Finite groups of local characteristic p - Fakultät für …€¦ · · 2012-04-01Groups of local...

Groups of local characteristic p

S. Astill B. Baumeister D. Bundy

A. Chermak N. Hebbinghaus A. Hirn

M. Mainardis U. Meierfrankenfeld G. Parmeggiani

C. Parker G. Pientka P. Rowley

M.R. Salarian A. Seidel B. Stellmacher

G. Stroth C. Wiedorn

Groups 2012Universitat BielefeldMarch 12th, 2012

Groups 2012, Bielefeld, March 12th, 2012 1 / 42

Definition

Let G be a group and p a prime.

I A p-local subgroup of G is the normalizer of a non-trivialp-subgroup of G .

I G has characteristic p if CG (Op(G )) ≤ Op(G ).

I G has local characteristic p if p divides |G | and all p-localsubgroups of G have local characteristic p.


Notation

From now on p is prime, G is a finite Kp-group of localcharacteristic p with Op(G ) = 1 and S is a Sylow p-subgroup of G .

Goal

Understand and classify the finite groups of local characteristic pwith Op(G ) = 1.

Disclaimer

For p odd we do not expect to be able to achieve a completeclassification. Some groups with a relatively small p-local structurewill remain unclassified.


Definition

Let L be a finite group. A p-reduced normal subgroup of L is anelementary abelian normal p-subgroup Y of L with

Op(L/CL(Y )) = 1.YL is the largest p-reduced normal subgroup of L.

Notation

C is a maximal p-local subgroup of G with NG (Ω1Z(S)) ≤ C and

E = Op(

F ∗p(CC (YC )

))We now distinguish two cases:

¬E ! There exist two distinct maximal p-local subgroups M1 andM2 with E ≤ M1 ∩M2.

E ! C is the unique maximal p-local subgroup of G containing C .


The ¬E !-case

In the ¬E ! we choose suitable subgroups L1 and L2 with

E ≤ L1 ∩ L2 and Op(〈L1, L2〉) = 1.

We then use the amalgam method to determine the structure of L1

and L2. Given L1 and L2 one should be able to identify G up toisomorphism.


If C is the unique maximal p-local subgroup of G containing S ,then either C is a strongly p-embedded subgroup of G or one canapply the local CGT -theorem to obtain a p-local subgroup of avery restricted structure. But we currently do not know whetherthis information will be enough to identify G .To avoid this problem we will assume from now on that S iscontained in at least two maximal p-local subgroups of G .


The E !-case

Definition

A p-subgroup Q of G is called large, if CG (Q) ≤ Q,(Q = Op(NG (Q))) and

NG (A) ≤ NG (Q) for all 1 6= A ≤ CG (Q)

Lemma

Suppose E lies in a unique maximal subgroup of G . Then Op(C ) isa large p-subgroup of G .


Theorem (Structure Theorem)

Let Q be a large p-subgroup of G and M be a p-local subgroup ofG with Q ≤ S ≤ G and Q 5 M. Put M = 〈QM〉,M = M/CM(YM) and I = [YM ,M

].Suppose that YM ≤ Q. Then one the following holds.

I M ∼= SLn(q), Sp2n(q) or Sp4(2)′ and I is the correspondingnatural module.

I There exists a normal subgroup K of M such that

I K = K1 × · · · × Kr , Ki∼= Sl2(q) and

YM = V1 × · · · × Vr

where Vi := [YM ,Ki ] is a natural Ki -module.I Q permutes the Ki ’s transitively.

I There exists a p-local subgroup M∗ of G with M ≤ M∗ andM∗ fulfills the previous case.


Suppose that YM Q. Then one of the following holds:

I There exists a normal subgroup K of M such thatK = K1 K2 with Ki

∼= SLmi (q), YM∼= V1 ⊗ V2 where Vi is a

natural module for Ki and M is one of K1,K2 or K1 K2.

