A Characteristic Subgroup for Fusion Systems Silvia Onofrei The - - PowerPoint PPT Presentation

A Characteristic Subgroup for Fusion Systems

Silvia Onofrei The Ohio State University

in collaboration with Radu Stancu

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ZJ-Theorem Let p be an odd prime. Let G be a Qd(p)-free finite group with CG(Op(G)) ≤ Op(G). Then Z(J(S)) is a normal subgroup of G.

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ZJ-Theorem Let p be an odd prime. Let G be a Qd(p)-free finite group with CG(Op(G)) ≤ Op(G). Then Z(J(S)) is a normal subgroup of G. Theorem [Kessar, Linckelmann, 2008] Let p be an odd prime and let W be a Glauberman functor. Let F be a fusion system on a finite p-group S. If F is a Qd(p)-free then F = NF (W (S)).

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Theorem [Stellmacher, 1990] Let S be a finite nontrivial 2-group. Then there exists a nontrivial characteristic subgroup W(S) of S which is normal in G, for every finite Σ4-free group G with S a Sylow 2-subgroup and CG(O2(G)) ≤ O2(G).

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Theorem [Stellmacher, 1990] Let S be a finite nontrivial 2-group. Then there exists a nontrivial characteristic subgroup W(S) of S which is normal in G, for every finite Σ4-free group G with S a Sylow 2-subgroup and CG(O2(G)) ≤ O2(G).

Theorem A (O-Stancu, 2008)

Let S be a finite 2-group. Let F be a Σ4-free fusion system over S. Then there exists a nontrivial characteristic subgroup W(S) of S with the property that F = NF (W(S))

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Main Theorem (O-Stancu, 2008)

Let p be a prime and let S be a finite p-group. There exists a Glauberman functor S − → 1 = W(S) char S with the additional property that for every Qd(p)-free fusion system F on S F = NF (W(S))

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Outline of the Talk

1

Background on Fusion Systems

2

H-Free Fusion Systems

3

Characteristic Functors

4

The Characteristic Subgroup W(S)

5

Proof of Theorem A

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Background on Fusion Systems

Let S be a finite p-group and let P,Q be subgroups of S. A Category F on S:

• bjects are the subgroups of S;

HomF (P,Q) ⊆ Inj (P,Q);

any ϕ ∈ HomF (P,Q) is the composite of an F-isomorphism followed by an inclusion.

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Background on Fusion Systems

Let S be a finite p-group and let P,Q be subgroups of S. A Category F on S:

• bjects are the subgroups of S;

HomF (P,Q) ⊆ Inj (P,Q);

any ϕ ∈ HomF (P,Q) is the composite of an F-isomorphism followed by an inclusion.

P and Q are F-conjugate if Q ≃ ϕ(P) for ϕ ∈ HomF (P,Q).

Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 6/24

Background on Fusion Systems

Let S be a finite p-group and let P,Q be subgroups of S. A Category F on S:

• bjects are the subgroups of S;

HomF (P,Q) ⊆ Inj (P,Q);

any ϕ ∈ HomF (P,Q) is the composite of an F-isomorphism followed by an inclusion.

P and Q are F-conjugate if Q ≃ ϕ(P) for ϕ ∈ HomF (P,Q). The subgroup Q of S is: fully F-centralized if |CS(Q)| ≥ |CS(Q′)| for all Q′ ≤ S which are F-conjugate to Q. fully F-normalized if |NS(Q)| ≥ |NS(Q′)| for all Q′ ≤ S which are F-conjugate to Q.

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Background on Fusion Systems

A fusion system on S is a category F on S such that: HomS(P,Q) ⊆ HomF (P,Q) for all P,Q ≤ S . AutS(S) is a Sylow p-subgroup of AutF (S). Every ϕ : Q → S such that ϕ(Q) is fully F-normalized extends to a morphism ϕ : Nϕ → S.

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Background on Fusion Systems

A fusion system on S is a category F on S such that: HomS(P,Q) ⊆ HomF (P,Q) for all P,Q ≤ S . AutS(S) is a Sylow p-subgroup of AutF (S). Every ϕ : Q → S such that ϕ(Q) is fully F-normalized extends to a morphism ϕ : Nϕ → S. For a morphism ϕ ∈ HomF (Q,S) Nϕ = {x ∈ NS(Q)|∃y ∈ NS(ϕ(Q)), ϕ(xu) = yϕ(u), ∀u ∈ Q} a subgroup with the property that: QCS(Q) ≤ Nϕ ≤ NS(Q).

