Realizability and tameness of fusion systems
aa r X i v : . [ m a t h . G R ] F e b TAMENESS OF FUSION SYSTEMS OF FINITE GROUPS
CARLES BROTO, JESPER M. MØLLER, BOB OLIVER, AND ALBERT RUIZ
Abstract.
The fusion system of a finite group at a prime p is a category that encodesall conjugacy relations among subgroups and elements of a given Sylow p -subgroup. Sucha fusion system is tame if there is some (possibly different) finite group that realizes it,with the property (very approximately) that all automorphisms of the fusion system areinduced by automorphisms of the group. In this paper, we prove that all fusion systems offinite groups are tame. This was already known for fusion systems of (known) finite simplegroups, and the arguments here consist mainly of a reduction to that case. In particular,this result depends on the classification of finite simple groups. Introduction
Let p be a prime. The fusion system of a finite group G over a Sylow p -subgroup S of G is the category F S ( G ) whose objects are the subgroups of G and whose morphismsare the homomorphisms between subgroups induced by conjugation in G , thus encoding G -conjugacy relations among subgroups and elements of S . With this as starting point andalso motivated by questions in representation theory, Puig defined the concept of abstractfusion systems (see [Pg] and Definition 1.1) and showed that they behave in many wayssimilarly to finite groups.The concept of linking systems associated to fusion systems was first proposed by Bensonin [Be3] and in unpublished notes, and was developed in detail by Broto, Levi, and Oliver[BLO2]. See Definition 1.6 for precise definitions. This was originally motivated by questionsinvolving classifying spaces of fusion systems and of the finite groups that they realize, butalso turned out to be important when studying many of the purely algebraic properties offusion systems.A fusion system F over a finite p -group is realized by a finite group G if F ∼ = F S ( G )for some S ∈ Syl p ( G ). We say that F is tamely realized by G if in addition, the naturalhomomorphism from Out( G ) to Out( L cS ( G )) is split surjective (Definitions 2.6 and 2.7).Here, L cS ( G ) is the linking system associated to G and to F . We say that F is tame if it istamely realized by some finite group.Tameness was originally defined in [AOV, § § Mathematics Subject Classification.
Primary 20D20. Secondary 20D05, 20D25, 20D45.
Key words and phrases. fusion systems, Sylow subgroups, automorphisms, wreath products, finite simplegroups.C. Broto is partially supported by MICINN grant MTM2016-80439-P and AGAUR grant 2017-SGR-1725.J. Møller was partially supported by the Danish National Research Foundation through the CopenhagenCentre for Geometry and Topology (DNRF151).B. Oliver is partially supported by UMR 7539 of the CNRS.A. Ruiz is partially supported by MICINN grant MTM2016-80439-P and AGAUR grant 2017-SGR-1725.
Tameness can also be interpreted topologically. For a finite group G , let BG ∧ p be theclassifying space of G completed at p in the sense of Bousfield and Kan, and let Out( BG ∧ p )be the set of homotopy classes of self homotopy equivalences of BG ∧ p . By [BLO1, TheoremB], Out( L cS ( G )) is isomorphic to Out( BG ∧ p ) for S ∈ Syl p ( G ). (See also [BLO2, Lemma 8.2]and [AOV, Lemma 1.14] for the proof that the group Out typ ( L cS ( G )) defined in [BLO1] isisomorphic to Out( L cS ( G )) as defined here.) Thus F S ( G ) is tamely realized by G exactlywhen the natural map from Out( G ) to Out( BG ∧ p ) is split surjective.We can now state our main theorem: Theorem A.
