.

5. p = 3, H ≅ 2.Alt(n) and V is the spin module.

6. p = 3, H ≅ P GU_{n}(2) and V is the Weil-module.

7. p = 3, H ≅ 2.Sp_{6}(2), 2.Ω_{8}(2), 2.J_{2}, 2.G_{2}(4), 2.Sz, 2.Co_{1} and V is known.

8. G = F^{∗}(H) ≅ ^{σ}G_{Φ}(𝔽) is a group of Lie type over the field 𝔽 with char 𝔽 = p. Moreover, if |A| > |𝔽| or if there exists a root subgroup R of H with A ∩ R ≠ 1 and A ≰ R, then V = V (λ_{i}) where λ_{i} is a fundamental weight with λ_{i}(α) = 1 for the highest long root α ∈ Φ.

Theorem 1.1.2 (FF-Module Theorem) Let H be a finite CK-group and V a faithful, irreducible F F -module for H over 𝔽_{p}. Suppose that F^{∗}(H) is quasisimple and that H is generated by the quadratic offenders on V. Then one of the following holds (where q is a power of p):

1. H ≅ SL_{n}(q), n ≥ 2; Sp_{2n}(q), n ≥ 2; SU_{n}(q), n ≥ 4; Ω_{2n}^{+}(q), n ≥ 3; Ω_{2n}^{−}(q), n ≥ 4; or Ω_{n}(q), n ≥ 7, n and q odd; and V is the corresponding natural module.

2. H ≅ SL_{n}(q), n ≥ 3 and V is the exterior square of a natural module.

3. H ≅ Ω_{7}(q), and V is the spin-module.

4. H ≅ Ω_{10}^{+}(q) and V is one of the two half-spin modules.

5. H ≅ O_{2n}^{±}(q), p = 2, n ≥ 3 and V is the natural module.

6. H ≅ G_{2}(q), p = 2 and |V | = q^{6}.

7. H ≅ Alt(n) or Sym(n), p = 2 and V is the natural module.

8. H ≅ Alt(7), p = 2 and |V | = 2^{4}.

9. H ≅ 3.Alt(6), p = 2 and |V | = 2^{6}.

Let V be an 𝔽_{p}H-module and S ∈ Syl_{p}(H). The group O^{p′}(C_{H}(C_{V} (S))) is called a point stabilizer for H on V. V is called p-reduced if O_{p}(H/C_{H}(V)) = 1.

Lemma 1.1.3 (Point Stabilizer Theorem) Let H be a finite CK- group, V a 𝔽_{p}H-module, L a point stabilizer for H on V and A ≤ O_{p}(L).

(a) If V is p-reduced, then |V/C_{V} (A)| ≥ |A/C_{A}(V)|.

(b) Suppose V is faithful and irreducible for H, F^{∗}(H) is quasi-simple, H = ⟨A^{H}⟩ and A is a non-trivial offender on V. Then H ≅ SL_{n}(q), Sp_{2n}(q), G_{2}(q) or Sym(n), where p = 2 in the last two cases, n = 2, 3 mod (4) in the last case, and q is a power of p. Moreover, V is the corresponding natural module.

Theorem 1.1.4 Let H be a finite CK-group with F^{∗}(H) quasi-simple. Let V be a faithful irreducible 𝔽_{p}H module. Suppose there exists 1 ≠ A ≤ T ∈ Syl_{p}(H) such that |A| > 2 and ⟨A^{L}⟩ acts quadratically on V for all S ≤ L < H. Then

(a) F^{∗}(H)A ≅ SL_{n}(q), Sp_{2n}(q), SU_{n}(q), G_{2}(q)′ or Sz(q), where p = 2 in the last two cases.

(b) Let I be an irreducible F^{∗}(H)A submodule of V. Then one of the following holds:

1. I is a natural module for F^{∗}(H)A.

2. p = 2, F^{∗}(H)A ≅ L_{3}(q), H induces a graph automorphism on F^{∗}(H) and I is the adjoint module.

3. p = 2, F^{∗}(H)(A) ≅ Sp_{6}(q) and I is the spin-module.

(c) Either A is contained in a long root subgroup of F^{∗}(H)A, or p = 2, F^{∗}(H)A ≅ Sp_{4}(q), A ≤ Z(S ∩ F^{∗}(H)A) and H induces a graph automorphism on F^{∗}(H).

The information given in the above theorem can be used to prove the following corollary, which is of great help in the local analysis.

Corollary 1.1.5 (Strong L-Lemma) Let L be a finite CK-group with O_{p}(L) = 1 and V a faithful 𝔽_{p}L-module. Suppose that there exists 1 ≠ A ≤ S ∈ Syl_{p}(L) such that

(∗) ⟨A^{P} ⟩ acts quadratically on V for every proper subgroup P < L satisfying A ≤ P and S ∩ P ∈ Syl_{p}(P).

Then

(a) L ≅ SL_{2}(p^{m}), Sz(2^{m}) or D_{2r}, where p = 2 in the last two cases and r is an odd prime.

(b) [V, L]C_{V} (L)/C_{V} (L) is a direct sum of natural modules for L.

Let H be a finite group, V a 𝔽_{p}H-module and A ≤ H. We say that A is cubic on V if [V, A, A, A] = 0. We say that V is a cubic 2F -module if H contains a non-trivial cubic 2F -offender. The following theorem is due to R. Guralnick and G. Malle [GM]:

Theorem 1.1.6 (The 2F -Module Theorem,I) Let H be a finite CK-group and V a faithful irreducible cubic 2F -module for H. Suppose that F^{∗}(H) is quasi-simple, but F^{∗}(H) is not a group of Lie-type in characteristic p. Then one of the following holds:

1. F^{∗}(H)/Z(F^{∗}(H)) ≅ Alt(n), p = 2 or 3 and one of the following holds.

1. V is the natural module.

2. H ≅ Alt(n), p = 2, n = 7 or 9 and V is a half-spin module.

3. H ≅ Sym(7), p = 2 and V is the spin-module.

4. F^{∗}(H) ≅ 2.Alt(5), p = 3 and V is the half spin module.

5. F^{∗}(H) ≅ 3.Alt(6) and |V | = 2^{6}.

2. F^{∗}(H) ≅ G_{2}(2)′, p = 2 and |V | = 2^{6}.

3. F^{∗}(H) ≅ 3.U_{4}(3), p = 2 and |V | = 2^{12}.

4. F^{∗}(H) ≅ 2.L_{3}(4), p = 3 and |V | = 3^{6}.

5. F^{∗}(H) ≅ Sp_{6}(2), p = 3 and |V | = 3^{7}.

6. F^{∗}(H) ≅ 2.Sp_{6}(2), p = 3 and |V | = 3^{8}.

7. F^{∗}(H) ≅ 2.Ω_{8}^{+}(2), p = 3 and |V | = 3^{8}.

8. F^{∗}(H) ≅ M_{12}, M_{22}, M_{23}, M_{24}, p = 2 and V is a non-trivial composition factor of dimension 10, 10, 11, 11 resp. of the natural permutation module.

9. F^{∗}(H) = 3.M_{22}, p = 2 and |V | = 2^{12}.

10. F^{∗}(H) = J_{2}, p = 2 and V is the 12-dimensional module which arises from the embedding into G_{2}(4).

11. F^{∗}(H) ≅ Co_{2} or Co_{1}, p = 2 and V is 22– resp. 24–dimensional module arising from the Leech Lattice

12. F^{∗}(H) ≅ M_{11} or 2.M_{12}, p = 3 and |V | = 3^{5} and 3^{6} respectively.

It is not known whether Case 11 in the preceding theorem really occurs. We tend to believe it does not.

1.2 An Example

To get an idea how these theorems are used in the local analysis we now discuss briefly a particular but fairly general situation.

Let G be as in the Main Hypothesis, that is a finite K_{p}-group of local characteristic p with trivial p-radical. Fix S ∈ Syl_{p}(G) and put Z := Ω_{1}Z(S). Let M_{1}, M_{2} ∈ L(S) and put F_{i} = F_{p}^{∗}(M_{i}). Suppose that

(i) F_{i}/O_{p}(F_{i}) is quasisimple, i = 1, 2,

(ii) O_{p}(⟨F_{1}, F_{2}⟩) = 1,

(iii) M_{i} = SF_{i}, i = 1, 2.

Let Z_{i} := ⟨Z^{M_{i}}⟩ and V_{i} := ⟨Z_{j}^{M_{i}}⟩ for i ≠ j. Note first that Z_{j} ≤ Z(O_{p}(M_{j})) ≤ S ≤ M_{i}, so Z_{i} and V_{i} are normal subgroups of M_{i}.

As an elementary consequence of (i) we get:

(1) Let U ≤ M_{i} and F_{i} ≤ N_{M_{i}}(U). Then either F_{i} ≤ U, or U ∩ O_{p}(M_{i}) ∈ Syl_{p}(U).

This property (1) together with (iii) applied to U = C_{M_{i}}(F_{i}/O_{p}(M_{i})) and U = C_{M_{i}}(V), V a non-central M_{i}-chief factor in O_{p}(M_{i}), gives:

(2) Suppose that V is a non-central M_{i}-chief factor in O_{p}(M_{i}). Then

C_{S}(V) = O_{p}(M_{i}) = C_{S}(F_{i}/O_{p}(M_{i})).

Next we show that one of the following cases holds:

(I) There exist g ∈ G and i ∈ {1, 2}, say i = 1, such that

[Z_{1}, Z_{1}^{g}] ≠ 1, [Z_{1}, Z_{1}^{g}] ≤ Z_{1} ∩ Z_{1}^{g} and Z_{1}Z_{1}^{g} ≤ M_{1} ∩ M_{1}^{g}.

(II) There exists an i ∈ {1, 2}, say i = 1, such that Z_{1} ≰ O_{p}(M_{2}), and (I) does not hold.

(III) V_{1} and V_{2} are elementary abelian, and (I) does not hold.

To see this, assume that (I) and (II) do not hold. Then V_{1}V_{2} ≤ O_{p}(M_{1}) ∩ O_{p}(M_{2}), and either (III) holds, or for some i ∈ {1, 2}, say i = 2, V_{2} is not abelian. In the latter case, there exists g ∈ M_{2} such that [Z_{1}, Z_{1}^{g}] ≠ 1. Since ⟨Z_{1}, Z_{1}^{g}⟩ ≤ O_{p}(M_{2}) ≤ M_{1} ∩ M_{1}^{g} we also have [Z_{1}, Z_{1}^{g}] ≤ Z_{1} ∩ Z_{1}^{g}. This gives (I) contrary to our assumption. We now discuss these three cases separately.

Assume case (I). We can choose the notation such that

|Z_{1}^{g}/C_{Z_{1}^{g}}(Z_{1})| ≥ |Z_{1}/C_{Z_{1}}(Z_{1}^{g})|,

so Z_{1}^{g} is a quadratic offender on Z_{1}.

Clearly [Z_{1}, M_{1}] ≠ 1 since [Z_{1}, Z_{1}^{g}] ≠ 1, so the definition of Z_{1} implies that M_{1} ≠ C_{M_{1}}(Z_{1})S. Thus [Z_{1}, O^{p}(M_{1})] ≠ 1 and there exists a non-central M_{1}- chief factor V = U/W of Z_{1}. From (2) we conclude that C_{Z_{1}^{g}}(Z_{1}) = C_{Z_{1}^{g}}(V) = Z_{1}^{g} ∩ O_{p}(M_{1}). It follows that

|V/C_{V} (Z_{1}^{g})| ≤ |U/C_{U}(Z_{1}^{g})| ≤ |Z_{1}/C_{Z_{1}}(Z_{1}^{g})| ≤ |Z_{1}^{g}/C_{Z_{1}^{g}}(Z_{1})| = |Z_{1}^{g}/C_{Z_{1}^{g}}(V)|.

Hence Z_{1}^{g} is a non-trivial quadratic offender on V, and the F F -Module Theorem gives the structure of F_{1}/O_{p}(M_{1}) and V.

Assume case (II). Then

[O_{p}(M_{2}), Z_{1}, Z_{1}] ≤ [O_{p}(M_{2}) ∩ Z_{1}, Z_{1}] = 1,

so Z_{1} is quadratic on every M_{2}-chief factor V of O_{p}(M_{2}). Hence (unless |Z_{1}O_{p}(M_{1})/O_{p}(M_{1})| = 2) the Quadratic Module Theorem applies to M_{2}/C_{M_{2}}(V) and A = Z_{1}C_{M_{2}}(V)/C_{M_{2}}(V). But in this case one also gets information about M_{1}:

Among all subgroups U ≤ M_{2} with Z_{1} ≤ U, U ∩ S ∈ Syl_{p}(U) and Z_{1} ≰ O_{p}(U) choose U minimal and set \overline{U} = U/O_{p}(U). Then for every proper subgroup O_{p}(U) ≤ P < U with S ∩ P ∈ Syl_{p}(P) and Z_{1} ≤ P we get that Z_{1} ≤ O_{p}(P). But this implies, since we are not in case (I), that X := ⟨Z_{1}^{P} ⟩ is abelian. Hence as above, since X is normal in S ∩ P, [O_{p}(P), X, X] = 1. This shows that the Strong L-Lemma 1.1.5 applies with L = \overline{U}, V = O_{p}(U)/Φ(O_{p}(U)) and A = \overline{Z}_{1}.