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Background on Fusion Systems

Lemma 1 Let F be a fusion system on S and let Q be a subgroup of S. a) There is ϕ ∈ HomF (NS(Q),S) such that ϕ(Q) is fully F-normalized. b) If Q is fully F-normalized, then ϕ(Q) is fully normalized, for any ϕ ∈ HomF (NS(Q),S).

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Background on Fusion Systems

The normalizer of Q in F is the category NF (Q): ⋄ objects: the subgroups of NS(Q); ⋄ morphisms: ϕ ∈ HomF (R,T), for which there exists ϕ ∈ HomF (QR,QT) such that

• ϕ|Q ∈ AutF (Q) and

ϕ|R = ϕ.

• Q fully F-normalized =

⇒ NF (Q) a fusion system on NS(Q) The centralizer of Q in F is the category CF (Q): ⋄ objects: the subgroups of CS(Q); ⋄ morphisms: ϕ ∈ HomF (R,T), for which there exists ϕ ∈ HomF (QR,QT) such that

• ϕ|Q = idQ and

ϕ|R = ϕ.

• Q fully F-centralized =

⇒ CF (Q) a fusion systems on CS(Q)

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Background on Fusion Systems

The category NS(Q)CF (Q): ⋄ objects: the subgroups of NS(Q); ⋄ morphisms: group homomorphisms ϕ : R → T, for which there exists ψ : QR → QT and x ∈ NS(Q) such that ψ|Q = cx and ψ|R = ϕ.

• Q fully F-centralized =

⇒ NS(Q)CF (Q) a fusion systems on CS(Q) Q F - if F = NF (Q). Op(F) - the largest normal subgroup in F. If Q F, F/Q is the fusion system on S/Q: ⋄ morphisms: for Q ≤ P,R ≤ S, a group homomorphism ψ : P/Q → R/Q is a morphism in F/Q if there is ϕ ∈ HomF (P,R) satisfying ψ(xQ) = ϕ(x)Q for all x ∈ P.

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Background on Fusion Systems

F-centric if CS(ϕ(Q)) ⊆ ϕ(Q), for all ϕ ∈ HomF (Q,S). F-radical if Op(AutF (Q)) = Inn(Q). F is constrained if Op(F) is F-centric. Theorem [BCGLO, 2005] If F is constrained then there exists a, unique up to isomorphism, finite p′-reduced p-constrained group L such that F = FS(L) and with S a Sylow p-subgroup of L.

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Background on Fusion Systems

F-centric if CS(ϕ(Q)) ⊆ ϕ(Q), for all ϕ ∈ HomF (Q,S). F-radical if Op(AutF (Q)) = Inn(Q). F is constrained if Op(F) is F-centric. Theorem [BCGLO, 2005] If F is constrained then there exists a, unique up to isomorphism, finite p′-reduced p-constrained group L such that F = FS(L) and with S a Sylow p-subgroup of L. If Q is fully F-normalized, F-centric F-radical then: NF (Q) = FNS(Q)(LF

Q )

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H-Free Fusion Systems

Let A B ≤ G be finite groups B/A is a section of G H is involved in G ⇐ ⇒ H is isomorphic to a section of G H is not involved in G ⇐ ⇒ G is H-free

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H-Free Fusion Systems

Let A B ≤ G be finite groups B/A is a section of G H is involved in G ⇐ ⇒ H is isomorphic to a section of G H is not involved in G ⇐ ⇒ G is H-free F is H-free - if H is not involved in any of the groups LF

Q

for all Q ≤ S, such that Q is F-centric, F-radical, fully F-normalized

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H-Free Fusion Systems

Let A B ≤ G be finite groups B/A is a section of G H is involved in G ⇐ ⇒ H is isomorphic to a section of G H is not involved in G ⇐ ⇒ G is H-free F is H-free - if H is not involved in any of the groups LF

Q

for all Q ≤ S, such that Q is F-centric, F-radical, fully F-normalized Proposition 1 [KL, 2008] Let F be a fusion system on a finite p-group S Q be a fully F-normalized subgroup of S NF (Q) If F is H-free = ⇒ NS(Q)CF (Q) are H-free F/Q for Q F

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Characteristic Functors

A positive characteristic functor is a map: if 1 = S is a p-group − → 1 = W(S) char S W(ϕ(S)) = ϕ(W(S)) for every ϕ ∈ Aut(S) A Glauberman functor is a positive characteristic functor and if S ∈ Sylp(L) where L is Qd(p)-free and CL(Op(L)) = Z(Op(L)) then W(S) L. Qd(p) = (Z/pZ×Z/pZ) : SL(2,p)