I (M, p, I ) is as given in the following table:

M p I

SLn(q) p nat

SLn(q) p∧2(nat)

SLn(q) p S2(nat)

SLn(q2) p nat⊗ natq

3 Alt(6), 3 Sym(6), 2 26

ΓSL2(4), Γ GL2(4) 2 nat

Sp2n(q) 2 nat

Ω±n (q) p nat

M p I

O+4 (2) 2 nat

Ω±10(q) 2 spin

E6(q) p q27

M11 3 35

2M12 3 36

M22 2 210

M24 2 211


Theorem (The H-Structure Theorem)

Suppose that Q is a large p-subgroup of G and let M be a p-localsubgroup of G with Q ≤ S ≤ G and YM Q. Then there existsH ≤ G such that MS ≤ H, Op(H) = 1 and H has the sameresidual type as one of the following groups:

I A group of Lie-type in characteristic p.

I For p = 2: M24,He,Co2, Fi22, Co1, J4, Fi24, Suz, B, M,U4(3) or G2(3).

I For p = 3: Fi24,Co3,Co1 or M.


Let Q = Op(C ). For L ≤ G put L = 〈Qg | g ∈ G ,Qg ≤ L〉. Inview of the H-structure theorem we assume from now on thatYM ≤ Q for all p-local subgroups M of G with S ≤ M.

Definition

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

Theorem (The P!-Theorem)

Let P ≤ G such that

(*) S ≤ P ≤ G , P is p-minimal, Op(P) 6= 1 and Q 6EP.

Put P∗ := POp(P) and Z0 := Ω1(Z (S ∩ P∗)). Then

I YP is a natural SL2(pm)-module for P∗.

I Z0 is normal in C .

I Either P is unique with respect to (*) or P ∼ q2SL2(q).


Theorem (The P!-Theorem)

Suppose that there exists more than one subgroup P of G suchthat S ≤ P, P is p-minimal, P NG (P) and Op(M) 6= 1, whereM = 〈P, P〉.Then p = 3 or 5 and M ∼ p3+3∗+3∗SL3(p) for any such P.


Theorem (The Isolated Subgroup Theorem)

Let H be a finite group, T ∈ Sylp(H) and P∗ be p-minimal

subgroup of H with T ≤ P∗. Put Y = 〈Op(P∗)H〉 and

L = 〈R | T ≤ R ≤ H,R is p-minimal,R 6= P∗.〉

Suppose that Op(L) Op(P∗) and P∗ is narrow. Then Y /Op(Y )is quasisimple.

Corollary

Put Y = 〈Op(P)C 〉. Then Y /Op(Y ) is quasisimple.


Theorem (The Small World Theorem.)

Let G be a finite group of local characteristic p with Op(G ) 6= 1.Then one of the following holds.

1. E is contained in at least two maximal p-local subgroups of G .

2. S is contained in a unique maximal p-local subgroup of G.

3. There exist p-minimal subgroups P1 and P2 of G withS ≤ P1 ∩ P2, Op(Pi ) 6= 1, P1 ≤ ES and Op(〈P1,P2〉) = 1.

4. There exists a p-local subgroup M of G with S ≤ M andYM Q.

5. There exists a p-minimal subgroup P of G with S ≤ P suchthat YP ≤ Q and 〈Y C

P 〉 is not abelian.


Theorem (The Rank 2 Theorem)

Suppose there exists p-minimal subgroups P1 and P2 of G withS ≤ P1 ∩ P2, P1 ≤ ES, Op(Pi ) 6= 1 and Op(〈P1,P2)〉) = 1. Thenone of the following holds:

I (P1,P2) is a weak BN-pair.

I The structure of P1 and P2 is as in one of the followinggroups.

I For p = 2: U4(3).2e , G2(3).2e , D4(3).2e , HS .2e , F3,F5.2

e or Ru.I For p = 3: D4(3n).3e , Fi23, F2.I For p = 5: F2.I For p = 7: F1.


