Saturated fusion systems with parabolic families Silvia Onofrei The Ohio State University AMS Fall Central Sectional Meeting, Saint Louis, Missouri, 19-21 October 2013
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
Basics on Chamber Systems 1 A chamber system over a set I is a nonempty set C whose elements are called chambers together with a family of equivalence relations ( i ;i I) on C indexed by I. 2 The i-panels are the equivalence classes with respect to i. 3 Two distinct chambers c and d are called i- adjacent if they are contained in the same i-panel: c i d 4 A gallery of length n connecting two chambers c 0 and c n is a sequence of chambers c 0 i1 c 1 i2... in 1 c n 1 in c n 5 The chamber system C is connected if any two chambers can be joined by a gallery. 6 The rank of the chamber system is the cardinality of the set I. 7 A morphism ϕ : C D between two chamber systems over I is a map on chambers that preserves i-adjacency: if c,d C and c i d then ϕ(c) i ϕ(d) in D. 8 Aut(C) is the group of all automorphisms of C (automorphism has the obvious meaning). 9 If G is a group of automorphisms of C then orbit chamber system C/G is a chamber system over I. Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 2/10
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: (F0) i I, F i is saturated, constrained, of F i -essential rank one; (F1) B := N F (S) is a proper subsystem of F i for all i I; (F2) F = F i ;i I and no proper subset {F j ;j J I} generates F ; (F3) F i F j = B for any pair of distinct elements F i and F j ; (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
Fusion - Chamber System Pairs Lemma (Onofrei, 2011) If G is a faithful completion of the diagram of groups A then: (P1) G := G i,i I = G j,j J I (P2) G i G j = B for all i j in I; (P3) B G i for all i I; (P4) g G B g = 1. 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
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 ji ), j i I. Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 5/10
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
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. [ ] [ ] Two chambers [α] and [β] are i-adjacent if τ G i B α = τ G i 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
Main Theorem: Sketch of the Argument There is an N G (P)-equivariant chamber system isomorphism f P : C P Rep(P,C) 0 given by f P (gb) = [c g 1 ] 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 Nϕ (P). The map Γ : Rep(N ϕ,c) 0 Rep(P,C) K 0 ( C P /C G (P) ) K 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
Main Theorem: Sketch of the Argument There is an N G (P)-equivariant chamber system isomorphism f P : C P Rep(P,C) 0 given by f P (gb) = [c g 1 ] 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 Nϕ (P). The map Γ : Rep(N ϕ,c) 0 Rep(P,C) K 0 ( C P /C G (P) ) K 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
An Application: Fusion Systems with Classical Parabolic Families Assume F contains a family of parabolic systems. Set U i = O p (F i ) and U ij = O p (F ij ). We say F contains a classical family of parabolic systems with diagram M if: For each i I, Out Fi (U i ) is a rank one finite group of Lie type in characteristic p. For each pair i,j I, Out Fij (U ij ) is either a rank two finite group of Lie type in characteristic p or it is a (central) product of two rank one finite groups of Lie type in characteristic p. The diagram M is a graph whose vertices are labeled by the elements of I, Out Fij (U ij ) is a product of two rank one Lie groups then the nodes i and j are not connected, Out Fij (U ij ) is a rank two Lie group the nodes i and j are connected by a bond of strength m ij 2, where m ij denotes the integer that defines the Weyl group. Proposition Let (F,C) be a fusion-chamber system pair with I 3. Assume that: (i). F contains a classical family of parabolic systems with diagram M ; (ii). M is a spherical diagram. Then F is the fusion system of a finite simple group of Lie type in characteristic p extended by diagonal and field automorphisms. Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 9/10
Thank You The End Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 10/10