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 Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 1/24

ZJ-Theorem Let p be an odd prime. Let G be a Qd ( p ) -free finite group with C G ( O p ( G )) ≤ O p ( G ) . Then Z ( J ( S )) is a normal subgroup of G. Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 2/24

ZJ-Theorem Let p be an odd prime. Let G be a Qd ( p ) -free finite group with C G ( O p ( G )) ≤ O p ( 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 = N F ( W ( S )) . Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 2/24

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 Σ 4 -free group G with S a Sylow 2 -subgroup and C G ( O 2 ( G )) ≤ O 2 ( G ) . Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 3/24

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 Σ 4 -free group G with S a Sylow 2 -subgroup and C G ( O 2 ( G )) ≤ O 2 ( G ) . Theorem A (O-Stancu, 2008) Let S be a finite 2 -group. Let F be a Σ 4 -free fusion system over S. Then there exists a nontrivial characteristic subgroup W ( S ) of S with the property that F = N F ( W ( S )) Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 3/24

Main Theorem (O-Stancu, 2008) Let p be a prime and let S be a finite p-group. There exists a Glauberman functor − → 1 � = W ( S ) char S S with the additional property that for every Qd ( p ) -free fusion system F on S F = N F ( W ( S )) Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 4/24

Outline of the Talk Background on Fusion Systems 1 H-Free Fusion Systems 2 Characteristic Functors 3 The Characteristic Subgroup W(S) 4 Proof of Theorem A 5 Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 5/24

Background on Fusion Systems Let S be a finite p -group and let P , Q be subgroups of S . A Category F on S : objects are the subgroups of S ; Hom F ( P , Q ) ⊆ Inj ( P , Q ) ; any ϕ ∈ Hom F ( P , Q ) is the composite of an F -isomorphism followed by an inclusion. 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 : objects are the subgroups of S ; Hom F ( P , Q ) ⊆ Inj ( P , Q ) ; any ϕ ∈ Hom F ( 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 ∈ Hom F ( 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 : objects are the subgroups of S ; Hom F ( P , Q ) ⊆ Inj ( P , Q ) ; any ϕ ∈ Hom F ( 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 ∈ Hom F ( P , Q ) . The subgroup Q of S is: fully F -centralized if | C S ( Q ) | ≥ | C S ( Q ′ ) | for all Q ′ ≤ S which are F -conjugate to Q . fully F -normalized if | N S ( Q ) | ≥ | N S ( Q ′ ) | for all Q ′ ≤ S which are F -conjugate to Q . Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 6/24

Background on Fusion Systems A fusion system on S is a category F on S such that: Hom S ( P , Q ) ⊆ Hom F ( P , Q ) for all P , Q ≤ S . Aut S ( S ) is a Sylow p -subgroup of Aut F ( S ) . Every ϕ : Q → S such that ϕ ( Q ) is fully F -normalized extends to a morphism � ϕ : N ϕ → S . Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 7/24

Background on Fusion Systems A fusion system on S is a category F on S such that: Hom S ( P , Q ) ⊆ Hom F ( P , Q ) for all P , Q ≤ S . Aut S ( S ) is a Sylow p -subgroup of Aut F ( S ) . Every ϕ : Q → S such that ϕ ( Q ) is fully F -normalized extends to a morphism � ϕ : N ϕ → S . For a morphism ϕ ∈ Hom F ( Q , S ) N ϕ = { x ∈ N S ( Q ) |∃ y ∈ N S ( ϕ ( Q )) , ϕ ( x u ) = y ϕ ( u ) , ∀ u ∈ Q } a subgroup with the property that: QC S ( Q ) ≤ N ϕ ≤ N S ( Q ) . Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 7/24

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 ∈ Hom F ( N S ( 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 ∈ Hom F ( N S ( Q ) , S ) . Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 8/24

Background on Fusion Systems The normalizer of Q in F is the category N F ( Q ) : ⋄ objects: the subgroups of N S ( Q ) ; ⋄ morphisms: ϕ ∈ Hom F ( R , T ) , for which there exists � ϕ ∈ Hom F ( QR , QT ) such that ϕ | Q ∈ Aut F ( Q ) and � � ϕ | R = ϕ . • Q fully F -normalized = ⇒ N F ( Q ) a fusion system on N S ( Q ) The centralizer of Q in F is the category C F ( Q ) : ⋄ objects: the subgroups of C S ( Q ) ; ⋄ morphisms: ϕ ∈ Hom F ( R , T ) , for which there exists � ϕ ∈ Hom F ( QR , QT ) such that ϕ | Q = id Q and � � ϕ | R = ϕ . • Q fully F -centralized = ⇒ C F ( Q ) a fusion systems on C S ( Q ) Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 9/24

Background on Fusion Systems The category N S ( Q ) C F ( Q ) : ⋄ objects: the subgroups of N S ( Q ) ; ⋄ morphisms: group homomorphisms ϕ : R → T , for which there exists ψ : QR → QT and x ∈ N S ( Q ) such that ψ | Q = c x and ψ | R = ϕ . • Q fully F -centralized = ⇒ N S ( Q ) C F ( Q ) a fusion systems on C S ( Q ) Q � F - if F = N F ( Q ) . O p ( 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 ∈ Hom F ( P , R ) satisfying ψ ( xQ ) = ϕ ( x ) Q for all x ∈ P . Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 10/24

Background on Fusion Systems F -centric if C S ( ϕ ( Q )) ⊆ ϕ ( Q ) , for all ϕ ∈ Hom F ( Q , S ) . F -radical if O p ( Aut F ( Q )) = Inn ( Q ) . F is constrained if O p ( 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 = F S ( L ) and with S a Sylow p-subgroup of L. Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 11/24

Background on Fusion Systems F -centric if C S ( ϕ ( Q )) ⊆ ϕ ( Q ) , for all ϕ ∈ Hom F ( Q , S ) . F -radical if O p ( Aut F ( Q )) = Inn ( Q ) . F is constrained if O p ( 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 = F S ( L ) and with S a Sylow p-subgroup of L. If Q is fully F -normalized, F -centric F -radical then: N F ( Q ) = F N S ( Q ) ( L F Q ) Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 11/24

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 Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 12/24

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 L F Q for all Q ≤ S , such that Q is F -centric, F -radical, fully F -normalized Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 12/24

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 L F 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 N F ( Q ) If F is H-free = ⇒ N S ( Q ) C F ( Q ) are H-free F / Q for Q � F Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 12/24

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 ∈ Syl p ( L ) where L is Qd ( p ) -free and C L ( O p ( L )) = Z ( O p ( L )) then W ( S ) � L . Qd ( p ) = ( Z / p Z × Z / p Z ) : SL ( 2 , p ) Silvia Onofrei (Ohio State University) A characteristic subgroup for fusion systems 13/24