Assume the classification of finite simple groups. Then for each prime p ,every realizable fusion system over a finite p -group is tame. When F is the fusion system of a finite simple group G , the tameness of F was shown in[AOV] when G is an alternating group, in [BMO2] when G is of Lie type, and in [O7] when G is sporadic. To reduce to those cases, we begin with the fusion system F S ( G ) of an arbitraryfinite group G , then reduce to the case where O p ( G ) = O p ′ ( G ) = 1 and O p ( F S ( G )) = 1,and then show how to get from the tameness of the fusion system of the generalized Fittingsubgroup F ∗ ( G ) (a product of simple groups) to the tameness of F S ( G ). See Theorem 5.4and its proof for details.The authors would like to thank the Universitat Aut`onoma de Barcelona and the Uni-versity of Copenhagen for its hospitality while the four authors met during early stages ofthis work; and also the French CNRS and BigBlueButton for helping us to meet virtuallyat frequent intervals to discuss this work during the covid-19 pandemic. Notation:
The notation used in this paper is mostly standard, with a few exceptions.Composition of functions and functors is always from right to left. Also, C n denotes a(multiplicative) cyclic group of order n , and E q is always an elementary abelian group oforder q (for q a prime power). When G is a (multiplicative) group, 1 ∈ G always denotes itsidentity element.When f : C −→ D is a functor, then for objects c, c ′ in C , we let f c,c ′ be the induced mapfrom Mor C ( c, c ′ ) to Mor D ( f ( c ) , f ( c ′ )), and also set f c = f c,c for short.When G is a group, we let S ( G ) denote the set of subgroups of G , and indicate conjugationby setting g x = c g ( x ) = gxg − and g H = c g ( H ) = gHg − for g, x ∈ G and H ≤ G . Also,Hom G ( P, Q ) (for
P, Q ≤ G ) is the set of (injective) homomorphisms from P to Q inducedby conjugation in G .Throughout the paper, p will always be a fixed prime.1. Fusion systems and linking systems
This is a background section intended to provide the reader with the necessary basicdefinitions and properties of fusion and linking systems that will be used throughout thepaper. Fusion systems and saturation were originally introduced by Puig, first in unpublishednotes, and then in [Pg]. Abstract linking systems were defined in [BLO2]. As a generalreference for the subject we refer to [AKO].1.1.
Fusion systems.
For a prime p , a fusion system over a finite p -group S is a categorywhose objects are the subgroups of S , and whose morphisms are injective homomorphismsbetween subgroups such that for each P, Q ≤ S : • Hom F ( P, Q ) ⊇ Hom S ( P, Q ); and
AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 3 • for each ϕ ∈ Hom F ( P, Q ), ϕ − ∈ Hom F ( ϕ ( P ) , P ).Here, Hom F ( P, Q ) denotes the set of morphisms in F from P to Q . We also write Aut S ( P ) forHom S ( P, P ), Iso F ( P, Q ) for the set of isomorphisms, Aut F ( P ) = Iso F ( P, P ), and Out F ( P ) =Aut F ( P ) / Inn( P ). For P ≤ S and g ∈ S , we set P F = { ϕ ( P ) | ϕ ∈ Hom F ( P, S ) } and g F = { ϕ ( g ) | ϕ ∈ Hom F ( h g i , S ) } (the sets of subgroups and elements F -conjugate to P and to g ).The following version of the definition of a saturated fusion system is the most convenientone to use here. (See Definitions I.2.2 and I.2.4 and Proposition I.2.5 in [AKO].) Definition 1.1.
Let F be a fusion system over a finite p -group S . (a) A subgroup P ≤ S is fully normalized (fully centralized) in F if | N S ( P ) | ≥ | N S ( Q ) | ( | C S ( P ) | ≥ | C S ( Q ) | ) for each Q ∈ P F . (b) The fusion system F is saturated if it satisfies the following two conditions: (I) (Sylow axiom) For each subgroup P ≤ S fully normalized in F , P is fully cen-tralized and Aut S ( P ) ∈ Syl p (Aut F ( P )) . (II) (extension axiom) For each isomorphism ϕ ∈ Iso F ( P, Q ) in F such that Q isfully centralized in F , ϕ extends to a morphism ϕ ∈ Hom F ( N ϕ , S ) where N ϕ = { g ∈ N S ( P ) | ϕc g ϕ − ∈ Aut S ( Q ) } . The above definition is motivated by fusion systems of finite groups. When G is a finitegroup and S ∈ Syl p ( G ), the p -fusion system of G is the category F S ( G ) whose objects arethe subgroups of S , and where Mor F S ( G ) ( P, Q ) = Hom G ( P, Q ) for each
P, Q ≤ S . For aproof that F S ( G ) is saturated, see, e.g., [AKO, Lemma I.1.2]. In general, a saturated fusionsystem F over a finite p -group S will be called realizable if F = F S ( G ) for some finite group G with S ∈ Syl p ( G ), and will be called exotic otherwise.We next list some of the terminology used to describe certain subgroups in a fusion system. Definition 1.2.