Theorem (Local Recognition of finite spherical buildigs)

Let Π be an irreducible spherical Coxeter diagram with index set Iwith |I | ≥ 2 and let ∆ and ∆∗ be thick buildings with Coxeterdiagram Π. Let c and c∗ be chambers of ∆ and ∆∗ respectively.Suppose that for each edge J = x , y of Π, there exists a specialisomorphism φJ from ∆J(c) to ∆∗J(c∗). Then there exists a specialisomorphism from ∆ to ∆∗.


Notation

Let F be a finite group, let L be a finite simple group of Lie typeof rank at least 3 and let ∆ be the associated spherical building, soL = Aut†(∆). Suppose as well the following:

I Π is the Coxeter diagram of ∆ and I is its index set.

I c is a fixed chamber in ∆.

I For T ⊆ J ⊆ I , LJ = Aut†(∆J(c)) and LJT = NLJ (∆T (c)).Thus LJ∅ is a Borel subgroup of LJ and LJT is the parabolicsubgroup of type ΠT of LJ containing LJ∅.

I D is a set of subsets of I of size at least two. A subset J of Iis called a D-set if J ⊆ D for some D ∈ D.

I For each D ∈ D, FD is a subgroup of F , φD : FD → LD is ahomomorphism and KD is its kernel.

I For J ⊆ D ∈ D, FDJ = φ−1D (LDJ), BD = FD∅ and

HDJ = Op(Op′(FDJ)).

I B = 〈BD | D ∈ D〉.


Hypothesis

I Each irreducible subset of I of size at most 2 is a D-set.

I The homomorphism φD is surjective for each D ∈ D.

I If D,E ∈ D and i ∈ D ∩E , then HDi = HEi . Thus for i ∈ I wecan define Hi = HDi , where D ∈ D with i ∈ D. For J ⊆ I , letHJ = 〈Hj | j ∈ J〉 and PJ = HJB (so H∅ = 1 and P∅ = B).

I If D,E ∈ D and i ∈ D then BE normalizes FDi .

I If i , j ∈ I and i , j is not a D-set, then HiHj = HjHi andHi 6= Hj .

I [KD ,FD ] ≤ Op(KD) for each D ∈ D.

I F = 〈FD | D ∈ D〉.I |Op(B)| ≥ |Op(L∅)|.I Op(F ) = 1.

I There exists D ∈ D with CF (Op(KD)) ≤ Op(KD).


Theorem (Local Recognition of Finite Groups of Lie-type)

Under the above Notation and Hypothesis

Op′(F ) ∼= L.


Theorem

Let M be a maximal p-local subgroup of G with S ≤ G and[YM ,M] Q. Suppose H ≤ G such that MS ≤ H,H = NG (F ∗(H)), F ∗(H) is a simple group of Lie type incharacteristic p and rank at least two and H ∩ C is not solvable.Then NG (A) ≤ H for all 1 6= A E S.


Theorem

Suppose H ≤ G such that S ≤ H, H = NG (F ∗(H)), F ∗(H) is asimple group of Lie type in characteristic p and rank at least twoand (if p is odd) F ∗(H) PSL3(pa), and CH(z) is soluble forsome 1 6= z ∈ Z (S). Then one of the following holds:

I NG (Q) = NH(Q);

I p = 2 and F ∗(G ) ∼= Mat11,Mat23,G2(3) or PΩ+8 (3); or

I p = 3 andF ∗(G ) ∼= PSU6(2),F4(2), 2E6(2),McL,Co2,Fi22,Fi23 or F2.


Theorem

Suppose that p is an odd prime and H is a strongly p-embeddedsubgroup of the finite group F . If F ∗(H) is a group of Lie type incharacteristic p of rank at least two, then F ∗(H) ∼= L3(p).