Set B̃ := Z_{1} ∩ O_{p}(U) and B := B̃^{x} for some x ∈ U \ N_{U}(S ∩ U). Note that U = ⟨Z_{1}, Z_{1}^{x}⟩ and so U normalizes B̃B and B̃ ∩ B ≤ Z(U). By 1.1.5, |Z_{1}/B̃| ≤ |B/B̃ ∩ B| and C_{B}(y) = B̃ ∩ B for every y ∈ Z_{1} \ B̃. It follows that

|Z_{1}/C_{Z_{1}}(B) ≤ |Z_{1}/B̃∩B| = |Z_{1}/B̃||B̃/B̃∩B| = |Z_{1}/B̃||B/B̃∩B| ≤ |B/B̃∩B|^{2}, so B/B̃ ∩ B is a 2F -offender on Z_{1}. Using (ii) we see that the 2F -Module Theorem applies to M_{1}/C_{M_{1}}(Z_{1}) and a non-central M_{1}-chief factor of Z_{1}.

Assume case (III). Note that ⟨C_{M_{i}}(V_{i}), M_{j}⟩ ≤ N_{G}(Z_{j}) and so by condition (ii)

F_{i} ≰ C_{M_{i}}(V_{i}) for every i ∈ I.

In particular by (1) C_{S}(V_{i}) ≤ O_{p}(M_{i}) for every i ∈ I. Since by property (ii) J(S) ≰ O_{p}(M_{1}) ∩ O_{p}(M_{2}) we may assume that

(∗∗) F_{1} ≰ C_{M_{1}}(V_{1}) and J(S) ≰ C_{M_{1}}(V_{1}).

Let D be the inverse image of O_{p}(M_{1}/C_{M_{1}}(V_{1})). Pick A ∈ A(S) such that A ≰ C_{M_{1}}(V_{1}). According to the Thompson Replacement Theorem we may assume that A acts quadratically on V_{1}. The maximality of A gives

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

and thus |V_{1}/C_{V_{1}}(A)| = |V_{1}/V_{1} ∩ A| ≤ |A/C_{A}(V_{1})|, so A is a quadratic offender on V_{1}. This looks promising, but A_{0} := A∩D might not centralize V_{1}. This is an obstacle for the application of the FF-Module Theorem to F_{1}A and non-central F_{1}A-chief factors of V_{1}.

Evidently |A_{0}/C_{A}(V_{1})| ≤ |A/A_{0}| or |A/A_{0}| ≤ |A_{0}/C_{A}(V_{1})|. In the first case

|V_{1}/C_{V_{1}}(A_{0})| ≤ |V_{1}/V_{1} ∩ A| ≤ |A/C_{A}(V_{1})| = |A/A_{0}||A_{0}/C_{A}(V_{1})| ≤ |A/A_{0}|^{2},

so in this case, using again (2), A/A_{0} is a quadratic 2F-offender on the noncentral F_{1}A-chief factors of V_{1}.

In the second case

|V_{1}/C_{V_{1}}(A_{0})| ≤ |V_{1}/V_{1}∩A| ≤ |A/C_{A}(V_{1})| = |A/A_{0}||A_{0}/C_{A_{0}}(V_{1})| ≤ |A_{0}/C_{A_{0}}(V_{1})|^{2},

so A_{0} is a quadratic 2F-offender on V_{1}. An elementary calculation then shows that there exists a quadratic 2F-offender on Z_{2}.

This concludes the discussion of the cases (I) – (III). In all cases the module theorems from 1.1 reveal the structure of F_{1}/O_{p}(F_{1}) or F_{2}/O_{p}(F_{2}).

2 The Local Analysis

In this part we discuss the p-local structure of G, where G is, according to our Main Hypothesis, a finite K_{p}-group of local characteristic p with trivial p-radical. We fix

S ∈ Syl_{p}(G), Z := Ω_{1}Z(S).

For further notation see the introduction.

The basic idea is to study the structure of L ∈ L by its action on elementary abelian normal subgroups contained in Z(O_{p}(L)) and by its interaction with other elements of L having a common Sylow p-subgroup. It is here where the module results of Part 1 are used.

The appropriate candidates for such normal subgroups in Z(O_{p}(L)) are the p-reduced normal subgroups, i.e. elementary abelian normal subgroups V of L with O_{p}(L/C_{L}(V)) = 1. Note that an elementary abelian normal subgroup V is p-reduced iff any subnormal subgroup of L that acts unipotently on V already centralizes V. Here are the basic properties of p-reduced normal subgroups. They include the fact that there exists a unique maximal p-reduced normal subgroup of L which we always denote by Y_{L}.

Lemma 2.0.1 Let L be a finite group of characteristic p and T ∈ Syl_{p}(L). Then

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

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

(c) Let T_{L} = C_{T} (Y_{L}) and L_{T} = N_{L}(T_{L}). Then L = L_{T} C_{L}(Y_{L}), T_{L} = O_{p}(L_{T}) and Y_{L} = Ω_{1}Z(T_{L}).

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

(e) Let V be p-reduced normal subgroup of L and K a subnormal subgroup of L. Then [V, O^{p}(K)] is a p-reduced normal subgroup of K.

Of course, the action of L on Y_{L} might be trivial, whence Y_{L} = Ω_{1}Z(T), T ∈ Syl_{p}(L). This leads to another notation. Let H be any finite group and T ∈ Syl_{p}(H). Then P_{H}(T) := O^{p′}(C_{H}(Ω_{1}Z(T))) is called a point stabilizer of H. In the above situation trivial action on Y_{L} implies that O^{p′}(L) = P_{L}(T). Here are some basic (but not entirely elementary) facts about point stabilizers.

Lemma 2.0.2 Let H be a finite group of local characteristic p, T ∈ Syl_{p}(H) and L a subnormal subgroup of H. Then

(a) (Kieler Lemma) C_{L}(Ω_{1}Z(T)) = C_{L}(Ω_{1}Z(T ∩ L))

(b) P_{L}(T ∩ L) = O^{p′}(P_{H}(T) ∩ L)

(c) C_{L}(Y_{L}) = C_{L}(Y_{H})

(d) Suppose L = ⟨L_{1}, L_{2}⟩ for some subnormal subgroups L_{1}, L_{2} of H. Then

(da) P_{L}(T ∩ L) = ⟨P_{L_{1}}(T ∩ L_{1}), P_{L_{2}}(T ∩ L_{2})⟩.

(db) For i = 1, 2 let P_{i} be a point stabilizer of L_{i}. Then ⟨P_{1}, P_{2}⟩ contains a point stabilizer of L.

It is evident that all elements of L(S) having a normal point stabilizer are contained in N_{G}(Z). Therefore, controlling N_{G}(Z), or better a maximal plocal subgroup containing N_{G}(Z), means controlling all elements of L ∈ L(S) with trivial action on Y_{L}. This point of view leads to the next definition and subdivision.

Let C̃ be a fixed maximal p-local subgroup of G containing N_{G}(Z). Put

E := O^{p}(F_{p}^{∗}(CC̃(YC̃))), Q := O_{p}(C̃)

The major subdivision is:

Non-E-Uniqueness (¬E!) : E is contained in at least two maximal p-local subgroups of G.

E-Uniqueness (E!) : C̃ is the unique maximal p-local subgroup containing E.

Another subdivision refers to the rank of G. Define the rank of G to be the minimal size of a non-empty subset Σ of P(S) with ⟨Σ⟩ ∉ L. If no such subset exists we define the rank to be 1. Note that rank G = 1 if and only if |M(S)| = 1. The cases rank G = 1 and rank G ≥ 2 are treated separately, so in the E!-case we will assume, in addition, that G has rank at least 2.

The subgroup ⟨M(S)⟩ is called the p-core of G (with respect to S). Note that G has a proper p-core if G has rank 1, so the rank 1 case can be treated in this more general context.

2.1 Pushing Up

Various times in the local analysis we will encounter a p-local subgroup L of G and a parabolic subgroup H of L such that N_{G}(O_{p}(H)) and L are not contained in a common p-local subgroup of G. In other words O_{p}(⟨L, N_{G}(O_{p}(H))⟩) = 1. In this section we provide theorems that allow, under additional hypotheses, to determine the shape of L.

For a p-group R we let PU_{1}(R) be the class of all finite CK-groups L containing R such

(a) L is of characteristic p,

(b) R = O_{p}(N_{L}(R))

(c) N_{L}(R) contains a point stabilizer of L.

Let PU_{2}(R) be the class of all finite CK- groups L containing R such that L is of characteristic p and

L = ⟨N_{L}(R), H | R ≤ H ≤ L, H ∈ PU_{1}(R)⟩.

Let PU_{3}(R) be the class of all finite CK-groups L such that (a) L is of characteristic p.

(b) R ≤ L and L = ⟨R^{L}⟩

(c) L/C_{L}(Y_{L}) ≅ SL_{n}(q), Sp_{2n}(q) or G_{2}(q), where q is a power of p and p = 2 in the last case.

(d) Y_{L}/C_{Y_{L}}(L) is the corresponding natural module.

(e) O_{p}(L) < R and N_{L}(R) contains a point stabilizer of L.

(f) If L/C_{L}(Y_{L}) ≅ G̸_{2}(q) then R = O_{p}(N_{L}(R)).

Let PU_{4}(R) be the class of all finite CK- groups L containing R

such that L is of characteristic p and

L = ⟨N_{L}(R), H | R ≤ H ≤ L, H ∈ PU_{3}(R)⟩.

For a finite p-group T let A(T) be the set of elementary abelian subgroups of maximal order in T, J(T) = ⟨A(T)⟩, the Thompson subgroup of T, and B(T) = C_{R}(Ω_{1}Z(J(T))), the Baumann subgroup of T. Recall that a finite group F is p-closed if O^{p′}(F) = O_{p}(F). The following lemma is a generalization of a well known lemma of Baumann, also the proof is similar to Baumann’s.

Lemma 2.1.1 (Baumann Argument) Let L be a finite group, R a p-subgroup of L, V := Ω_{1}Z(O_{p}(L)), K := ⟨B(R)^{L}⟩, Ṽ = V/C_{V} (O^{p}(K)), and suppose that each of the following holds:

(i) O_{p}(L) ≤ R and L = ⟨J(R)^{L}⟩N_{L}(J(R)).

(ii) C_{K}(Ṽ) is p-closed.

(iii) |Ṽ /CṼ (A)| ≥ |A/C_{A}(Ṽ)| for all elementary abelian subgroups A of R.

(iv) If U is an F F -module for L/O_{p}(L) with Ṽ ≤ U and U = C_{U}(B(R))Ṽ, then U = C_{U}(O^{p}(K))Ṽ.

Then O_{p}(K) ≤ B(R).

Using the Point Stabilizer Theorem 1.1.3 and the Baumann Argument 2.1.1 one can prove

Lemma 2.1.2 Let R be a p-group. Then PU_{2}(R) ⊆ PU_{4}(B(R)).

Similarly,

Lemma 2.1.3 Let L be a finite p-minimal CK- group of characteristic p. Let T ∈ Syl_{p}(L). Then either L centralizes Ω_{1}Z(T) (and so P_{L}(T) is normal in L) or L ∈ PU_{4}(B(T)).

If R is a group and Σ is a set of groups containing R we define

O_{R}(Σ) = ⟨T ≤ R | T ⊴ L, ∀L ∈ Σ⟩

So O_{R}(Σ) is the largest subgroup of R which is normal in all L ∈ Σ.

Theorem 2.1.4 Let R be a finite p-group with R = B(R) and Σ a subset of PU_{3}(R). Suppose O_{R}(Σ) = 1. Then there exists L ∈ Σ such that O^{p}(L) has one of the following shapes: (where q is a power of p.)

1. q^{n}SL_{n}(q)′;

2. q^{2n}Sp_{2n}(q)′, p odd;

3. q^{1+2n}Sp_{2n}(q)′, p = 2;

3. 2^{6}G_{2}(2)′, p = 2;

4. q^{1+6+8}Sp_{6}(q), p = 2;

5. 2^{1+4+6}L_{4}(2), p = 2; or

6. q^{1+2+2}SL_{2}(q)′, p = 3.

Examples for above configurations can be found in SL_{n+1}(q), L_{p^{n}}(r)( with q = p | r − 1), Sp_{2n+2}(q), Ru, F_{4}(q), Co_{2} and G_{2}(q), respectively.

We are currently working on determining the shapes of all L ∈ Σ, not only of one. We expect all elements L ∈ Σ to have one of the structures of the previous theorem, accept for one additional possibility namely L/O_{p}(L) ≅ SL_{2}(q) and all non-central chief-factors for L on O_{p}(L) are natural. For a given R, the number of such chief-factors is bounded. But as R varies it cannot be bounded.