Let F be a fusion system over a finite p -group S . For a subgroup P ≤ S , (a) P is F -centric if C S ( Q ) ≤ Q for each Q ∈ P F ; (b) P is F -radical if O p (Out F ( P )) = 1 ; (c) P is F -quasicentric if for each Q ∈ P F which is fully centralized in F , the centralizerfusion system C F ( Q ) (see Definition 1.4(b)) is the fusion system of the group C S ( Q ) ; (d) P is weakly closed in F if P F = { P } ; (e) P is strongly closed in F if for each x ∈ P , x F ⊆ P ; (f) P is normal in F ( P E F ) if each ϕ ∈ Hom F ( Q, R ) (for Q, R ≤ S ) extends to amorphism ϕ ∈ Hom F ( P Q, P R ) such that ϕ ( P ) = P ; and (g) P is central in F if each ϕ ∈ Hom F ( Q, R ) (for Q, R ≤ S ) extends to a morphism ϕ ∈ Hom F ( P Q, P R ) such that ϕ | P = Id P .Let F cr ⊆ F c ⊆ F q denote the sets of F -centric F -radical, F -centric, and F -quasicentricsubgroups of S , respectively, or (depending on the context) the full subcategories of F withthose objects. Let O p ( F ) ≥ Z ( F ) denote the (unique) largest normal and central subgroups,respectively, in F . The following result is one of the versions of Alperin’s fusion theorem for fusion systems.
CARLES BROTO, JESPER M. MØLLER, BOB OLIVER, AND ALBERT RUIZ
Theorem 1.3.
Let F be a saturated fusion system over a finite p -group S . Then eachmorphism in F is a composite of restrictions of automorphisms of subgroups that are F -centric, F -radical, and fully normalized in F .Proof. This follows from [AKO, Theorem I.3.6] (the same statement but for F -essentialsubgroups), together with [AKO, Proposition I.3.3(a)] (all F -essential subgroups are F -centric and F -radical). Alternatively, the result as stated here is shown directly (withoutmention of essential subgroups) in [BLO2, Theorem A.10]. (cid:3) Definition 1.4.
Let F be a saturated fusion system over a finite p -group S , and let Q ≤ S be a subgroup. (a) Let N F ( Q ) be the normalizer fusion system of Q in F : the fusion subsystem over N S ( Q ) whose morphisms are those ϕ ∈ Hom F ( P, R ) (for P, R ≤ N S ( Q ) ) that extend to ϕ ∈ Hom F ( P Q, RQ ) with ϕ ( Q ) = Q . (b) Let C F ( Q ) be the centralizer fusion system of Q in F : the fusion subsystem over C S ( Q ) whose morphisms are those ϕ ∈ Hom F ( P, R ) (for P, R ≤ C S ( Q ) ) that extend to ϕ ∈ Hom F ( P Q, RQ ) with ϕ | Q = Id . If Q is fully normalized (fully centralized) in F then N F ( Q ) ( C F ( Q )) is a saturated fusionsystem (see [BLO2, Proposition A.6] or [AKO, Theorem I.5.5]). Note that Q E F if andonly if N F ( Q ) = F , and Q is central in F (i.e., Q ≤ Z ( F )) if and only if C F ( Q ) = F .Quotient fusion systems, defined as follows, will be needed in Sections 4 and 5. Definition 1.5.