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Characteristic Functors

Proposition 2 [KL, 2008 ] Let F be a fusion system on a finite p-group S. Let W be a positive characteristic functor. Assume: for any 1 = Q < S which is F-centric, F-radical, fully F-normalized NF (Q) = NNF (Q)(W(NS(Q))) Then F = NF (W(S))

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A characteristic subgroup

Notation: S a finite p-group J(S) Thompson subgroup An embedding is a pair (ϕ,E ) where ϕ ∈ Aut(S) E is a category on ϕ(S) = S Let C denote the family of all embeddings of S. A nonempty subclass U of C is characteristically closed if (ϕα,E ) ∈ U whenever (ϕ,E ) ∈ U and α ∈ Aut(S). UJ - the class of embeddings (ϕ,E ) which satisfy: UJ is characteristically closed J(ϕ(S)) = J(S) is normal in E E is a Qd(p)-free fusion system

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A characteristic subgroup

Notation: A(S) = Ω(Z(S)) B(S) = Ω(Z(J(S))) Let: W0 := A(S) ≤ B(S) and assume that for i ≥ 1 the subgroups W0,W1,...Wi−1 with A(S) = W0 < W1 < ...Wi−1 ≤ B(S) are defined. If ϕ(Wi−1) E for all (ϕ,E ) ∈ UJ then set W(S) := Wi−1 If ϕi(Wi−1) Ei for some (ϕi,Ei) ∈ UJ define Wi := ϕ−1

i

• ϕi(Wi−1)Ei
• Silvia Onofrei (Ohio State University)

A characteristic subgroup for fusion systems 16/24

A characteristic subgroup

the recursive definition terminates after a finite number of steps A(S) = W0 < W1 ... < Wi < ... < Wn =: W(S) ≤ B(S) W(S) it is independent of the pairs (ϕi,Ei) ϕ(W(S)) E for all (ϕ,E ) ∈ UJ W(ϕ(S)) = ϕ(W(S)) for any ϕ ∈ Aut(S) W(S) is a characteristic subgroup of S if S = 1 then W(S) = 1 S − → W(S) is a positive characteristic functor.

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A characteristic subgroup

S − → W(S) is a Glauberman functor J(S) is a characteristic, centric subgroup E with (ϕ,E ) ∈ UJ is a constrained fusion system on S E = EL(S) and satisfying the following conditions:

S is a Sylow p-subgroup of L CL(Op(L)) ≤ Op(L)

E is Qd(p) free − → L is Qd(p)-free

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Lemma

Let W be a fully F-normalized subgroup of S and suppose that there are two fusion subsystems F1 and F2 of F such that F = F1,F2. If moreover F1 = NF1(W) and F2 = NF2(W). Then F = NF (W).

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Proof of Theorem A

Let S a finite 2-group F a Σ4-free fusion system on S W(S) the characteristic subgroup of S If F = FS(S) then NF (W(S)) = F Suppose by induction that all proper Σ4-free subsystems G of F all Σ4-free quotient systems G = F/Q with 1 = Q F satisfy NG (W(S)) = G

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Proof of Theorem A

⋄ If O2(F) = 1 then: NF (P) ⊂ F for every fully F-normalized 1 = P ≤ S which by induction gives: NF (P) = NNF (P)(W(NS(P))) and by Proposition

2 it follows:

F = NF (W(S)) ⋄ Suppose O2(F) = 1.

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Proof of Theorem A

⋄ Set Q := O2(F) and R := QCS(Q). ⋄ If Q = R then Q is F-centric and F is constrained. Thus F = FS(LQ) with CLQ(O2(LQ)) ≤ O2(LQ). Since F is Σ4-free, the group LQ is Σ4-free. By Stellmacher’s theorem: W(S) LQ. Thus W(S) FS(LQ) and F = NF (W(S)). ⋄ Suppose Q = R.

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Proof of Theorem A

⋄ If SCF (Q) = F then F/Q is a proper quotient system of F which is Σ4-free. By induction: F/Q = NF/Q(W(S/Q)). Let U be the inverse image of W(S/Q) in S. Proposition 3.6 ?: F = NF (U). As U F it follows U ≤ Q. Contradiction given that W(S/Q) = 1. ⋄ SCF (Q) = F. ⋄ F = SCF (Q),NF (R) = F1,F2. With SCF (Q) ⊂ F and NF (R) ⊂ F. Induction gives: W(S) SCF (Q) and W(S) NF (R). W(S) is fully F-normalized. NF1(W(S)),NF2(W(S)) ⊆ NF (W(S)) ⊆ F. Therefore F = NF (W(S)).

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T H E E N D

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