About the proof: Using elements A ∈ A(R) and their interaction with the Y_{L}’s, L ∈ Σ one shows that there exist L, M ∈ Σ such that ⟨Y_{L}^{M}⟩ is not abelian. The fact that ⟨Y_{L}^{M}⟩ is not abelian allows us to pin down the structure of L and M. (Compare this with the cases (I) and (II) in 1.2).

Theorem 2.1.5 (The Pushing Up Theorem) Let R be a finite p-group, 1 ≤ i ≤ 4, and Σ a subset of PU_{i}(R) with O_{R}(Σ) = 1. If i = 3 or 4 suppose that R = B(R). Then the shape of ⟨B(R)^{L}⟩ will be known for all L ∈ Σ.

Given 2.1.2, the Pushing Up Theorem should be a straight forward but tedious consequence of 2.1.4. The details still need to be worked out.

2.2 Groups with a Proper p-Core

Recall from the introduction that a proper subgroup M < G is strongly pembedded if M is not a p′-group but M ∩ M^{g} is a p′-group for every g ∈ G \ M. The following lemma is well known and elementary to prove:

Lemma 2.2.1 Let H be a finite group, T a Sylow p-subgroup of H and M a proper subgroup of H with p dividing |M|. Put K := ⟨N_{G}(A) | 1 ≠ A ≤ T ⟩. Then M is strongly p-embedded iff N_{G}(A) ≤ M for all non-trivial p-subgroups A of M and iff M contains a conjugate of K. In particular, H has a strongly p-embedded subgroup if and only if p divides |H| and K is a proper subgroup of M.

Note that the group K from the proceeding lemma contains the p-core of H with respect to T. Thus if our G has a strongly p-embedded subgroup then G also has a proper p-core. We say that G satisfies CGT if G has proper p-core but no strongly p-embedded subgroups.

2.2.1 Strongly p-embedded subgroups

Suppose that G has a strongly p-embedded subgroup. If p = 2, we can apply Bender’s theorem [Be]:

Theorem 2.2.2 (Bender) Let H be a finite group with a strongly 2-embedded subgroup. Then one of the following holds:

1. Let t be an involution in H. Then H = O(H)C_{H}(t) and t is the unique involution in C_{H}(t).

2. O^{2′}(H/O(H)) ≅ L_{2}(2^{k}), U_{3}(2^{k}) or Sz(2^{k}).

If p ≠ 2 we end our analysis without a clue.

2.2.2 CGT

Suppose that G satisfies CGT. Let M := ⟨M(S)⟩ be the p-core with respect to S. According to CGT, M is a proper subgroup of G, but M is not strongly p-embedded. Thus there exists g ∈ G \ M such that |M ∩ M^{g}|_{p} ≠ 1. Evidently we can choose g such that M ∩ S^{g} is a Sylow p-subgroup of M ∩ M^{g}. Thus |S^{g} ∩ M|_{p} ≠ 1. If S^{g} ≤ M, then S^{gm} = S for some m ∈ M. Since N_{G}(S) ≤ M we obtain the contradiction to g ∉ M. Thus S^{g} ≰ M. Also S^{g} ∈ L.

Among all L ∈ L satisfying L ≰ M we choose L such that |L ∩ M|_{p} is maximal. Then |L ∩ M|_{p} ≥ |S^{g} ∩ M|_{p} ≠ 1. Let T ∈ Syl_{p}(L ∩ M) and without loss T ≤ S.

If T = S we get that L ∈ L(S) and so by the definition of M, L ≤ M, a contradiction. Thus T ≠ S. Let C be a non-trivial characteristic subgroup of T. Then N_{S}(T) ≤ N_{G}(C) and so |M ∩ N_{G}(C)|_{p} > |M ∩ L|. Hence the maximal choice of |M ∩ L|_{p} implies N_{G}(C) ≤ M. In particular, N_{L}(C) ≤ M ∩ L. For C = T we conclude that T ∈ Syl_{p}(L). We can now apply the following theorem with L in place of H:

Theorem 2.2.3 (Local C(G,T)-Theorem) Let H be a finite K_{p}-group of characteristic p, T a Sylow p-subgroup of H, and suppose that

C(H, T) := ⟨N_{H}(C) | 1 ≠ C a characteristic subgroup of T ⟩

is a proper subgroup of H. Then there exists an H-invariant set D of subnormal subgroups of H such that

(a) H = ⟨D⟩C(H, T)

(b) [D_{1}, D_{2}] = 1 for all D_{1} ≠ D_{2} ∈ D.

(c) Let D ∈ D, then D ≰ C(H, T) and one of the following holds:

1. D/Z(D) is the semi-direct product of SL_{2}(p^{k}) with a natural module for SL_{2}(p^{k}). Moreover O_{p}(D) = [O_{p}(D), D] is elementary abelian.

2. p = 2 and D is the semi-direct product of Sym(2^{k} + 1) with a natural module for Sym(2^{k} + 1).

3. p = 3, D is the semi-direct product of O_{3}(D) and SL_{2}(3^{k}), Φ(D) = Z(D) ≤ O_{3}(D) has order 3^{k}, and both [Z(O_{3}(D)), D] and O_{3}(D)/Z(O_{3}(D)) are natural modules for D/O_{p}(D).

For p = 2 the local C(G, T)-theorem was proved by Aschbacher in [Asch] without using the K_{2}-hypothesis. For us it will be consequence of the 2.1.5. Using the local C(G, T) theorem and that G is of local characteristic p it is not difficult to show:

Theorem 2.2.4 Suppose that G fulfills CGT. Let M be a p-core for G and L ∈ L such that |L ∩ M|_{p} is maximal with respect to L ≰ M. Then there exists a normal subgroup D of L such that D/Z(D) ≅ q^{2}SL_{2}(q) and C_{L}(D) ≤ O_{p}(L).

Using the preceding theorem, A. Hirn is currently trying to show that for p = 2, G cannot fulfill CGT.

2.3 ¬E!

In this section we assume that we are in the ¬E!-case, so E is contained in C̃ and at least one other maximal p-local subgroup of G. To illustrate this situation we look at a few examples.

Let p = 2, q = 2^{k} and G = F_{4}(q)⟨σ⟩ where σ induces a graph automorphism of order 2. ( Yes, G is not of local characteristic 2, only of parabolic characteristic 2. But as we mostly look at subgroups containing a Sylow p-subgroup, or at least a large part of the Sylow p-subgroup, it is difficult for us to detect that G is not of local characteristic p.)

Note that G′ is a group of Lie-type with Dynkin-diagram

∘  ∘  ∘  ∘.

Also S is only contained in two parabolic subgroups, namely the σ-invariant B_{2}- and A_{1} × A_{1}-parabolic. Trying to treat this amalgam would not be easy. To determine E, note that Z(S) has order q and is contained in the product of the highest long root group and the highest short root group. It follows that E ≤ G′ and E is essentially the B_{2}-parabolic. So E is contained in the B_{3}- and C_{3}-parabolic, and G will be identified by the (Sp_{6}(q), Sp_{6}(q))-amalgam.

As a second example consider G = E_{8}(q) ≀ Sym(p^{k}) (Again this is a group of parabolic characteristic p, but not of local characteristic p.) Here E helps us to find p-local subgroups which are not of characteristic p. Let H be the normalizer of a root subgroup in E_{8}(q), i.e. the E_{7}-parabolic. Then C̃ is H ≀ Sym(p^{k}), and E is essentially a direct product of p^{k} copies of H. Hence, E is contained in a p-local subgroup L which is a direct product of p^{k} − 1 copies of H and E_{8}(q), so L is not of characteristic p.

As a final example consider p = 2 and G = M_{24}. Then C̃ = 2^{4}L_{4}(2) and so CC̃(YC̃) = O_{2}(C̃) and E = 1. It seems that E is not of much use in this case, but E = 1 can only occur if C̃/O_{p}(C̃) acts faithfully on YC̃. Together with the fact that C̃ contains N_{G}(Z), the E = 1-situation can be handled with the amalgam method.

To summarize, the ¬E!-case detects situations which allow a treatment via the amalgam method. The general idea is to find a p-subgroup R and a set Σ of subgroups of G containing R such that we can apply the Pushing Up Theorem 2.1.5 to (R, Σ).

To get started we choose a subgroup X of C̃ such that X is the point stabilizer of some subnormal subgroup X̃ of C̃ and such that X is maximal with respect to M(EX) ≠ {C̃}. By assumption M(E) ≠ C̃ so such a choice is possible. For L ∈ L(EX) let SC̃(L) be the largest subnormal subgroup of C̃ contained in L. We choose L such that in consecutive order

L1. L ∈ L(EX) with L ≰ C̃.

L2. |C̃ ∩ L|_{p} is maximal.

L3. SC̃(L) is maximal.

L4. C̃ ∩ L is maximal.

L5. L is minimal.

Let R = O_{p}(C̃ ∩ L). Consider the following two conditions:

(PU-L) NC̃(R) ≰ L ∩ C̃.

¬ (PU-L) NC̃(R) = L ∩ C̃

If (PU-L) holds we define H := NC̃(R). Note here that L ∩ C̃ < H.

If ¬ (PU-L) holds we choose a C̃ ∩ L-invariant subnormal subgroup N of C̃ minimal with respect to N ≰ L and put H = N(L ∩ C̃).

Note that in both cases H ≰ L and H ∩ L = C̃ ∩ L, since C̃ ∩ L ≤ H ≤ C̃. Let T be a Sylow p-subgroup of H ∩ L such that T ∩ X is a Sylow p-subgroup of X. Without loss T ≤ S.

Lemma 2.3.1 (a) O_{p}(⟨H, L⟩) = 1.

(b) N_{G}(Ω_{1}Z(T)) ≤ C̃.

(c) T is a Sylow p-subgroup of L and H ∩ L contains a point stabilizer of L.

(d) If ¬(PU-L) holds, then O_{p}(N_{H}(R)) = R and Q ≤ R. In particular, H is of characteristic p.

Proof: Suppose (a) is false. Then there exists a p-local subgroup L^{∗} of G with ⟨H, L⟩ ≤ L^{∗}. Since L ≤ L^{∗}, L^{∗} fulfills all the assumptions on L (except for the minimality of L, our last choice). But H ≤ L^{∗} and so C̃ ∩ L < C̃ ∩ L^{∗} contradicting (L4). This proves (a).

We claim that EX ≤ N_{G}(Ω_{1}Z(T)). Since X̃ is subnormal in C̃, and T contains a Sylow p-subgroup of X and so of X̃, we conclude that T is a Sylow psubgroup of ⟨X̃, T ⟩. Thus by the Kieler Lemma 2.0.2, X ≤ CX̃(Ω_{1}Z(T ∩ X)) ≤ C_{G}(Ω_{1}Z(T)) ≤ N_{G}(Ω_{1}Z(T)). Similarly E ≤ N_{G}(Ω_{1}Z(T)).

By the choice of C̃, N_{G}(Ω_{1}Z(S)) ≤ C̃. Thus to prove (b) we may assume T ≠ S. Since N_{S}(T) ≤ NC̃(Ω_{1}Z(T)) we conclude that |C̃ ∩ N_{G}(Ω_{1}Z(T))|_{p} > |C̃ ∩ L|_{p}. If (L1) holds for N_{G}(Ω_{1}Z(T)) we obtain a contradiction to (L2). Thus N_{G}(Ω_{1}Z(T)) ≤ C̃ and (b) holds.

By (b) N_{L}(T) ≤ H and so T is a Sylow p-subgroup of L. Hence (c) follows from (b).

Suppose ¬ (PU-L) holds. Then

L ∩ C̃ ≤ N_{H}(R) ≤ NC̃(R) = L ∩ C̃, and
N_{Q}(R)) ≤ O_{p}(NC̃(R)) = O_{p}(C̃ ∩ L) = R,

so R = O_{p}(N_{H}(R)) and Q ≤ R. This is (d). □

Proposition 2.3.2 Suppose ¬E! and that (PU-L) holds. Set Σ = L^{H}. Then Σ ⊆ PU_{1}(R) and O_{R}(Σ) = 1.

Proof: By 2.3.1(c) and since R is normal in H, Σ ⊆ PU_{1}(R). As H and L both normalize O_{R}(Σ) we get from 2.3.1(a) that O_{R}(Σ) = 1. □

In view of the preceding proposition the (PU-L)-case can be dealt with via the Pushing Up Theorem 2.1.5. The ¬ (PU-L)-case is more complicated. As a first step we show

Lemma 2.3.3 Suppose ¬E! and ¬ (PU-L). Put Σ = {H, L}. Then O_{R}(Σ) = 1 = O_{B(R)}(Σ) and L ∈ PU_{1}(R) ⊆ PU_{4}(B(R)). If H ∩ L contains a point stabilizer of H, then Σ ⊆ PU_{1}(R) ⊆ PU_{4}(B(R)).

