Groups of local characteristic p

Barbara Baumeister Andy Chermak Andreas Hirn Mario Mainardis

Ulrich Meierfrankenfeld Gemma Parmeggiani

Chris Parker Peter Rowley

Bernd Stellmacher Gernot Stroth G is a finite K_{p}-group, and p a fixed prime.

G has characteristic p if C_{G}(O_{p}(G)) ≤ O_{p}(G).

p-local subgroup: Normalizer of a nontrivial p-subgroup.

G has local characteristic p if all p-local subgroups of G have characteristic p.

Object of the talk: Describe the current status of the project to understand and classify the finite groups of local characteristic p with O_{p}(G) = 1.

Disclaimer: For p odd we do not expect to be able to achieve a complete classification. Some groups with a relatively small p-local structure will remain unclassified. In particular, we currently have no idea how to treat the case where G has a strongly p-embedded subgroup.

Motivation

1. We are trying to understand why the plocal subgroups of the finite simple groups look the way they do.

2. We hope that the classification of the groups of local characteristic 2 will serve as the first step in a third generation proof for the classification of the finite simple groups.

Future plans

1. Understand and classify all groups of parabolic characteristic p.

(Here a parabolic subgroup of G is a subgroup which contains a Sylow p-subgroup. And G is of parabolic characteritic p if all plocal, parabolic subgroups of G have characteristic p.)

2. Classify all finite simple groups which are not of parabolic characteristic 2.

Characteristics of the simple groups

Groups of Lie-Type

Let G be a finite simple group of Lie type defined over a field of characteristic r. If p = r, then G is of local characteristic p. If p ≠ r and a Sylow p-subgroup of G is not cyclic, then G is usually not of parabolic characteristic p. Some exceptions: U_{3}(3) ≅ G_{2}(2)′ , Sp_{4}(2)′ ≅ L_{2}(9), P Ω_{5}(3) ≅ Ω_{6}^{−}(2), L_{3}(4) and U_{4}(3) all have local characteristics 2 and 3. L_{4}(3) has parabolic characteristics 2 and 3.

Alternating groups

The alternating groups usually have no local characteristic. But Alt(p^{n} +ϵ),ϵ ≤ 2 has parabolic characteristic p.

Simple groups with parabolic characteristic p which are neither sporadic nor a group of Lie-type defined over a field of characteristic p nor have cyclic Sylow p-subgroups. (No guaranty that the list is correct)

group  prime local char. ?
Alt(p^{n} + ϵ), 0 ≤ ϵ ≤ 2, n > 1  any  no
L_{2}(r), r ± 1 = 2^{k}  2  yes
G_{2}(3)  2  yes
L_{4}(3)  2  no
D_{4}(3)  2  no

Characteristics of the sporadics

Group local char.

parabolic char.

M_{11} 2, 3 2, 3
M_{12}  2, 3

J_{1}

M_{22} 2 2
J_{2}  2
M_{23} 2 2
HS  2
J_{3}  2  2
M_{24}  2  2
McL  3, 5  3, 5
He  2
Ru  2, 5
Suz  2
ON 7 7
Co_{3}  3, 5
Co_{2} 2 3, 5
F i_{22}  2, 3
HN  2, 3, 5
Ly  5  5
T h  2, 5  2, 3, 5
F i_{23}  3
Co_{1}  2, 3, 5
J_{4} 2, 11 2, 11
F i′_{24}  2, 3, 7
B  2, 3, 5
M  2, 3, 5, 7, 13

Here we only listed cases with non-cyclic Sylow psubgroup.

Notation

G is a group of local characteristic p with O_{p}(G) = 1. L = L_{G} = {L ≤ G | C_{G}(O_{p}(L)) ≤ O_{p}(L)} Note that L contains all the p-local subgroups of G. M is the set of maximal members of L ( by inclusion), i.e., the set of maximal p-local subgroups of G. If T is a set of subgroups of G and A ≤ G, then T (A) = {T ∈ T | A ≤ T } and T_{A} = {T ∈ T | T ≤ A}. S is a Sylow p-subgroup of G. Z = Ω_{1}Z (S). p-core of G with respect to S: ⟨M(S)⟩.

The Pushing Up Theorem

Let H be a finite group and T ∈ Syl_{p}(H). The group

P_{H}(T) := O^{p′}(C_{H}(Ω_{1}Z (T)))

is called the point-stabilizer of H with respect to T. Theorem Let T be a p-group and let Σ be a set of groups such that for all L ∈ Σ

i) L is of characteristic p. ii) T ≤ L and T = O_{p}(N_{L}(T)). iii) N_{L}(T) contains a point stabilizer of L. Suppose that no non-trivial subgroup of T is normal in all L ∈ Σ. Then there exist L ∈ Σ and H ≤ L with B(T) ≤ H such O^{p}(H) has one of the following structures q^{n}SL_{n}(q)′; q^{2n}Sp_{2n}(q)′, p odd; q^{1+2n} Sp_{2n}(q)′, p = 2; 2^{6}G_{2}(2)′, p = 2; q^{1+6+8} Sp_{6}(q), p = 2; 2^{1+4+6} L_{4}(2), p = 2; or q^{1+2+2} SL_{2}(q)′, p = 3. (where q is a power of p)

Strongly p-embedded subgroups

We say that H is a strongly p-embedded subgroup of G if H ≠ G and H ∩ H^{g} is a p′-group for all g ∈ G \ H.