The following groups have been characterized by their p-localstructure:

p G

2 Aut(G2(3))

2 Ω+8 (3)

3 Mat12

3 SL3(3)

3 Ω+8 (2)

3 Fi22,Fi23,Fi24,Fi ′24

3 Co3

3 U6(2)

p G

3 Alt(8)

3 McL

3 F2

3 Co1

3 F4(2)

3 E6(2)

5 Ly

3, 5, 7 F1


Let H be a finite group and V finite dimensional FpH-module

Definition

Let A be a subgroup of H such that A/CA(V ) is an elementaryabelian p-group. A is a best offender of H on V if|B| · |CV (B)| ≤ |A| · |CV (A)| for every B ≤ A.

Definition

The normal subgroup of H generated by the best offenders of H onV is denoted by JH(V ).A JH(V )- component is non-trivial subgroup K of JH(V ) minimalwith respect to K = [K , JH(V )].


Theorem (FF-Module Theorem, Guralnick-Malle)

Let M be a finite group with F∗(M) quasisimple and V a faithfulsimple FpM-module. Suppose that M = JM(V ).Then (M, p,V ) is one of the following:

M p V

SLn(q) p nat

Sp2n(q) p nat

SUn(q) p nat

Ωεn(q) p nat

Oε2n(q) 2 nat

G2(q) 2 q6

SLn(q) p∧2(nat)

M p V

Spin7(q) p Spin

Spin+10(q) p Spin

3.Alt(6) 2 26

Alt(7) 2 24

Sym(n) 2 nat

Alt(n) 2 nat


Theorem (J-Module Theorem)

Let M be a finite CK-group, V a faithful, reduced FpM-module.Put J = JV (M) and let J = JV (M) be the set of JV -componentsof V . Put W = [V ,J ]CV (J )/CV (J ) and let K ∈ J .

I K is either quasisimple or p = 2 or 3 and K ∼= SL2(p)′.

I [V ,K , L] = 0 for all K 6= L ∈ J .

I W =⊕

K∈J [W ,K ].

I JpJ ′ = Op(J) = F∗(J) =×J .

I W is a semisimple FpJ-module.


Theorem (J-Module Theorem, continued)

Let JK = J/CJ([W ,K ]). Then K ∼= Op(JK ) and one of thefollowing holds:

I [W ,K ] is a simple K -module and (JK , [W ,K ]) fullfills theassumptions and so also the conclusion of FF-ModuleTheorem

I JK and [W ,K ] are as follows (where N denotes a naturalmodule and N∗ its dual):

JK [W ,K ] conditions

SLn(q) N r ⊕ N∗s√

r +√

s ≤√

n

Sp2n(q) N r r ≤ n+12

SUn(q) N r r ≤ n4

Ωεn(q) N r r ≤ n−2

4

Oε2n(q) N r p = 2, r ≤ 2n−2

4


Definition (The Fitting Submodule)

Let F be a field, H a finite group and V a finite dimensionalFH-module.

I radV (H) is the intersection of the maximal FH-submodules ofV

I Let W be an FH submodule of V and N E H. Then W is N-quasisimple if W is H-reduced, W / radW (H) is simple forFH, W = [W ,N] and N acts nilpotently on radW (H).

I SV (H) is the sum of all simple FH-submodules of V .I EH(V ) := CF∗(H)(SV (H)).I W is a component of V if either W is a simpleFH-submodule with [W ,F∗(H)] 6= 0 or W is anEH(V )-quasisimple FH-submodule.

I The Fitting submodule FV (H) of V is the sum of allcomponents of V .

I RV (H) :=∑

radW (H), where the sum runs over allcomponents W of V


Theorem

I The Fitting submodule FV (H) is H-reduced.

I RV (H) is a semisimple FF∗(H)-module.

I RV (H) = radFV (H)(H).

I FV (H)/RV (H) is a semisimple FH-module

Theorem

Let V be faithful and H-reduced. Then also FV (H) andFV (H)/RV (H) are faithful and H-reduced.


Nearly Quadratic Modules

Definition

Let F be a field, A a group and V an FA-module. Then V is anearly quadratic FA-module (and A acts nearly quadratically onV ) if [V ,A,A,A] = 0 and[V ,A]+CV (A) = [vF,A]+CV (A) for every v ∈ V \[V ,A]+CV (A).