Proof: By 2.3.1(a) O_{R}(Σ) = 1 and by 2.3.1(c), L ∈ PU_{1}(R). If H ∩ L contains a point stabilizer of H, then by 2.3.1(d) H ∈ PU_{1}(R). By 2.1.2, PU_{1}(R) ⊆ PU_{4}(B(R)). Also O_{B(R)}(Σ) ≤ O_{R}(Σ) = 1 and all parts of the lemma have been verified. □

The preceding lemma is the main tool in the proof of:

Proposition 2.3.4 Suppose ¬E!, ¬ (PU-L) and Y_{H} ≤ O_{p}(L). Then H ∈ PU_{4}(B(R)).

Outline of a Proof: Suppose that H ∉ PU_{4}(B(R)). Note that by 2.3.1(d) Q ≤ O_{p}(H) ≤ R, so Y_{H} ≤ Q. Since H ∉ PU_{4}(B(R)), B(R) is not normal in H and so B(R) ≰ O_{p}(H). The definition of H (and the minimal choice of N) shows that N = [N, B(R)]. As H ∉ PU_{1}(R), we get [Y_{H}, N] ≠ 1 and thus also [Y_{H}, B(R)] ≠ 1. It follows that Y_{H} is an F F -module for H. Let \overline{H} := H/C_{H}(Y_{H}). Then there exist subnormal subgroups K_{1}, …, K_{m} of N such that \overline{K}_{i} is quasi-simple and

\overline{N} = \overline{K}_{1} × ⋯ × \overline{K}_{m} and [Y_{H}, N] = ⨁_{i=1}^{m}[Y_{H}, K_{i}].

(Note here that H ∉ PU_{1}(R) rules out the case where \overline{N} is solvable.) We show next:

(∗) Let 1 ≠ x ∈ C_{S}((S ∩ K_{i})C_{S}(K_{i}/O_{p}(K_{i})). Then M(C_{G}(x)) = {C̃}.

Since K_{i} and X̃ are subnormal in C̃ and K_{i} ≰ X̃ we get [K_{i}, X̃] ≤ O_{p}(K_{i}). By 2.0.2 and the choice of x, C_{G}(x) contains a point stabilizer of ⟨K_{i}, X̃⟩. Suppose that C_{G}(x) ≤ L^{∗} for some L^{∗} ∈ L with L^{∗} ≰ C̃. Then the maximal choice of X implies that X contains a point stabilizer of K_{i}. But then by 2.0.2(d), H ∩ L contains a points stabilizer of H, which contradicts 2.3.3 and H ∉ PU_{1}(R). So (∗) holds.

We apply the amalgam method to (H, L) using the standard notation as it is given in [DS]. For α = Hg put K_{αi} = K_{i}^{g} and C̃_{α} = C̃^{g}.

Suppose that b is even and (α, α′) is a critical pair with α = H. Typically we will find 1 ≠ x ∈ [Y_{α}, Y_{α′}] such that x is centralized by a Sylow p-subgroup of K_{αi} and K_{α′j}. Thus (∗) implies C̃_{α} = C̃_{α′}. But this contradicts Y_{α} ≤ O_{p}(C̃_{α}) and Y_{α} ≰ O_{p}(G_{α′}).

A typical case where one cannot find such an x is, when \overline{K}_{i} ≅ Ω_{2n}^{±}(2) and A := [[Y_{α}, K_{αi}], Y_{α′}] has order 2 (for some i) and M(C_{G}(A)) ≠ {C̃}. Then there exists C_{G}(A) ≤ L^{∗} ∈ L with L^{∗} ≰ C̃. It is easy to see that K_{i}R^{∗} ∈ PU_{4}(R^{∗}). Using the Pushing Up Theorem 2.1.5 one derives a contradiction.

Suppose that b is even, but α ≠ H for every critical pair (α, α′). One then proves that Y_{H}Y_{L} is normal in L and O_{p}(⟨O_{p}(H)^{L}⟩) ≤ O_{p}(H). Another application of 2.1.5 gives a contradiction.

So b is odd and without loss α = H. Let α′ + 1 ∈ △(α′) with Y_{α} ≰ G_{α′+1}. One usually gets that Y_{α} ∩ Q_{α′} ∩ Q_{α′+1} contains an element x as in (∗). This forces Y_{α′+1} ≤ C̃_{α′} ∩ G_{α+1} ≤ G_{α}. This allows us to find y ∈ Y_{α′+1} ∩ Q_{α} with C_{G}(y) ≤ C̃_{α′+1}. Hence Y_{α} ≤ G_{α′+1}, a contradiction. □

The propositions in this section together with the Pushing Up Theorem leave us with the following open problem:

2.3.1 The open ”¬E!, b = 1”-Problem

Suppose ¬E!, ¬(PU-L), Y_{H} ≰ O_{p}(L) and H ∉ PU_{4}(B(R)). Determine the shapes of H and L.

2.4 E!

The way we usually use E! is through an intermediate property called Q-Uniqueness.

(Q!) C_{G}(x) ≤ C̃ for all 1 ≠ x ∈ C_{G}(Q)

Lemma 2.4.1 E! implies Q!.

Proof: Since C̃ is a maximal p-local subgroup, N_{G}(Q) = C̃. Thus x ∈ C_{G}(Q) = CC̃(Q). Since C̃ is of characteristic p we conclude x ∈ Z(Q). Without loss |x| has order p and thus x ∈ Ω_{1}Z(Q). Note that EQ/Q has no p-chief-factors and so Ω_{1}Z(Q) = Y_{EQ}. By 2.0.2(c)

C_{E}(Y_{EQ}) = C_{E}(Y_{E}) = C_{E}(YC̃) = E

Thus E ≤ C_{G}(x) and E! implies C_{G}(x) ≤ C̃. □

The reader might want to verify that L_{n}(q) is an example of a group which fulfills Q! but not E!.

In this section we assume Q! and that G has rank at least two. For L ∈ L define L^{∘} = ⟨Q^{g} | g ∈ G, Q^{g} ≤ L⟩.

Lemma 2.4.2 Suppose Q!.

(a) C̃^{∘} = Q, in particular, any p-subgroup of G contains at most one conjugate of Q.

(b) If L ∈ L with Q ≤ O_{p}(L), then L ≤ C̃. In particular, if 1 ≠ X ≤ Z(Q) then N_{G}(X) ≤ C̃.

(c) If Q_{1}, Q_{2} ∈ Q^{G} with Z(Q_{1}) ∩ Z(Q_{2}) ≠ 1, then Q_{1} = Q_{2}.

(d) Let L ∈ L with Q ≤ L. Then

(da) L^{∘} = ⟨Q^{L^{∘}}⟩

(db) L = L^{∘}(L ∩ C̃).

(dc) [C_{L}(Y_{L}), L^{∘}] ≤ O_{p}(L).

(dd) If L acts transitively on Y_{L}^{♯}, then L^{∘} = N_{G}(Y_{L})^{∘}.

(de) If L^{∘} ≠ Q, then C_{Y_{L}}(L^{∘}) = 1.

Proof: (a) Let g ∈ G with Q^{g} ≤ C̃. We may assume that Q^{g} ≤ S. Then Z(S) ≤ C_{G}(Q^{g}) and thus S ≤ C_{G}(x) ≤ C̃^{g} for 1 ≠ x ∈ Z(S). Since N_{G}(S) ≤ N_{G}(Ω_{1}Z(S)) ≤ C̃ we conclude that S is in a unique conjugate of C̃, so C̃ = C̃^{g} and Q = Q^{g}.

(b) By (a) Q = O_{p}(L)^{∘} ⊴ L and so L ≤ N_{G}(Q) = C̃. By Q! we have C_{G}(X) ≤ C̃, so Q ≤ O_{p}(C_{G}(X)) ≤ O_{p}(N_{G}(X)) and we are done.

(c) As ⟨Q_{1}, Q_{2}⟩ ≤ C_{G}(Z(Q_{1})∩Z(Q_{2})), we get from Q! and (a) that Q_{1} = Q_{2}.

(d) By (a) each Sylow p-subgroup of L^{∘} contains a unique G-conjugate of Q. Thus Sylow’s Theorem gives

{Q^{g} | Q^{g} ≤ L} = Q^{L^{∘}} = Q^{L},

in particular (da) holds and by the Frattini argument L = L^{∘}N_{L}(Q). Then also (db) holds since N_{L}(Q) ≤ C̃. Note that C_{Y_{L}}(Q) ≠ 1, so C_{L}(Y_{L}) ≤ L ∩ C̃ by Q!. Thus

[C_{L}(Y_{L}), Q] ≤ C_{L}(Y_{L}) ∩ Q ≤ O_{p}(C_{L}(Y_{L})) ≤ O_{p}(L),

and (dc) follows from (da).

Let Q^{g} ≤ N_{G}(Y_{L}). Then there exists 1 ≠ x ∈ C_{Y_{L}}(Q^{g}). If L is transitive on Y_{L}^{#}, then x is also centralized by an L-conjugate of Q. On the other hand, by Q! and (a) C_{G}(x) contains a unique conjugate of Q. Hence Q^{g} ≤ L and N_{G}(Y_{L})^{∘} = L^{∘}.

(de) follows immediately from Q! and (a). □

2.4.1 The Structure Theorem

In this section we assume Q! and that G has rank at least two. Our goal is to determine the action of L on Y_{L} for all L ∈ L(S) with L ≰ C̃.

For this let M^{‡}(S) be the set of all M ∈ M(S) such that

M(L) = {M} for all L ∈ L_{M}(S) with M = LC_{M}(Y_{M})

To explain the relevance of this set we define a partial ordering on a certain subset of L(S). For L ∈ L define L^{†} = LC_{G}(Y_{L}) and so L = L^{†} iff C_{G}(Y_{L}) ≤ L. Then clearly Y_{L} is a p-reduced normal subgroup of L^{†} and so Y_{L} ≤ Y_{L^{†}}. Thus C_{G}(Y_{L^{†}}) ≤ C_{G}(Y_{L}) ≤ L^{†}. We conclude that every L ∈ L is contained in a member of

L^{†} = {L ∈ L | C_{G}(Y_{L}) ≤ L}

For L_{1}, L_{2} ∈ L^{†}(S) we define

L_{1} ≪ L_{2} ⇔ L_{1} = (L_{1} ∩ L_{2})C_{G}(Y_{L_{1}})

The following lemma has an elementary proof:

Lemma 2.4.3 (a) ≪ is a partial ordering on L^{†}(S).

(b) M^{‡}(S) is precisely the set of maximal elements in L^{†}(S) with respect to ≪.

(c) If L, H ∈ L(S) with L^{†} ≪ H^{†}, then Y_{L} ≤ Y_{H^{†}} and L^{∘} ≤ H^{∘}. □

Let L ∈ L(S) with L ≰ C̃. As we have said earlier, we want to determine the action of L on Y_{L}. This will be done using a particular point of view based on the following elementary observations.

By the preceding lemma L^{†} ≪ M for some M ∈ M^{‡}(S), so

L = (L ∩ M)C_{L}(Y_{L}) = (L ∩ M)(L ∩ C̃)

since C_{L}(Y_{L}) ≤ C̃; in particular, also M ≰ C̃.

It is easy to see that

L = ⟨P_{L}(S)⟩N_{L}(S) = ⟨P_{L}(S)⟩(L ∩ C̃),

so there exists P ∈ P_{L}(S) with P ≰ C̃.

According to these observations it suffices to study the action of M on Y_{M}, where M ∈ M^{‡}(P) for a given P ∈ P(S) with P ≰ C̃. This point of view allows a case subdivision that requires another definition:

For L ∈ L(S) we write gb(L) = 1 if Y_{M} ≰ Q for some M ∈ L(L), and gb(L) > 1 otherwise. In the above discussion we now distinguish the cases gb(P) > 1 and gb(P) = 1. These two cases are treated in the next two sections. We remark that of the actual groups have gb(P) = 1. Indeed among the groups of Lie Type in characteristic p, only ^{2}F_{4}(2^{k}), ^{3}D_{4}(q) and (for p ≠ 3) G_{2}(q) fulfill E!, rank G > 1 and gb(P) > 1.

We further set

L^{∘} = {L ∈ L | O^{p}(L) ≤ L^{∘}} and P^{∘} = P ∩ L^{∘}.

Note that for P ∈ P(S), P ∈ P^{∘} iff P ≰ C̃.

2.4.1.1 The Structure Theorem for Y_{M} ≤ Q

In this section we discuss a proof of the following theorem:

Theorem 2.4.4 (M-Structure Theorem for Y_{M} ≤ Q) Suppose Q! and that P ∈ P^{∘}(S) with gb(P) > 1. Let M ∈ M^{‡}(P). Then one of the following two cases holds for \overline{M} := M/C_{M}(Y_{M}) and M_{0} := M^{∘}C_{S}(Y_{M}):

(a) (aa) \overline{M_{0}} ≅ SL_{n}(p^{k}) or Sp_{2n}(p^{k}) and C\overline{M} (\overline{M_{0}}) ≅ C_{q}, q|p^{k} − 1, or \overline{M} ≅ Sp_{4}(2) and \overline{M_{0}} ≅ Sp_{4}(2)′ (and p = 2),

(ab) [Y_{M}, M^{∘}] is the corresponding natural module for \overline{M_{0}},

(ac) C_{M_{0}}(Y_{M}) = O_{p}(M_{0}), or p = 2 and M_{0}/O_{2}(M_{0}) ≅ 3Sp_{4}(2)′.