An elementary argument shows that G has a strongly p-embedded subgroup if and only if ⟨N_{G}(T) | 1 ≠ T ≤ S⟩ is a proper subgroup of G.

Bender classified all groups with a strongly 2-embedded subgroup.

For p ≠ 2 no such theorem exists ( independent from the CFSG).

The Open “Strongly p-embedded”-Problem

Determine all groups (of local characteristic p) with a strongly p-embedded subgroup and non-cyclic Sylow p-groups.

Proper p-core

Suppose now that G has no strongly pembedded subgroup but the p-core H := ⟨M(S)⟩ = ⟨N_{G}(T) | 1 ≠ T ⊴ S⟩ is a proper subgroup of G.

Choose L ∈ L such that, in consecutive order, L ≰ H, |L ∩ H|_{p} maximal, and L is minimal. An application of the Pushing Up Theorem gives us that O^{p}(L) ∼ q^{ϵ}q^{2}SL_{2}(q)′, ϵ ∈ {0, 1}.

For p = 2, Andreas Hirn is currently trying to obtain a contradiction in this situation.

The case G = ⟨M(S)⟩

From now on we assume that G is equal to its pcore.

The basic idea here is to determine the structure of sufficiently many members L of L(S) to be able to identify a geometry on which G acts.

Let H and Ĥ be finite groups and T and T̂ Sylow p-subgroups of H and Ĥ, respectively. We say that H has residual parabolic type Ĥ if there exists a subset Λ of L_{H}(T) with H = ⟨Λ⟩ and an inclusion preserving bijection L_{H}(T̂) → Λ, L̂ ↦ L such that for

all L̂ ∈ L_{H}(T), L/O_{p}(L) ≅ L̂/O_{p}(L̂).

Often the residual parabolic type of a group is enough to identify it. So one of our main tasks is to derive information about L/O_{p}(L) for at least some members of L(S). Our favorite method for this is to study the action of L on p-reduced normal subgroups, i.e. elementary abelian normal p-subgroups Y of L with

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

Y_{L} is the largest p-reduced subgroup of L.

Modules

Let H be a finite group, V a p-reduced 𝔽_{p}H-module and A an elementary abelian p-subgroup of V with [V, A] ≠ 1.

If |V/C_{V} (A)| ≤ |A/C_{A}(V)|, then A is an offender on V, and V is a FF-module for G.

If (i) A is an offender on C_{V} (a), for all a ∈ A \ C_{A}(V) (ii) [V, A, A, A] = 1 and (iii) |V/C_{V} (A)| ≤ |A/C_{A}(V)|^{2}, then A is near offender on V, and V is a near FF-module for G.

If [V, A, A] = 1, then A is quadratic on V, and V is a quadratic module for G.

Note that FF- and near FF-modules are special cases of 2F-modules (|V/C_{V} (A)| ≤ |A/C_{A}(V)|^{2}). So a list of FF-modules and near 2F-modules for quasi-simple groups can be easily obtained once the work of Guralnick and Malle on 2F-modules is complete.

Unfortunately the action of L on Y_{L} does not yield any information about C_{L}(Y_{L}). An elementary argument shows that Z := Ω_{1}Z (S) ≤ Y_{L} and so C_{L}(Y_{L}) ≤ C_{G}(Z).

So to make up for this misfortune we also study the group N_{G}(Z). For this we pick

C̃ ∈ M with N_{G}(Z) ≤ C̃.

For a group H, define F_{p}^{∗}(H) by F_{p}^{∗}(H)/O_{p}(H) = F^{∗}(H/O_{p}(H)).

To work with a group which is a little bit more manageable than C̃ we define

E := O^{p}(F_{p}^{∗}(CC̃(YC̃))).

We now distinguish two cases:

E-uniqueness (E!): M(E) = {C̃}

and

non E-uniqueness (¬E!): |M(E)| ≥ 2.

¬E!, an example

Here is an example for the ¬E! case which illustrates why we look at overgroups of E despite the fact that these overgroups might not contain a Sylow p-subgroup.

