Saturated fusion systems with parabolic families Silvia Onofrei The - PowerPoint PPT Presentation

Saturated fusion systems with parabolic families Silvia Onofrei The Ohio State University AMS Fall Central Sectional Meeting, Saint Louis, Missouri, 19-21 October 2013

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families Basics on Fusion Systems A fusion system F over a finite p -group S is a category whose: • objects are the subgroups of S , • morphisms are such that Hom S ( P , Q ) ⊆ Hom F ( P , Q ) ⊆ Inj ( P , Q ) , every F -morphism factors as an F -isomorphism followed by an inclusion. Let F be a fusion system over a finite p -group S . A subgroup P of S is fully F -normalized if | N S ( P ) | ≥ | N S ( ϕ ( P )) | , for all ϕ ∈ Hom F ( P , S ) ; F -centric if C S ( ϕ ( P )) = Z ( ϕ ( P )) for all ϕ ∈ Hom F ( P , S ) ; F -essential if Q is F -centric and S p ( Out F ( P )) = S p ( Aut F ( P ) / Aut P ( P )) is disconnected. The fusion system F over a finite p -group S is saturated if the following hold: • Sylow Axiom • Extension Axiom The normalizer of P in F is the fusion system N F ( P ) on N S ( P ) ϕ ∈ Hom N F ( Q , R ) if ∃ � ϕ ∈ Hom F ( PQ , PR ) with � ϕ ( P ) = P and � ϕ | Q = ϕ . The fusion system F is constrained if F = N F ( Q ) for some F -centric subgroup Q � = 1 of S . The group G has (finite) Sylow p -subgroup S if S is a finite p -subgroup of G and if every finite p -subgroup of G is conjugate to a subgroup of S . F S ( G ) is the fusion system on S with Hom F ( P , Q ) = Hom G ( P , Q ) , for P , Q ≤ S . Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 1/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families Basics on Chamber Systems A chamber system over a set I is a nonempty set C whose elements are called chambers 1 together with a family of equivalence relations ( ∼ i ; i ∈ I ) on C indexed by I . The i -panels are the equivalence classes with respect to ∼ i . 2 Two distinct chambers c and d are called i - adjacent if they are contained in the same i -panel: 3 c ∼ i d A gallery of length n connecting two chambers c 0 and c n is a sequence of chambers 4 c 0 ∼ i 1 c 1 ∼ i 2 ... ∼ i n − 1 c n − 1 ∼ i n c n The chamber system C is connected if any two chambers can be joined by a gallery. 5 6 The rank of the chamber system is the cardinality of the set I . A morphism ϕ : C → D between two chamber systems over I is a map on chambers 7 that preserves i -adjacency: if c , d ∈ C and c ∼ i d then ϕ ( c ) ∼ i ϕ ( d ) in D . Aut ( C ) is the group of all automorphisms of C (automorphism has the obvious meaning). 8 If G is a group of automorphisms of C then orbit chamber system C / G is a chamber system over I . 9 Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 2/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families Fusion Systems with Parabolic Families A fusion system F over a finite p -group S has a family { F i ; i ∈ I } of parabolic subsystems if: ∀ i ∈ I , F i is saturated, constrained, of F i -essential rank one; (F0) (F1) B := N F ( S ) is a proper subsystem of F i for all i ∈ I ; F = � F i ; i ∈ I � and no proper subset { F j ; j ∈ J ⊂ I } generates F ; (F2) F i ∩ F j = B for any pair of distinct elements F i and F j ; (F3) (F4) F ij := � F i , F j � is saturated constrained subsystem of F for all i , j ∈ I . Proposition (Onofrei, 2011) If F contains a family of parabolic subsystems then there are: • p ′ -reduced p-constrained finite groups B , G i , G ij with B = F S ( B ) , F i = F S ( G i ) , F ij = F S ( G ij ) , ∀ i , j ∈ I; • injective homomorphisms ψ i : B → G i , ψ ij : G i → G ij such that ψ ji ◦ ψ j = ψ ij ◦ ψ i , ∀ i , j ∈ I. In other words, A = { ( B , G i , G ij ) , ( ψ i , ψ ij ); i , j ∈ I } is a diagram of groups. The proof is based on: • [BCGLO, 2005]: Every saturated constrained fusion system F over S is the fusion system F S ( G ) of a finite group G that is p ′ -reduced O p ′ ( G ) = 1 and p -constrained C G ( O p ( G )) ≤ O p ( G ) , and if we set U := O p ( F ) then 1 − → Z ( U ) − → G − → Aut F ( U ) − → 1. • [Aschbacher, 2008]: If G 1 and G 2 are such that F = F S ( G 1 ) = F S ( G 2 ) then there is an isomorphism ϕ : G 1 → G 2 with ϕ | S = Id S . Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 3/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families Fusion - Chamber System Pairs Lemma (Onofrei, 2011) If G is a faithful completion of the diagram of groups A then: G := � G i , i ∈ I � � = � G j , j ∈ J � I � (P1) (P2) G i ∩ G j = B for all i � = j in I; B � = G i for all i ∈ I; (P3) ∩ g ∈ G B g = 1 . (P4) Hence ( G ; B , G i , i ∈ I ) is a parabolic system of rank n = | I | . The chamber system C = C ( G ; B , G i , i ∈ I ) is defined as follows: • the chambers are cosets gB for g ∈ G ; • two chambers gB and hB are i -adjacent if gG i = hG i where g , h ∈ G . G acts chamber transitively, faithfully on C by left multiplication. Definition (Onofrei, 2011) A fusion - chamber system pair ( F , C ) consists of: ⋄ a fusion system F with a family of parabolic subsystems { F i ; i ∈ I } ; ⋄ a chamber system C = C ( G ; B , G i , i ∈ I ) with G a faithful completion of A . Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 4/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families Main Theorem on Fusion - Chamber System Pairs Theorem (Onofrei, 2011) Let ( F , C ) be a fusion-chamber system pair. Assume the following hold. ( i ) . C P is connected for all p-subgroups P of G. ( ii ) . If P is F -centric and if R is a p-subgroup of Aut G ( P ) , then ( C P / C G ( P )) R is connected. Then F = F S ( G ) is a saturated fusion system over S. Sketch of the Argument Step 1 : S is a Sylow p-subgroup of G. Since C P is connected, C P � = / 0 and ∃ g ∈ G such that gB ∈ C P , thus P ≤ gBg − 1 and since gSg − 1 ∈ Syl p ( gBg − 1 ) , ∃ h ∈ G such that hPh − 1 ≤ S . Step 2 : F is the fusion system given by conjugation in G, this means F = F S ( G ) . Clearly F ⊆ F S ( G ) . F S ( G ) ⊆ F follows from the fact that every morphism in F S ( G ) is a composite of morphisms ϕ 1 ,..., ϕ n with ϕ i ∈ F S ( G j i ) , j i ∈ I . Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 5/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families Main Theorem: Sketch of the Argument Step 3 : Every morphism in F is a composition of restrictions of morphisms between F -centric subgroups. Recall F i is saturated, constrained and of essential rank one. If E i is F i -essential then E i is F -centric. Alperin-Goldschmidt Theorem: Each morphism ϕ i ∈ F i can be written as a composition of restrictions of F i -automorphisms of S and of automorphisms of fully F i -normalized F i -essential subgroups of S . Hence we may use: [BCGLO, 2005]: It suffices to verify the saturation axioms for the collection of F -centric subgroups only. Step 4 : The Sylow Axiom: For all F -centric P that are fully F -normalized, Aut S ( P ) ∈ Syl p ( Aut F ( P )) . Proposition [Stancu, 2004]: Assume that • Aut S ( S ) is a Sylow p -subgroup of Aut F ( S ) ; • The Extension Axiom holds for all F -centric subgroups P . Then if Q is F -centric and fully F -normalized then Aut S ( Q ) is a Sylow p -subgroup of Aut F ( Q ) . Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 6/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families Main Theorem: Sketch of the Argument Step 5 : The Extension Axiom: Let P be F -centric. For any ϕ ∈ Hom F ( P , S ) there is a morphism � ϕ ∈ Hom F ( N ϕ , S ) such that � ϕ | P = ϕ . PC S ( P ) ≤ N ϕ ≤ N S ( P ) For P ≤ S we introduce a new chamber system Rep ( P , C ) as follows: The chambers are the elements of Rep ( P , B ) := Inn ( B ) \ Inj ( P , B ) ; [ α ] ∈ Rep ( P , B ) denotes the class of α ∈ Inj ( P , B ) The i -panels are represented by the elements of Rep ( P , B , G i ) := { [ γ ] ∈ Rep ( P , G i ) : γ ∈ Inj ( P , G i ) with γ ( P ) ≤ B } . Let τ K H denote the inclusion map of the group H into the group K . � � � � τ G i τ G i Two chambers [ α ] and [ β ] are i -adjacent if B ◦ α = B ◦ β in Rep ( P , G i ) . N G ( P ) acts on Rep ( P , C ) via g · [ α ] = [ α ◦ c g − 1 ] for g ∈ N G ( P ) . Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 7/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families Main Theorem: Sketch of the Argument There is an N G ( P ) -equivariant chamber system isomorphism f P : C P − → Rep ( P , C ) 0 f P ( gB ) = [ c g − 1 ] given by that induces an isomorphism on the orbit chamber systems C P / C G ( P ) − → Rep ( P , C ) 0 where Rep ( P , C ) 0 is the connected component of Rep ( P , C ) that contains [ τ B P ] , affords the action of Aut G ( P ) = N G ( P ) / C G ( P ) . For ϕ ∈ Hom F ( P , S ) let K = N ϕ / Z ( P ) = Aut N ϕ ( P ) . � C P / C G ( P ) � K → Rep ( P , C ) K Γ : Rep ( N ϕ , C ) 0 − 0 ≃ The map is onto. • that is induced by the restriction N ϕ → P • between the connected component of Rep ( N ϕ , C ) that contains [ τ B N ϕ ] and the fixed point set of K acting on Rep ( P , C ) 0 Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 8/10