(b) (ba) P = M_{0}S, Y_{M} = Y_{P}, and there exists a unique normal subgroup P^{∗} of P containing O_{p}(P) such that

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

(bc) Q permutes the subgroups K_{i} of (bb) transitively,

(bd) O^{p}(P) = O^{p}(P^{∗}) = O^{p}(M_{0}), and P^{∗}C_{M}(Y_{P}) is normal in M,

(be) either C_{M^{0}}(Y_{P}) = O_{p}(M_{0}), or p = 2, r > 1, K_{i} ≅ SL_{2}(2), and

C_{M_{0}}(Y_{P})/O_{2}(M_{0}) = Z(M_{0}/O_{2}(M_{0})) is a 3-group.

A second look at the situation discussed in section 1.2 (with M_{1} corresponding to M) might help the reader to appreciate the conclusion of the Structure Theorem. In section 1.2 we have assumed that F^{∗}(M_{1}/O_{p}(M_{1})) is quasisimple. Here we get a similar statement as a conclusion in part (a), and part (b) shows that only for “small groups” it is not true (in fact, this case later will be ruled out in the P!-Theorem).

In section 1.2 we found that Y_{M_{1}} is an FF-module or a 2F -module for M_{1}, where the second case is basically ruled out here by the hypothesis Y_{M} ≤ Q. But in the FF-module case a glance at the FF-Module Theorem 1.1.2 shows that by far not all possible groups actually occur in the conclusion of the Structure Theorem. In the following we want to demonstrate, using the groups Sym(I) and G_{2}(2^{k}) as examples, how these additional groups are ruled out.

Suppose that \overline{M} ≅ Sym(I), |I| ≥ 9, p = 2, and Y := [Y_{M}, M] is the noncentral irreducible constituent of the natural permutation module for Sym(I). To describe the action of M on Y let V be a GF (2)-vector space with basis v_{k}, k ∈ I, and set v_{J} = ∑_{k∈J} v_{k} for every J ⊆ I. Then Sym(I) acts on V via v_{k} ↦ v_{kx}, x ∈ Sym(I). Let V_{e} := \{v_{J} | J ⊆ I, |J| even\} and \overline{V}_{e} = V_{e} + ⟨v_{I}⟩/⟨v_{I}⟩. Then \overline{V}_{e} is the irreducible constituent meant above, so Y = \overline{V}_{e}.

Assume first that Q does not act transitively on I. Then there exists a proper Q-invariant subset J of I with |J| ≤ |I \ J| and \overline{v}_{J} ∈ C_{Y} (Q). Hence Q! gives C_{M}(\overline{v}_{J}) ≤ C̃ and Q ≤ O_{2}(C_{M}(\overline{v}_{J})). Note that C\overline{M} (a_{J}) ≅ Sym(J)×Sym(I \J) (respectively Sym(I) ≀ C_{2}, if |J| = |I \ J|). By 2.4.2(b) \overline{Q} ≠ 1, we conclude that O_{2}(Sym(J)) ≠ 1 or O_{2}(Sym(I \ J)) ≠ 1. Since |I| ≥ 9, |J| ≤ |I \ J|, and O_{2}(Sym(n)) = 1 for all n ≥ 5, we get that |J| ∈ {2, 4} and O_{2}(Sym(I \J)) = 1. Thus \overline{Q} ≤ Sym(J), and Q centralizes every \overline{v}_{J^{∗}} for J^{∗} ⊆ I \ J. Choose such an J^{∗} with |J^{∗}| = 2. Then C\overline{M} (\overline{v}_{J^{∗}}) = Sym(J^{∗}) × Sym(I \ J^{∗}) and \overline{Q} ≤ O_{2}(C\overline{M} (\overline{v}_{J^{∗}}))∩O_{2}(Sym(J)). We conclude that \overline{Q} = 1 since O_{2}(Sym(I \J^{∗})) = 1. But this is impossible by 2.4.2(b).

Assume now that Q is transitive on I. Let J be an orbit of a maximal subgroup of Q that contains the stabilizer of a point. Then |J| = \frac{1}{2}|I| and \overline{Q} centralizes \overline{v}_{J} since \overline{v}_{J} = \overline{v}_{I\J}. Now a similar argument as above leads to a contradiction.

As a second example let p = 2, \overline{M} ≅ G_{2}(q), q = 2^{k}, and Y := [Y_{M}, M] be the module of order q^{6}. In addition, suppose that there exists g ∈ G with Y Y^{g} ≤ M ∩ M^{g} and [Y, Y^{g}] ≠ 1. Then it is easy to see that |\overline{Y^{g}}| = q^{3} and |[Y, Y^{g}]| = q^{3}. Let 1 ≠ x ∈ [Y, Y^{g}]. Since M act transitively on Y, there exist h ∈ M such that [x, Q^{h}] = 1 From Q! applied to Q^{h}, C_{G}(x) ≤ C̃^{h}. From the hypothesis Y_{M} ≤ Q we get Y ≤ Q^{h} and so Y ≤ O_{2}(C_{G}(x)); in particular

Y ≤ O_{2}(C_{M^{g}}(x)) for all 1 ≠ x ∈ [Y, Y^{g}].

This contradicts the action of M^{g} on Y^{g}.

Outline of a proof for 2.4.4: Let H be minimal in M with S ≤ H and M = HC_{M}(Y_{M}). Then by definition of M^{‡}(S), M is the unique maximal p-local subgroup containing H. Let Y = Y_{H}(= Y_{M}).

We consider the following cases:

(a) [The Orthogonal Case] p = 2, \overline{H} ≅ O_{2n}^{ϵ}(2), [Y, H] is the natural module and C_{H}(y) ≰ M for every non-singular element y ∈ [Y, H].

(b) [The Symmetric Case] (a) does not hold and there exists g ∈ G with Y Y^{g} ≤ H ∩ H^{g} and [Y, Y^{g}] ≠ 1.

(c) [The Non Abelian Asymmetric Case] Neither (a) nor (b) holds and there exists L ∈ L with O_{p}(H) ≤ L and Y ≰ O_{p}(L).

(d) [The Abelian Asymmetric Case] None of (a),(b) or (c) holds.

In the following we show how these cases arise from the amalgam method and how they are dealt with.

Choose P_{1} ∈ PC̃(S) with P_{1} ≰ M and P_{1} minimal. Since M is the unique maximal p-local containing H, O_{p}(⟨H, P_{1}⟩) = 1 and we can apply the amalgam method to the pair (H, P_{1}). For notation see [DS].

Assume that b is even. Let (α, α′) be a critical pair. Then Q! shows that G_{α} ∼ H, and we obtain g ∈ G with Y Y^{g} ≤ H∩H^{g} and [Y, Y^{g}] ≠ 1. Hence either the Symmetric Case or the Orthogonal Case holds. Note that the symmetry in H and H^{g} allows to assume that Y^{g} is an offender on Y.

Suppose that there exists 1 ≠ x ∈ [Y, Y^{g}] that is p-central in both, H and H^{g}. Then our hypothesis Y ≤ Q and Q! imply that Y^{g} ≤ O_{p}(C_{H}(x)), and (after a technical reduction to one component of H/C_{H}(Y)) the Point Stabilizer Theorem 1.1.3 applies. This gives the desired conclusion since the preceding discussion already ruled out Sym(n) and G_{2}(2^{k}).

Suppose now that [Y, Y^{g}] does not contain such an element. Then (again omitting a reduction to components) the FF-Module Theorem 1.1.2 shows that Case (b) of the Structure Theorem holds, or that |[Y, Y^{g}]| = 2. The latter possibility leads to the Orthogonal Case.

Assume now that b is odd and (α, α′) is a critical pair. Then again G_{α} ∼ H. If there exists 1 ≠ x ∈ Y_{α} with [x, O^{p}(G_{α′})] = 1, then Y_{α} ≰ O_{p}(C_{G_{α′}} (x)) and the Non-Abelian Asymmetric Case (or (a) or (b)) hold.

Suppose that [x, O^{p}(G_{α′})] ≠ 1 for all x ∈ Y_{α}^{#} (in the actual proof we do not use O^{p}(G_{α′}) but a possibly smaller subgroup of G_{α′}). Using the action of Y_{α} on V_{α′} one can show the existence of a strong offender on Y_{α}. Here an offender A on a module V is called strong, if C_{V} (a) = C_{V} (A) for all a ∈ A \ C_{A}(V). This rules out most of the cases of the FF-module Theorem 1.1.2, and we get what we want, (except that it does not rule out SL_{n}(q) on a direct sum of natural modules, a case which we will not discuss here).

This leaves us with the Orthogonal Case or the Non-Abelian Asymmetric Case. In the Orthogonal Case we choose L minimal with C_{H}(x) ≤ (L∩H)C_{H}(Y) and L ≰ M, where x is a non-singular vector (i.e. a non-p-central element) in [Y, H]. Let z be a non-zero singular vector in [Y, H] perpendicular to x, so z is p-central in H. Then [z, Q^{h}] = 1 for some h ∈ H. Let Q_{z} := Q^{h}. We now show that O_{p}(⟨Q_{z}, L⟩) = 1, [Q_{z}, C_{L}(z)] ≤ Q_{z} ∩ L, and that z and y are not conjugate in G. Then 2.1.5 gives the shape of L, and one obtains a contradiction.

It remains to discuss the Non-Abelian Asymmetric Case. Let U ∈ L(O_{p}(H)) with Y ≰ O_{p}(U) such that first |U ∩H|_{p} is maximal and then U is minimal. Let T ∈ Syl_{p}(U ∩ H). If N_{G}(T) ≰ M, then considering the amalgam (H, N_{G}(T)) we obtain g ∈ G with Y Y^{g} ∈ H ∩ H^{g} and [Y, Y^{g}] ≠ 1. But this contradicts the assumptions of the Non-Abelian Asymmetric Case. Hence N_{G}(T) ≤ M, in particular T is a Sylow p-subgroup of U. If Q ≰ U we can apply 2.1.5 and get a contradiction. So Q ≤ U. Since Y ≤ Q but Y ≰ O_{p}(U), we have U ≰ C̃, and 2.4.2(de) implies C_{Y_{U}} (U) = 1.

Let T ≤ X < U . Then by minimality of U, Y ≤ O_{p}(X). Since O_{p}(X) ≤ T ≤ H we get ⟨Y^{X}⟩ ≤ H. Hence ⟨Y^{X}⟩ is abelian since we are not in the symmetric case. So 1.1.4 gives the structure of U/O_{p}(U) and Y_{U}. Moreover, in most cases we can conclude that Y_{U} is a strong dual offender on Y and in all cases we get some strong dual offender on Y. Here a group A is called a strong dual offender on a module V if A acts quadratically on V and [v, A] = [V, A] for all v ∈ V \ C_{V} (A). The existence of a strong dual offender on Y together with the FF-Module Theorem 1.1.2 gives the desired conclusion. □

2.4.1.2 The Structure Theorem for Y_{M} ≰ Q

In this section we outline a proof of the following theorem. (It might be worthwhile to mention that given E! we do not need to assume in this section that G is of local characteristic p but only that G is of parabolic characteristic p.)

Theorem 2.4.5 (M-Structure Theorem for Y_{M} ≰ Q) Let M ∈ M(S) with M^{∘} maximal and put \overline{K} = F^{∗}(M^{∘}/C_{M^{∘}}(Y_{M})). Suppose E!, Y_{M} ≰ Q and that M^{∘}S is not p-minimal, then one of the following holds

1. \overline{K} is quasisimple and isomorphic to SL_{n}(q), Ω_{n}^{±}(q), or E_{6}(q). In case of \overline{K} ≅ SL_{n}(q), or E_{6}(q) no element in M induces a diagram automorphism.

2. \overline{K} ≅ SL_{n}(q)′ ∘ SL_{m}(q)′.

3. p = 2 and \overline{K} ≅ Alt(6), 3Alt(6), Sp_{8}(2), M_{22}, or M_{24}

4. p = 3 and \overline{K} ≅ M_{11} or M_{12}

Moreover, the module Y_{M} is a 2F–module with quadratic or cubic offender and contains a module V as in the table below.