Let p = 2 and G = F_{4}(q).2, where the 2 induces a graph automorphism. We would like to identify G via the F_{4}-building

∘  ∘  ∘  ∘

But due to the graph automorphisms, not all of the parabolics of F_{4}(q) are contained in parabolics of G. Now E ≤ F_{4}(q), namely E is the ∘ ∘-parabolic. So E is contained in two different maximal parabolics M_{1} and M_{4} of F_{4}(q).

Let Σ = {M_{1}, M_{4}} and R = O_{2}(M_{1} ∩ M_{4}). Then it is not too difficult to see that R and Σ fulfill the assumption of the Pushing Up Theorem.

¬E!, a second example

Consider G = E_{8}(q)≀Sym (p^{k}). Here E helps us to detect that G is not of local 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 a direct product of p^{k} copies of H. Hence, E is contained in the p-local subgroup L which is a direct product of p^{k} − 1 copies of H and E_{8}(q).

¬E!

The general idea of the ¬E! case is to find a subgroup R of G and Σ ⊆ L(RE) such that we can apply the Pushing Up Theorem to R and Σ. For this we make the following choices: X is a point-stabilizer of some subnormal subgroup of C̃, such that X is maximal with respect to M(EX) ≠ {C̃}. Next choose L such that in consecutive order: L ∈ L(EX) with L ≰ C̃. |C̃ ∩ L|_{p} is maximal. SC̃(L) is maximal, here SC̃(L) is the largest subnormal subgroup of C̃ contained in L. C̃ ∩ L is maximal. L is minimal.

Define R = O_{p}(L ∩ C̃). The following two situations need to be treated differently:

(PU-L): NC̃(R) ≠ L ∩ C̃.

(¬ PU-L) : NC̃(R) = L ∩ C̃.

In the (PU-L)-Case put H = NC̃(R) and Σ = L^{H}. A short and elementary argument shows that we can apply the Pushing Up Theorem.

The (¬ PU-L)-Case is more difficult. Here we choose an C̃ ∩ L invariant subnormal subgroup N of C̃ minimal with respect to N ≰ L. Put H = N(C̃ ∩ L) and Σ = (H, L). If Y_{H} ≤ O_{p}(L) a rather lengthy amalgam type argument shows that the Pushing Up Theorem^{∗} can be applied. This leaves us with

∗Actually one needs a stronger (not yet finished) version of the Pushing Up Theorem than stated above

The Open “¬E!, b = 1”-Problem In the ¬E! and (¬ PU-L) Case, determine the structure of H and L if Y_{H} ≰ O_{p}(L).

E!

We usually apply E! through an intermediate property we call Q-uniqueness. Let Q = O_{p}(C̃).

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

An application of Thompson’s P ×Q-Lemma shows that [x, E] = 1 for all x ∈ Ω_{1}Z (Q). Hence E ≤ C_{G}(x) and so E! implies C_{G}(x) ≤ C̃. Thus

E! implies Q!

Elementary consequences of Q!

For L ∈ L define L^{∘} = ⟨Q^{g} | g ∈ G, Q^{g} ≤ L⟩. Lemma 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 (a) L^{∘} = ⟨Q^{L^{∘}}⟩

(b) L = L^{∘}(L ∩ C̃).

(c) [C_{L}(Y_{L}), L^{∘}] ≤ O_{p}(L).

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

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

To state our first Structure Theorem we need a few more definitions.

A finite group L is p-minimal if a Sylow p-subgroup of L is contained in a unique maximal subgroup of L but is not normal in L.

P ∈ L is a minimal parabolic subgroup if P is parabolic and p-minimal.

P denotes the set of minimal parabolics of G.

For T ⊆ L let T^{∘} = {T ∈ T | O^{p}(T) ≤ T^{∘}}.

It is an easy consequence of the definitions that if P ∈ P(S), then P ∈ P^{∘} if and only if P ≰ C̃.

Let P ∈ P^{∘}(S). We say that gb(P) > 1 if Y_{M} ≤ Q for all M ∈ L(P). Otherwise we say gb(P) = 1.

The Structure Theorem for Y_{M} ≤ Q

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

1.

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

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

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

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

(b) \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,

(c) Q permutes the subgroups K_{i} of (b) transitively,

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

(e) either C_{M^{∘}}(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.

The Structure Theorem for Y_{M} ≰ Q.

Theorem Let M ∈ L(S) with M^{∘} maximal. Assume that Y_{M} ≰ Q. Set K = F^{∗}(M^{∘}S/C_{M^{∘}S}(Y_{M})). Then one of the following holds:

1. K is quasisimple and isomorphic to SL(n, q), Sp(2n, q)′, Ω^{±}(n, q), or E_{6}(q), q a power of p. In case of K ≅ SL_{n}(q) or E_{6}(q) no element in M^{∘}S induces diagram automorphisms.

2. K ≅ SL_{n}(q)′ ∗ SL_{m}(q)′, q a power of p. Further Y_{M} is the tensor product module.

3. p = 2 and K ≅ 3A_{6}, M_{22} or M_{24} .