Theorem

Let F be field, H a group and V be a faithful semisimpleFH-module. Let Q be the set of nearly quadratic, but notquadratic subgroups of H. Suppose that H = 〈Q〉. Then thereexists a partition (Qi )i∈I of Q such that

I H =⊕

i∈I Hi , where Hi = 〈Qi 〉.I V = CV (H)⊕

⊕i∈I [V ,Hi ].

I For each i ∈ I , [V ,Hi ] is a simple FHi -module.


TheoremLet H be a finite group, and V a faithful simple FpH-module.Suppose that H is generated by nearly quadratic, but not quadraticsubgroups of H. Let W a Wedderburn-component for Fp F∗(H) inV and K := Z(EndF∗(H)(W )). Then W is a simpleFpF∗(H)-module and one of the following holds for H,V ,W ,Kand (if V = W ) H/CH(K)

H V W K H/CH (K)

(C2 o Sym(m))′ Fm3 F3 F3 − m ≥ 3,m 6= 4

SLn(F2) o Sym(m) (Fn2)m Fn2 F2 − m ≥ 2, n ≥ 3

Wr(SL2(F2),m) (Fn2)m Fn2 F4 − m ≥ 2

Frob(39) F27 V F27 C3

Γ GLn(F4) Fn4 V F4 C2 n ≥ 2

ΓSLn(F4) Fn4 V F4 C2 n ≥ 2

SL2(F2)× SLn(F2) F22 ⊗ Fn2 V F4 C2 n ≥ 3

3. Sym(6) F34 V F4 C2

SLn(K) SLm(K) Kn ⊗ Km V any 1 n,m ≥ 3

SL2(K) SLm(K) K2 ⊗ Km V K 6= F2 1 m ≥ 2

SLn(F2) o C2 Fn2 ⊗ Fn2 V F2 1 n ≥ 3

(C2 o Sym(4))′ F43 V F3 1

SU3(2)′ F34 V F4 1

F∗(H) = Z(H)K ? V ? 1

K quasisimple

Moreover, in case SU3(2) case, H is not generated by abelian, nearly quadratic subgroups.(Here Wr(L,m) is the normal closure of Sym(m) in L o Sym(m))


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Preprints

A. Chermak, Finite Groups generated by a pair of minimalparabolic subgroups, Part 1

A. Chermak, C. Parker, Rank 2 groups, a piece of the action.

U. Meierfrankenfeld, B. Stellmacher, The Small World Theorem forgroups of local characteristic p with E !

U. Meierfrankenfeld, B. Stellmacher, Point-stabilizer amalgams.

U. Meierfrankenfeld, B. Stellmacher, G. Stroth, The TrichotomyTheorem for groups of local characteristic p with ¬E !.

U. Meierfrankenfeld, G. Stroth, The b = 1 case for groups of localcharacteristic p with 6= E !.


Preprints

U. Meierfrankenfeld, G. Stroth, The H-structure Theorem forgroups of local characteristic p.

U. Meierfrankenfeld, G. Stroth, An identification of Ω+8 (3).

U.Meierfrankenfeld, G. Stroth, R. Weiss, Local identification ofspherical buildings and finite simple groups of Lie type.

C. Parker, G. Stroth, An identification theorem for groups withsocle PSU6(2), arXiv:1102.5392.

C. Parker, G. Stroth, F4(2) and its automorphism group,arXiv:1108.1661.

C. Parker, G. Stroth, An improved 3-local characterisation of McLand its automorphism group, arXiv:1201.1077v1.


C. Parker, M. Salarian, G. Stroth, A characterisation of almostsimple groups with socle E6(2) or M(22), Arxiv:1108.1894.

C. Parker, G. Stroth, Groups which are almost groups of Lie typein characteristic p, arXiv:1110.1308.

C. Parker, G. Stroth, An identification theorem for the sporadicsimple groups F2 and M(23), arXiv:1201.3229


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