K  prime  module  example
SL_{n}(q)  p  ext. square  Ω_{2n}(q)
SL_{n}(q) p odd sym. square Sp_{2n}(q)
SL_{n}(q^{2})  p  V (λ_{1}) ⊗ V (λ_{1}^{σ}) SU_{2n}(q)
SL_{3}(2)  2  natural  G_{2}(3).2
Alt(6)  2  natural  Suz
3Alt(6)  2  6-dim  M_{24}
Sp_{8}(2)  2  8-dim  F_{2}
Ω_{n}^{±}(q)  p  natural  Ω_{n+2}^{±}(q)
Ω_{10}^{±}(q)  p  half spin  E_{6}(q)
E_{6}(q)  p  V (λ_{1})  E_{7}(q)
M_{11}  3  5-dim  Co_{3}
2M_{12}  3  6-dim  Co_{1}
M_{22}  2  10-dim  M(22)
M_{24}  2  11-dim  M(24)

The proof of the above theorem corresponds to the discussion of the Cases (I) and (II) in section 1.2. Let L ∈ LC̃ be minimal with Y_{M} ≤ S ∩ L ∈ Syl_{p}(L) and Y_{M} ≰ O_{p}(L). Note that such a choice is possible since Y_{M} ≰ Q. Let Y_{M} ≤ P < L and S ∩ P ∈ Syl_{p}(P). Then by the minimal choice of L, Y_{M} ≤ O_{p}(P) and so ⟨Y_{M}^{P}⟩ ≤ O_{p}(P) ≤ S ≤ M. We now consider the following two cases separately:

(1F) There exists g ∈ G such that 1 ≠ [Y_{M}, Y_{M}^{g}] ≤ Y_{M} ∩ Y_{M^{g}} and Y_{M}Y_{M}^{g} ≤ M ∩ M^{g}.

(2F) ⟨Y_{M}^{P}⟩ is abelian for all Y_{M} ≤ P < L with S ∩ P ∈ Syl_{p}(P).

In the 1F-Case, possibly after replacing g be g^{−1}, we may assume that A := Y_{M}^{g} is a quadratic offender on Y_{M}.

In the 2F-Case 1.1.5 can be used to get a cubic 2F -offender on Y_{M} as in case (III) of 1.2.

The FF-module Theorem1.1.2 and the 2F-module Theorem 1.1.6 now allow us to identify the components (or solvable variants of components) of M^{∘}/C_{M^{∘}}(Y_{M}) which are not centralizes by A. In the iF-case one can show that A centralizes all but i of the components. Let K be the product of the components not centralized by A. By 2.4.2(de), M^{∘}/C_{M^{∘}}(Y_{M}) acts essentially faithful on [Y_{M}, K]. This allows also to obtain information about all of M^{∘}/C_{M^{∘}}(Y_{M}) .

2.4.2 The P!-Theorem

In this section we assume Q! and that G has rank at least 2. Note that this implies that P^{∘}(S) ≠ ∅. We investigate the members of P^{∘}(S), and distinguish the two cases ⟨P^{∘}(S)⟩ ∉ L and ⟨P^{∘}(S)⟩ ∈ L. Detailed proofs for the following two theorems can be found in [PPS].

Theorem 2.4.6 (The P! Theorem,I) Suppose Q! hold and ⟨P^{∘}(S)⟩ ∉ L. Then

(a) p is odd.

(b) Q = B(S), C̃ = N_{G}(B(S)) and |Q| has order q^{3}, q a power of p.

(c) P^{∘} ∼ q^{2}SL_{2}(q) for all P ∈ P^{∘}(S)

Outline of a Proof: Let L = N_{G}(B(S)). By our assumption not every element of P^{∘}(S) is in L. We first investigate an element P ∈ P^{∘}(S) with P ≰ L. Observe that Q! implies that Ω_{1}Z(X) = 1 for every X ∈ P^{∘}(S), so by 2.1.3 P ∈ PU_{4}(B(S)); i.e.

(∗) P = ⟨N_{P} (B(S)), P_{0} | P_{0} ≤ P, P_{0} ∈ PU_{3}(B(S))⟩.

An application of 2.1.4 and a short argument show that for the groups P_{0} in (∗):

(1) Y_{P} = O_{p}(P_{0}) = O_{p}(P),

(2) P_{0}/Y_{P}, Y_{P} is a natural SL_{2}(q)-module for P_{0}/Y_{P} (q = p^{m}), and | B(S)| = q^{3}.

(3) P_{0} is normal in P, and P = SP_{0}.

Suppose that p = 2. Then |A(S)| = 2, so L = SO^{2}(L) ≤ N_{G}(A) for all A ∈ A(S). It is now easy to see that there exist exactly two maximal 2-local subgroups containing S. One of them is C̃ and so ⟨P^{∘}(S)⟩ is contained in the other. But this contradicts our hypothesis.

So p is odd. Suppose that Q ≰ B(S). Then (ii) shows that q = p^{pk} for some integer k ≥ 1. Moreover, [Y_{P} , Q] has order at least p^{(2p−1)k}. Let V = ⟨[Y_{P} , Q]C̃⟩. Note that an elementary abelian p-subgroup of S not contained in B(S) has order at most p^{2k+1}. Since p > 2, (2p − 1)k > 2k + 1 and so V ≤ B(S).

In particular, Z(V) ≤ C_{B(S)}([Y_{P} , Q]) ≤ Y_{P}. It follows that either V ≤ Y_{P} or [V, Y_{P}] = Z(V) = Z(B(S)). In both cases ⟨Y C̃ ; P⟩ acts trivially on the series 1 ≤ Z(V) ≤ V ≤ Q, so Y_{P} ≤ Q since C̃ has characteristic p. As Y_{P} is not normal in C̃ we get B(S) = ⟨Y C̃ ; P⟩ ≤ Q. In particular B(S) = B(Q), so C̃ = L and Q ≤ O_{p}(N_{P} (B(S))) = B(S).

We have proved that Q ≤ B(S), so by 2.4.2(b) L ≤ C̃. In particular P ≰ L for every P ∈ P^{∘}(S), and (1) – (3) hold for every P ∈ P^{∘}(S). It remains to prove that Q = B(S).

Suppose that Q ≠ B(S). Again as C̃ is of characteristic p, we get that Z(B(S)) < Q. Note that N_{P} (B(S)) acts irreducibly on Y_{P} /Z(B(S)) and B(S)/Y_{P}.

It follows that either Y_{P} ≤ Q or Y_{P} ∩Q = Z(B(S)). The first case gives Q = Y_{P} contrary to our assumption. The second case shows, with an argument as above using the series 1 ≤ Ω_{1}Z(Q) ≤ Q, that Ω_{1}Z(Q) ≠ Z(B(S)), so Q is elementary abelian of order q^{2}.

For every P ∈ P^{∘}(S) let t_{P} be an involution in P that maps onto the central involution of P_{0}/Y_{P}. Then t_{P} normalizes B(S) and so also Q. We conclude that t_{P} inverts Z(B(S)) and B(S)/Q and centralize Q/Z(B(S)). There exists X ∈ P^{∘}(S) with Y_{X} ≠ Y_{P}. Let u = t_{P} t_{X}. Then u centralizes Z(B(S)), B(S)/Q and Q/Z(B(S)). So u induces a p-element on B(S) and since N_{G}(B(S)) has characteristic p, B(S)⟨u⟩ is a p-group. By (1)-(3) we conclude that u ∈ B(S) and t_{P} B(S) = t_{X} B(S). But then Y_{P} = [B(S), t_{P}] = [B(S), t_{X}] = Y_{X}, a contradiction.

We have shown that Q = B(S), and the lemma is proved. □

We say that P! holds in G provided that:

(P!-1) There exists a unique P ∈ P^{∘}(S).

(P!-2) P^{∘}/O_{p}(P^{∘}) ≅ SL_{2}(q), q a power of p.

(P!-3) Y_{P} is a natural module for P^{∘}.

(P!-4) C_{Y_{P}} (S ∩ P^{∘}) is normal in C̃.

Theorem 2.4.7 (The P! Theorem,II) Suppose that

(i) Q! holds and G has rank at least 2.

(ii) P is a maximal element of P^{∘}(S) and gb(P) > 1.

(iii) M := ⟨P^{∘}(S)⟩ ∈ L

Then P! holds in G.

Outline of a Proof: Applying the Structure Theorem 2.4.4 to some M̃ ∈ M^{‡}(M) it is fairly easy to see that P = M. In case (a) of the Structure Theorem 2.4.4 P^{∘}/O_{p}(P^{∘}) ≅ SL_{2}(q) and Y_{p} is the natural P^{∘}/O_{p}(P^{∘})-module. In this case we define Z_{0} := C_{Y_{P}} (P^{∘} ∩S) (= Ω_{1}Z(S ∩P^{∘})). In case (b) of the Structure Theorem we define Z_{0} := C_{Y_{P}} (S ∩ P^{∗}), where P^{∗} is as given there.

The main step in the proof of the P!-Theorem is to show that Z_{0} is normal in C̃. Suppose not and let P̃ ∈ PC̃(S) be minimal with Z_{0} ⋬P̃. Another application of the Structure Theorem shows that O_{p}(⟨P, P̃⟩) = 1. So we can apply the amalgam method to the pair (P, P̃).

For γ = P̃g put C̃_{γ} = C̃^{g}. Let (α, α′) be a critical pair. Suppose that α ∼ P̃ ∼ α′. Then both Q_{α} and Q_{α′} contain a conjugate of Q. Since 1 ≠ [Z_{α}, Z_{α′}] ≤ Z(Q_{α}) ∩ Z(Q_{α′}) we conclude from 2.4.2(c) that C̃_{α} = C̃_{α′} and so Z_{α}Z_{α′} ≤ ZC̃_{α}. Thus [Z_{α}, Z_{α′}] = 1, a contradiction.

So we may assume that α = P. Since Y_{P} ≤ Q ≤ O_{p}(P̃) we have b > 1. Suppose that b = 2. By the Structure Theorem Q ( and so also Q_{β}) acts transitively on the ”components” of G_{α}/Q_{α}. Hence Z_{0} = [Z_{α}, Z_{α′}]. This is used to show that Z_{0} ⊴ G_{β}, a contradiction.

Thus b ≥ 3. A lengthy amalgam argument now leads to contradiction.

We have established that Z_{0} is normal in C̃. In Case (a) of the Structure Theorem we are done. So suppose that Case (b) of the Structure Theorem holds. Since N_{P} (Z_{0}) ≤ C̃, Q ≤ O_{p}(N_{P} (Z_{0})). Since Q acts transitively on the components we conclude that q = p = 2.

Note that M̃ is the unique maximal 2-local subgroup of G containing P. Suppose that N_{G}(B(S)) ≰ M̃. Then O_{2}(⟨P, N_{G}(B(S))⟩ = 1 and 2.1.4 gives a contradiction. Hence N_{G}(B(S)) ≤ M̃. Since B(S) ≤ C_{G}(Z_{0}) and Z_{0} is normal in C̃, the Frattini argument implies C̃ = (C̃ ∩ M̃)C_{G}(Z_{0}).

Let K be the one of the Sym(3)-components of P/O_{p}(P), T a subgroup of index 2 in S with N_{S}(K) ≤ T, X = ⟨([Y_{P} , K] ∩ Z_{0})^{T} ⟩ and L = N_{G}(X). Then ⟨K, T, C_{G}(Z_{0})⟩ ≤ L. Since Q acts transitively on the components of P/O_{p}(P), Q ≰ T and P = ⟨L ∩ P, Q⟩. Thus O_{2}(⟨Q, L⟩) = 1. Suppose that T is not a Sylow 2-subgroup of L. Since T is of index 2 in a Sylow 2-subgroup of G, N_{L}(T) contains a Sylow 2-subgroup of L and G. But N_{L}(T) ≤ N_{G}(B(S)) ≤ M̃ and so N_{L}(T) contains a Sylow 2-subgroup of M̃. One concludes that P ≤ L, a contradiction.

Thus T is a Sylow 2-subgroup of L. Since C_{L}(Ω_{1}Z(T)) ≤ C_{G}(Ω_{1}Z(S)) ≤ C̃ and |Q/Q ∩ T | = 2 we get C_{L}(Ω_{1}Z(T)) ⊴ QC_{L}(Ω_{1}Z(T)). So we can apply 2.1.5 to Σ = L^{Q} and R = O_{p}(C_{L}(Ω_{1}Z(T))). A little bit of more work gives a contradiction. □

2.4.3 The P̃! Theorem

Suppose that G fulfills Q! and P!. We say that P̃! holds in G provided that

(P̃!-1) There exists at most one P̃ ∈ P(S) such that P̃ does not normalize P^{∘} and M := ⟨P, P̃⟩ ∈ L.

(P̃!-2) If such a P̃ exists then,

(a) M ∈ L^{∘}.

(b) M^{∘}/C_{M^{∘}}(Y_{M}) ≅ SL_{3}(q), Sp_{4}(q) or Sp_{4}(2)′

(c) Y_{M} is a corresponding natural module.

In this section we outline a proof of the following theorem from [MMPS]:

Theorem 2.4.8 (The P̃! Theorem) Suppose Q! and gb(P) > 1 for some P ∈ P^{∘}(S). Then one of the following is true:

1. G fulfills P̃!.

2. Let P̃ ∈ P(S) with P̃ ≰ N_{G}(P^{∘}) and M := ⟨P, P̃⟩ ∉ L. Then

(a) p = 3 or 5.