4. p = 3 and K ≅ M_{11} or 2M_{12} .

5. M^{∘}S is a minimal parabolic.

Further Y_{M} is a near F F -module, and except for case 5, Y_{M} contains a M^{∘}S submodule V as described on the next slide.

K  prime  module  example
SL_{n}(q)  p  ext. square  Ω_{2n}(q)
SL_{n}(q)  p  sym. square  Sp_{2n}(q)
SL_{n}(q^{2})  p  V (λ_{1}) ⊗ V (λ_{1}^{σ}) SU_{2n}(q)
A_{6}  2  natural  Suz
3A_{6}  2  6-dim  M_{24}
Sp_{8}(2)  2  8-dim  B
Ω_{n}^{±}(q)  p  natural  Ω_{n+2}^{±} (q)
Ω_{10}^{±} (q)  2  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_{2}
M_{22}  2  10-dim  M(22)
M_{24}  2  11-dim  M(24)

The P!-Theorems

The P!-Theorem,I Suppose that Q! holds 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). We say that P -Uniqueness (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.

The P!-Theorem,II Suppose that (i) Q! holds. (ii) There exists P ∈ P^{∘}(S) with gb(P) > 1. (iii) M := ⟨P^{∘}(S)⟩ ∈ L Then P! holds in G.

The P̃! Theorem

Suppose Q! and P! and let P be the unique member of P^{∘}(S). 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.

The P̃! Theorem Suppose Q! and that 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.

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, which is impossible under our current assumption that G is equal to its p-core.

Elementary consequences of P! and P̃! Lemma Suppose E!, P!, P̃! and that G has rank at least three. Let L = N_{G}(P^{∘}) and H = (L ∩ C̃)E. Then

(a) There exists a unique P̃ ∈ P_{H}(S) with P̃ ≰ L. Moreover, P̃ ≤ ES.

(b) P̃/O_{p}(P̃) ∼ SL_{2}(q).p^{k}.

(c) H has a unique p-component K.

(d) H = K(L ∩ H), L ∩ H is a maximal subgroup of H and O_{p}(H ∩ L) ≠ O_{p}(H).

(e) Let D = C_{H}(K/O_{p}(K)). Then D/O_{p}(H) is isomorphic to a section of the Borel subgroup of Aut(SL_{2}(q)).

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

(g) Let \overline{V} = V/Z_{0}. Then H ∩ L contains a point-stabilizer for H on \overline{V}.

(h) ⟨H, L⟩ ∉ L.

The Small World Theorem Suppose Q! and let P ∈ P^{∘}(S). We say that gb(P) = 2 if gb(P) > 1 and ⟨(Y_{P})^{E}⟩ is not abelian. 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.

3. Neither 1. nor 2. hold and (a) There exists a unique M ∈ M(S)

with C̃ ≠ M ≠ N_{G}(P^{∘}).

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

(c) C̃ has a unique p-component K and K/O_{p}(K) ≅ SL_{3}(q), Sp_{4}(q) or G_{2}(q).

The Open Rank 3 Problem Rule out Case 3 of the Small World Theorem.

The rank 2 Case

Rank 2 Theorem, I Suppose E!, P!, P̃! and that G has rank 2. Choose P̃ ∈ P(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. (P^{∘}N_{H}(P^{∘}), H) is a weak BN-pair.

3. The structure of P and P̃ is as 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} , F_{2}.

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

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

In Case 2. one can apply the Delgado-Stellmacher Weak-BN Pair paper. Which leave us in the rank 2 Case with

The Open “rank 2, gb(P)=1” Problem

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

gb(P)=2

The Open “gb(P)=2” Problem

Suppose E!, P!, P̃! and that ⟨Y_{P}^{E}⟩ is not abelian. Determine the structure of P and E.

The “gb(P) = 2”-Problem is actually just a special case of the symplectic amalgams treated by Parker and Rowley. But since the assumptions of the “gb(P) = 2”-Problem are stronger than for symplectic amalgams, we believe that a significantly shorter proof should be possible.

The H-Structure Theorem ( for p=2) Supppose E!, rank G ≥ 3 and that there exists M ∈ M(S) with M^{∘} maximal and Y_{M} ≰ Q. If p = 2, then there exists M^{∘}S ≤ H ≤ G with O_{p}(H) = 1 such that H is of parabolic type H^{∗} where H^{∗} is one of the following groups:

1. A group of Lie-Type in characteristic p with Lie-rank at least three.

2. M_{24} , He, Co_{2}, M(22).2^{e}, Co_{1}, J_{4}, M(24)′.2^{e}, Suz, F_{2} or F_{1}.

3. U_{4}(3).2^{e}.

Moreover, M^{∘}S has the same structure as its corresponding group in H^{∗}.