(b) M/O_{p}(M) ≅ SL_{3}(p)

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

Outline of a Proof: We may assume that P̃! does not hold. Then there exists P_{1} ∈ P(S) such that M_{1} = ⟨P, P_{1}⟩ ∈ L and P_{1} ≰ N_{G}(P^{∘}). The Structure Theorem 2.4.4 shows that M_{1}/O_{p}(M_{1}) ≅ SL_{3}(q) or Sp_{4}(q) (or some variant of Sp_{4}(2)) and that Y_{M_{1}} is a corresponding natural module. In particular, if P_{1} were unique P̃! would hold. Hence we can choose P_{2} having the same properties as P_{1} and P_{1} ≠ P_{2}. Define M_{2} = ⟨P, P_{2}⟩. The Structure Theorem also implies that ⟨M_{1}, M_{2}⟩ ∉ L and so we can apply the amalgam method to (M_{1}, M_{2}). Fairly short and elementary arguments show that b ≤ 2. In the b = 1 case one easily gets M′_{i} ≅ 2^{4}Sp_{4}(2)′ and then obtains a contradiction to Y_{M_{i}} ≤ Q. Fairly routine arguments in the b = 2 case show that M_{i} ∼ q^{3+3}SL_{3}(q) or q^{3+3+3}SL_{3}(q). A little extra effort rules out the second of this possibilities. But the proof that q = 3 or 5 in the remaining case currently is a rather tedious commutator calculation. □

The next lemma collects some information about C̃/O_{p}(C̃) which can be easily obtained using Q!, P! and P̃!:

Lemma 2.4.9 Suppose Q!, P!, P̃! and that G has rank at least three. Let L = N_{G}(P^{∘}). Then

(a) N_{G}(T) ≤ L ∩ C̃ for all O_{p}(C̃ ∩ L) ≤ T ⊴ S.

(b) There exists a unique P̃ ∈ PC̃(S) with P̃ ≰ L.

(c) P̃/O_{p}(P̃) ∼ SL_{2}(q).p^{k}.

(d) C̃/Q has a unique component K/Q. Moreover, P̃ ≤ KS.

(e) C̃ = K(L ∩ C̃), L ∩ C̃ is a maximal subgroup of C̃ and O_{p}(C̃ ∩ L) ≠ Q.

(f) Let Z_{0} = C_{Y_{P}} (S ∩ P^{∘}) and V = ⟨Y C̃ ; P⟩. Then Z_{0} ⊴ V and V ≤ Q.

(g) Let D = CC̃(K/O_{p}(K)). Then D is the largest normal subgroup of C̃ contained in L and D/Q is isomorphic to a section of the Borel subgroup of Aut(SL_{2}(q)).

(h) Let \overline{V} = V/Z_{0}. Then

(ha) [\overline{V} , Q] = 1

(hb) CC̃(\overline{V}) ≤ D and CC̃(\overline{V}) ∩ CC̃(Z_{0}) = Q.

(hc) Let 1 ≠ X ≤ Y_{P} /Z_{0}. Then NC̃(X) ≤ C̃ ∩ L.

(hd) C̃ ∩ L contains a point-stabilizer for C̃ on \overline{V}.

(g) ⟨C̃, L⟩ ∉ L.

2.4.4 The Small World Theorem

Given Q! and P ∈ P^{∘}(S). We say that gb(P) = 2 if gb(P) > 1 and ⟨Y_{P}^{E}⟩ is not abelian. If neither gb(P) = 1 nor gb(P) = 2 for P we say that gb(P) is at least three.

Theorem 2.4.10 (The Small World Theorem) Suppose E! and let P ∈ P^{∘}(S). Then one of the following holds:

1. G has rank 1 or 2.

2. gb(P) = 1 or gb(P) = 2.

Outline of a Proof: Assume that G has rank at least three and that gb(P) is at least three . In the exceptional cases of the P!-theorems (2.4.7, 2.4.6) one easily sees that gb(P) = 1. Thus P! holds. Also in the exceptional case of the P̃! -Theorem 2.4.8 one gets gb(P) = 1 or gb(P) = 2. Thus P̃! holds. We proved

Step 1 P! and P̃! hold.

2.4.9 gives us a good amount of information about E. We use the notation introduced in 2.4.9.

Since ⟨C̃, L⟩ ∉ L, we can apply the amalgam method to the pair (C̃, L). A non-trivial argument shows

Step 2 One of the following holds:

1. O_{p}(C̃ ∩ L)/Q contains a non-trivial quadratic offender on \overline{V}.

2. There exists a non-trivial normal subgroup A of C̃ ∩ L/Q and normal subgroups Y_{P} ≤ Z_{2} ≤ Z_{3} ≤ V of C̃ ∩ L such that:

(a) A and V/Z_{3} are isomorphic as 𝔽_{p}C_{C̃∩L}(Y_{P})-modules.

(b) |Z_{3}/Z_{2}| ≤ |A|.

(c) [\overline{V} , A] ≤ \overline{Z_{2}} ≤ C\overline{V} (A). In particular, A is a quadratic 2F -offender.

(d) [\overline{x}, A] = \overline{Y_{P}} for all x ∈ Z_{3} \ Z_{2}.

(e) Let Z_{1} = ⟨Y P̃ ; P⟩. Then Y_{1} ≤ Z_{2} and \overline{Z_{1}} is a natural SL_{2}(q)-module for P̃ ∩ CC̃(Z_{0}).

We remark that 1. and 2. of Step 2 correspond to the b > 3- and b = 3-Case for the amalgam (C̃, L).

Let X = C_{V} (O^{p}(K)). Using 1.1.1, 1.1.3 and 2.4.9 (and the Z^{∗}-theorem [Gl]) to deal with the case |A| = 2) it is not too difficult to derive

Step 3 K/O_{p}(K) ≅ SL_{n}(q), (n ≥ 3), Sp_{2n}(q)′,(n ≥ 2) or G_{2}(q)′,( p = 2). Moreover, V/X is the natural module for K/O_{p}(K) and C̃ ∩L contains a pointstabilizer for C̃ on V/W.

An amalgam argument now shows that X = Z_{0}. In particular, K acts transitively on \overline{V}. Hence all elements in V are conjugate under K to an element of Y_{P}. From this it is not to difficult to show that b = 3 in the amalgam (C̃, L). Finally also the b = 3 case leads to a contradiction.

□

We finish this section with

2.4.4.1 The open ”gb = 2”-Problem

Suppose P ∈ P^{∘}(S), gb(P) = 2 and that G has rank at least three. Determine the shape of C̃ and P

Note that by the definition of gb(P) = 2, Y_{P} ≤ Q and ⟨Y C̃ ; P⟩ is not abelian. So it should be possible to treat the gb = 2 problem with the methods of Parker/Rowley from [PR].

2.4.5 Rank 2

In this section we consider the case where Q! holds and G has rank 2. The general idea is to show that ⟨P, P̃⟩ is a weak BN-pair and then apply the Delgado-Stellmacher weak BN-pair Theorem [DS]. More precisely we try to characterize the situations where no weak BN-pair can be found. The following theorem has been proved in [Ch1] and [Ch2]

Theorem 2.4.11 (The Rank 2 Theorem) Suppose Q!, P!, and P̃! and that G has rank 2. Choose P̃ ∈ PC̃(S) such that

(i) ⟨P, P̃⟩ ∉ L.

(ii) H := ⟨P ∩ C̃, P̃⟩ is minimal with respect to (i).

(iii) P̃ is minimal with respect to (i) and (ii).

Then one of the following holds:

1. Y_{P} ≰ O_{p}(P̃)

2. (N_{H}(P^{∘})P^{∘}, H) is a weak BN-pair.

3. (H, P) has the same shape as a suitable pair of parabolic subgroups in one of the following groups.

1. For p = 2, U_{4}(3).2^{e}, G_{2}(3).2^{e}, D_{4}(3).2^{e}, HS.2^{e}, F_{3}, F_{5}.2^{e} or Ru.

2. For p = 3, D_{4}(3^{n}).3^{e}, F i_{23} or F_{2}.

3. For p = 5, F_{2}.

4. For p = 7, F_{1}.

We will not go into the details of this proof. It is a rather technical application of the amalgam method applied to the pair (N_{H}(P^{∘})P^{∘}, H).

The Rank 2 Theorem leaves as in the rank 2 case with the following open problem.

2.4.5.1 The open ”Rank 2, gb=1”-Problem

Suppose Q! holds and there exists P, P̃ ∈ P(S) such that ⟨P, P̃⟩ ∉ L, P ∈ P^{∘}(S) and gb(P) = 1. Determine the structure of P.

2.4.6 gb = 1

In this section we assume E! and that G has rank at least 3. We investigate the case where Y_{M} ≰ Q for some M ∈ M(S) with M^{∘} maximal. Put M_{0} = M^{∘}S. The Structure Theorem 2.4.5 tells us the action of M^{∘}/O_{p}(M^{∘}) on Y_{M}.

But we can get a lot more information. Let us consider one example. Suppose \overline{M_{0}} = F^{∗}(M_{0}/O_{p}(M_{0})) ≅ SL_{n}(q) and Y_{M} is the natural module. Then \overline{M_{0}} has the following Dynkin diagram

∘  ∘  ∘ ⋯ ∘  ∘  ∘

We have that C\overline{M_{0}}(Ω_{1}(Z(S)) ∩ Y_{M}) is a maximal parabolic, which then by

E! is in C̃. Hence there is a unique minimal parabolic P in M_{0} which is not in C̃. Notice that most of our groups we aim at are groups of Lie type in which C̃ is a maximal parabolic. So there is a unique P ∈ P^{∘}(S). Hence we are going to approach this situation.

But this unique P does not exist in general, as one can see in Case 2 of the structure theorem 2.4.5, in Case 3 with M^{∘}/O_{2}(M^{∘}) ≅ 3A_{6} and Y_{M} a 6-dimensional module, and in Case 4 with M^{∘}/O_{3}(M^{∘}) ≅ M_{11} and Y_{M} a 5dimensional module. Hence one cannot expect a similar theorem as 2.4.7 for ”gb = 1”.

To be able to state the theorems in this section we need to introduce some notation:

Let H^{∗} be a finite group. We say the G is of identification type H^{∗} provided that:

(I1) There exist T^{∗} ∈ Syl_{P} (H^{∗}) and I^{∗} ⊆ P_{H^{∗}}(T^{∗}) with H^{∗} = ⟨I^{∗}⟩.

(I2) There exists H ≤ G with C_{G}(H) = 1 = O_{p}(H) and M_{0} ≤ N_{G}(H).

(I3) Let T = S ∩ H. Then there exists I ⊆ P_{H}(T) with H = ⟨I⟩.

(I4) There exists a bijection I → I^{∗}, L ↦ L^{∗} such that for all J ⊂ I,

⟨J⟩/O_{p}(⟨J⟩) ≅ ⟨J^{∗}⟩/O_{p}(⟨J^{∗}⟩).

(I5) There exist M^{∗}, C^{∗} ∈ L_{H^{∗}}(T^{∗}) such that M_{0} ∩ H has the same structure as M^{∗} and C̃ ∩ H has the same structure as C̃^{∗}.

Theorem 2.4.12 Suppose E!, gb(P) = 1, rank G > 2 and P^{∘}(S) ≠ {P }. If p = 2 then G is of identification type M_{24}, He or L_{n}(q).

So suppose from now on that P^{∘}(S) = {P }.

Here is another observation. Let P ∈ P_{M_{0}}^{∘}(S). Then in our example P corresponds to an end node of the Dynkin diagram of M_{0}. Hence (in most cases) there is a unique P̃ in P_{M_{0}}(S) with P̃ ≰ N_{G}(P^{∘}). Let us consider the group G we aim at, a group of Lie type. Then again in most cases P corresponds to an end node of the Dynkin diagram of G and there exists a unique minimal parabolic in L(S) not normalizing P^{∘}.

Unfortunately this is not true in general, for example if Y_{M} is the exterior square of the natural SL_{n}(q)–module. To analyze this situation, we consider P_{1} ≠ P_{2} in P(S) such that P_{i} does not normalize P^{∘} for i = 1, 2. Let L = ⟨P_{1}, P, P_{2}⟩. The case O_{p}(L) = 1 should be approachable with the amalgam method, (see the open problem at the end of the section).

So suppose that L ∈ L. From the structure theorem we conclude that L^{∘}/C_{L^{∘}}(Y_{L}) ≅ SL_{n}(q), n ≥ 4 (on the exterior square), M_{24} (on a 11-dimensional module) or M_{22} (on a 10-dimensional module.) These cases lead to the different groups in our next theorem.

Theorem 2.4.13 Suppose E!, rank G > 2, P(S) = {P } and gb(P) = 1. Furthermore, assume that there exist P_{1} ≠ P_{2} ∈ P(S) with P_{i} ≰ N_{G}(P^{∘}) and ⟨P_{1}, P, P_{2}⟩ ∈ L. Then G is of identification type Ω_{n}^{±}(q) or ( for p = 2) Co_{2}, M(22), Co_{1}, J_{4}, or M(24)′

From now on we can assume that there is a unique P in P^{∘}(S) and a unique P̃ ∈ P_{M_{0}}(S) which does normalize P^{∘}.

Theorem 2.4.14 Suppose E!, rank G > 2, P(S) = {P }, gb(P) = 1 and that there exists a unique P̃ ∈ P(S) with P̃ ≰ N_{G}(P^{∘}). If p = 2, then G is of identification type U_{n}(q), ^{2}E_{6}(q), E_{6}(q), E_{7}(q), Sz,F_{2} or F_{1}.

In the remainder of this section, we will illustrate in some examples the basic ideas of the proof of the theorems. All the examples will be for p = 2.

Example 2.4.15 Let K = F^{∗}(M_{0}/O_{2}(M_{0})) ≅ M_{24} and assume that Y_{M} contains an 11-dimensional submodule V with |Y_{M} : V | ≤ 2. Assume further that V is the module in which L = C_{K}(C_{V} (S)) ≅ 2^{6}3Sym(6). Then G is of identification type J_{4} or M(24)′.

For L we have the following series in V

1 < V_{1} < V_{2} < V,

where |V_{1}| = 2, V_{2}/V_{1} is the 6–dimensional 3Sym(6)–module and V/V_{1} is the 4–dimensional Sym(6)–module. As C̃ ∩ M/O_{2}(M_{0}) contains L, we see that QO_{2}(M_{0})/O_{2}(M_{0}) is the elementary abelian subgroup of order 2^{6} in L.

Suppose V ≤ Q. By 1.1.2 V is not an F F –module and we conclude that W = ⟨(Y_{M} ∩Q)C̃⟩ is elementary abelian. Hence W ≤ O_{2}(M_{0}), i.e. [Y_{M}, W] = 1. But as Y_{M} ≰ Q and [Q, Y_{M}] ≤ W this contradicts CC̃(Q) ≤ Q.

So we have V ≰ Q. This now gives V_{2} = [V, Q] and V_{1} = [V_{2}, Q]. Define W := ⟨V C̃ ; 2⟩. Then V acts quadratically and nontrivially on W. Further from M_{0} we see that for any x ∈ V we have |[W/V_{1}, x]| ≤ 2^{4}. Let L_{1} be the preimage of L in M_{0}. Then L_{1}/O_{2}(L_{1}) ≅ 3Sym(6). Let U = ⟨V C̃⟩Q. Then C_{U}(W)Q/Q ≤ Z(U/Q). Hence as |[W/V_{1}, x]| ≤ 2^{4} for x ∈ V, we see that [F (U/Q), V] = 1. So there is some component U_{1} of U/Q containing L′_{1}/Q. If U_{1} is a group of Lie type defined over a field of characteristic 2 we see that it has to be F_{4}(2) or Sp_{2n}(2), for some n. But in both cases the Sp_{4}(2)–parabolic has no elementary abelian normal subgroup of order 16. So U_{1} is not a group of Lie type defined over a field of characteristic 2. As V Q/Q acts quadratically an application of 1.1.1 yields that U_{1} ≅ 3U_{4}(3) or 3M_{22}. This now tells us that V_{1} is normal in U and that W/V_{1} involves exactly one nontrivial irreducible module, which is 12–dimensional. As [V, Q] ≤ W, we see that [U, Q] ≤ W. This shows that L_{1}/O_{2}(L_{1}) possesses exactly three nontrivial chief-factors in O_{2}(L_{1}), two of them 6–dimensional and one 4-dimensional. Since L_{1} has a a 4–dimensional and a 6-dimensional factor in V and a 6-dimensional factor in QO_{2}(M_{0})/O_{2}(M_{0}), we get that [F^{∗}(M_{0}/O_{2}(M_{0})), O_{2}(M_{0})] ≤ V. This shows that O_{2}(M_{0}) = Y_{M} and so O_{2}(M_{0}) = V or |O_{2}(M_{0}) : V | = 2.

In both cases we get that Q is extraspecial of order 2^{13} and that C̃/Q is an automorphism group of 3M_{22} or 3U_{4}(3). In the former case we have C̃/Q ≅ 3Aut(M_{22}) and so G is of identificationtype J_{4}. So assume the latter case. Then we have that C̃/Q ≅ 3U_{4}(3).2, or 3U_{4}(3).4. Now M_{0} has a geometry with diagram

∘ ∘ ∼ ∘

and C̃ has one with diagram

∘ ∘ ∼ ∘.

The intersection is the geometry for L_{1}. Let P ∈ P_{M_{0}}^{∘}(S). Then P centralizes the foursgroup on which P_{0} acts nontrivially. Hence ⟨P_{0}, P ⟩ = P_{0}P. This shows that we have a geometry with diagram

∘  ∘ ∼ ∘  ∘

and that G is of identification type M(24)′.

Example 2.4.16 Let K = F^{∗}(M_{0}/O_{2}(M_{0})) ≅ Ω_{10}^{+}(q), q a power of 2, and assume that Y_{M} contains V the half spin module. Then G is of identification type E_{6}(q).

Let L = C_{K}(C_{V} (S)). Then L ∼ q^{\binom{n}{2}}L_{n}(q) and QO_{2}(M_{0})/O_{2}(M_{0}) = O_{2}(L). Note that V has the following L–series

1 < V_{1} < V_{2} < V,

where |V_{1}| = q, |V_{2}/V_{1}| = q^{10}, |V/V_{2}| = q^{5}. As in 2.4.15 V ≰ Q, and so V_{2} = [V, Q] = V ∩ Q. Now |V/V ∩ Q| = q^{5} and V/V ∩ Q is a natural module for L_{1}/O_{2}(L_{1}), where L_{1} is the pre-image of L. We can now proceed as in 2.4.15. Let U be as before, then we again see that [F (U/Q), V] = 1. Let U_{1} be a component of U/Q containing O^{2}(L_{1}/Q). Because of quadratic action and the fact that |V Q/Q| ≥ 32, we get with 1.1.1 that U_{1} is a group of Lie type in characteristic two and the list of possible groups U_{1} and the corresponding modules W in Q. As L_{1} induces in some W on C_{W} (O_{2}(O^{2}(L_{1}))) the 10 - dimensional module, we see that W is not a V (λ) where λ belongs to an end node of the Dynkin diagram of U_{1}. Hence the possible groups U_{1} are SL_{n}(q), Sp_{2n}(q) or U_{n}(q). Further for t ∈ V we have [W, t] ≤ C_{W} (O_{2}(O^{2}(L_{1}))) and C_{L_{1}}(t) has to act on this group. This shows that |[W, t]| = q^{6} or q^{4}. This in the first place shows that U_{1} ≅ SL_{n}(q) and then that W = V (λ_{2}) or V (λ_{3}). In both cases we have U_{1} ≅ SL_{6}(q), as |[W, t]| ≤ q^{6}. Moreover, [W, t] is not the natural C_{L}(t)–module in the case |[W, t]| = q^{6}. If we have W = V (λ_{2}), then W/C_{W} (V) is a 5-dimensional L_{1}–module, but there is no such module in O_{2}(L). So we have that W = V (λ_{3}). Now we see that L/O_{2}(L) induces in QO_{2}(M_{0}) exactly two 10-dimensional and one 5-dimensional module, as [V, Q] ≤ ⟨V C̃ ; 2⟩. But in V this group induces one 10-dimensional module and one 5-dimensional one. Further in QO_{2}(M_{0})/O_{2}(M_{0}) we see another 10-dimensional module. This shows [K, O_{2}(M_{0})] = V. Again Y_{M} = O_{2}(M_{0}) and so Y_{M} = V. Now we see that M^{∘} ≅ q^{16}Ω_{10}^{+}(q) and U ≅ q^{1+20}SL_{6}(q). The intersection is the SL_{5}(q)- parabolic. Now in this case we are in the situation of 2.4.14, so any minimal parabolic not in M_{0} normalizes P^{∘}.

We try to show that H = ⟨M^{∘}, U⟩ has a parabolic system with an E_{6} – diagram.

∘  ∘  ∘∘  ∘  ∘

We have that M_{0} = M^{∘}S and so there might be some field automorphism involved. But these field automorphisms are also field automorphisms on L, so they induce field automorphisms on U/O_{2}(U). This shows that U and M^{0} have a common Sylow 2–subgroup, and so G is of identificationtype E_{6}(q).

2.4.6.1 The open ”P̃!,gb=1”-Problem 2

Suppose E!, rank G > 2, P^{∘}(S) = {P }, gb(P) = 1 and that there exist P_{1}, P_{2} ∈ P(S) such that for i = 1 and 2:

(i) P_{i} ≰ N_{G}(P^{∘}).

(ii) M_{i} := ⟨P, P_{i}⟩ ∈ L

(iii) ⟨M_{1}, M_{2}⟩ ∉ L.

Determine the shape of M_{1} and M_{2}.

As a starting point towards a solution of the preceding problem we observe

Lemma 2.4.17 Suppose E! and P^{∘}(S) = {P }. Let P̃ ∈ P(S) with P̃ ≠ P, L := ⟨P, P̃⟩ ∈ L and P̃ ≰ N_{G}(P^{∘}). Then L ∈ L^{∘} and L^{∘}/C_{L^{∘}}(Y_{L}) ≅ SL_{3}(q), Sp_{4}(q), Ω_{5}(q) (and p odd), Alt(6) (and p = 2), or 2.M_{12} (and p = 3).

Note that in all cases of the preceding lemma L/O_{p}(L) has a weak BN-pair of rank 2. Hence [StTi] provides a solution to the above open problem. But we believe that our stronger assumptions allow for a shorter solution.

3 The Global Analysis

We have not yet devoted much time to this part of the project, but here are some thoughts.

The main tool to identify the group G is via a diagram geometry for a nonlocal parabolic subgroup H of G. Usually we will not only know the diagram but also the group induced on each of the residues and so the isomorphism type of each of the residues. This allows to identify the geometry and then the group H.

For example if the diagram is the diagram of the spherical building of rank at least four, then the isomorphism type of the residues uniquely determines the building. This follows from the classification of spherical buildings, but can actually be proved using only a small part of the theory of buildings.

For many of diagrams which we encounter, classification results are available in the literature. At this time we have not decided which of these results we will quote and which ones we will revise as part of our program.

The situation when M(S) = {M_{1}, M_{2}} is different. If (M_{1}, M_{2}) is a weak BN-pair associated to a BN-pair of rank 2 defined over a not to small field, one tries to recover the Weyl-group. For p odd, this probably requires a K-group assumptions not only for the p-local subgroups but also for some 2-locals. Once the Weyl-group has been identified, H can be recognized as a group of Lie-type, see [BS].

Suppose (M_{1}, M_{2}) is not associated to a BN-pair of rank 2. If p = 2, the knowledge of the parabolic subgroups often allows to determine the order of G by counting involutions. The actual identification will be done by some ad hoc methods depending on the group. If p is odd, the group is probably better left unidentified.

After the group H is identified, one still needs to deal with situations where H ≠ G. Usually our choice of the group H will allow us to show that H = is the p-core with respect to S, but some exceptions will have to be dealt with. The strongly p-embedded situation has been discussed before. If G has rank 1 the CGT-theorem 2.2.3 will limit the structure of H. For p = 2 this y hopefully will lead to a contradcition, while for p ≠ 2 we might not be able to identify G.

References

[Asch] M. Aschbacher, A Factorization Theorem for 2-constrained Groups, Proc. London. Math. Soc. (3) 43 (1981), 450-477.

[BBSM] B. Baumeister, U. Meierfrankenfeld et.al, The big book of small modules, in preparation.

[Be] H. Bender, Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläßt J. Algebra 17 (1971) 527 - 554.

[Br] R. Brauer, On the structure of groups of finite order, in : Proc. Internat. Congr. Math. Vol 1, 1954, 209 - 217.

[BS] C.D. Bennett, S. Shpectorov, A remark on a theorem of J. Tits, Proc. Amer. Math. Soc. 129, (2001) 2571-2579.

[Ch1] A. Chermak, Finite groups generated by a pair of minimal parabolic subgroups: Part I, preprint.

[Ch2] A. Chermak, Finite groups generated by a pair of minimal parabolic subgroups: Part II, draft.

[DS] A. Delgado, B. Stellmacher Weak BN-pairs of rank 2, in A. Delgado, D. Goldschmidt, B. Stellmacher, Groups and graphs: new results and methods, DMV Seminar, 6. Birkhäuser Verlag, Basel, (1985) 244 pp.

[FeTh] W. Feit, J. Thompson, Solvability of groups of odd order. Pacific J. Math. 13 (1963) 775–1029.

[Gl] G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966) 403–420.

[Gor] D. Gorenstein,Finite Groups, Chelsea (1980) New York.

[GM] R. Guralnick, G. Malle, Classification of 2F-Modules, I, submitted.

[MMPS] M. Mainardis, U. Meierfrankenfeld, G. Parmeggiani, B. Stellmacher, The P̃!-Theorem, preprint.

[PR] C. Parker, P. Rowley, Symplectic Amalgams,Springer Monographs in Mathematics, Springer Verlag (2002) London. 360 pp.

[PPS] C. Parker, G. Parmeggiani, B. Stellmacher, The P!-Theorem, submitted.

[StTi] B. Stellmacher, F. Timmesfeld,Rank 3 amalgams, Mem. Amer. Math. Soc. 136, 1998, 123pp.