Let F be a fusion system over a finite p -group S , and assume T E S isstrongly closed in F . Then F /T is defined to be the fusion system over S/T where
Hom F /T ( P/T, Q/T ) = (cid:8) ϕ/T (cid:12)(cid:12) ϕ ∈ Hom F ( P, Q ) (cid:9) for P, Q ≤ S containing T . Here, ϕ/T ∈ Hom(
P/T, Q/T ) sends gT to ϕ ( g ) T . Note that F /T = N F ( T ) /T in all cases. It is not hard to see that if H E G are finitegroups, S ∈ Syl p ( G ), and T = S ∩ H ∈ Syl p ( H ), then F S ( G ) /T = F S/T ( G/H ). We refer to[AKO, § II.5] for more details and for some of the other properties of these quotient systems.1.2.
Linking systems.
Before defining linking systems, we need to introduce more notation.If
P, Q ≤ G are subgroups of a finite group G , the transporter set T G ( P, Q ) is defined bysetting T G ( P, Q ) = { g ∈ G | g P ≤ Q } . The transporter category of G is the category T ( G ) whose objects are the subgroups of G ,and whose morphisms sets are the transporter sets:Mor T ( G ) ( P, Q ) = T G ( P, Q ) . Composition in T ( G ) is given by multiplication in G . If H is a set of subgroups of G , then T H ( G ) ⊆ T ( G ) denotes the full subcategory with object set H .The following definition of linking system taken from [AKO, Definition III.4.1]. Definition 1.6.
Let F be a fusion system over a finite p -group S . A linking system as-sociated to F is a triple ( L , δ, π ) where L is a finite category, and δ and π are a pair offunctors T Ob( L ) ( S ) δ −−−−→ L π −−−−→ F that satisfy the following conditions: AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 5 (A1) Ob( L ) is a set of subgroups of S closed under F -conjugacy and overgroups, and contains F cr . Each object in L is isomorphic (in L ) to one which is fully centralized in F . (A2) δ is the identity on objects, and π is the inclusion on objects. For each P, Q ∈ Ob( L ) such that P is fully centralized in F , C S ( P ) acts freely on Mor L ( P, Q ) via δ P and rightcomposition, and π P,Q : Mor L ( P, Q ) → Hom F ( P, Q ) is the orbit map for this action. (B) For each
P, Q ∈ Ob( L ) and each g ∈ T S ( P, Q ) , π P,Q sends δ P,Q ( g ) ∈ Mor L ( P, Q ) to c g ∈ Hom F ( P, Q ) . (C) For all ψ ∈ Mor L ( P, Q ) and all g ∈ P , the diagram P ψ / / δ P ( g ) (cid:15) (cid:15) Q δ Q ( π ( ψ )( g )) (cid:15) (cid:15) P ψ / / Q commutes in L .When the functors δ and π are understood, we refer directly to the category L as a linkingsystem. A centric linking system associated to F is a linking system L associated to F such thatOb( L ) = F c .Linking systems associated to a fusion system were originally motivated by centric linkingsystems of finite groups. For a finite group G , a p -subgroup P ≤ G is p -centric in G if Z ( P ) ∈ Syl p ( C G ( P )); equivalently, if C G ( P ) = Z ( P ) × O p ( C G ( P )) and O p ( C G ( P )) has orderprime to p . For S ∈ Syl p ( G ), the centric linking system of G over S is consists of the category L cS ( G ) whose objects are the subgroups of S which are p -centric in G and whose morphismsets are given byMor L cS ( G ) ( P, Q ) = T G ( P, Q ) /O p ( C G ( P )) (all P, Q ∈ Ob( L cS ( G ))),together with functors T Ob( L cS ( G )) ( S ) δ −→ L cS ( G ) π −−→ F S ( G ) defined in the obvious way.When G is a finite group and S ∈ Syl p ( G ), then P ≤ S is F -centric (see Definition1.2) if and only if P is p -centric in G (see [BLO1, Lemma A.5]). Moreover, F S ( G ) is alwayssaturated (see [AKO, Theorem I.2.3]), and ( L cS ( G ) , δ, π ) is a centric linking system associatedto F S ( G ).Some of the basic properties of linking systems are listed in the next proposition. Proposition 1.7.
Let ( L , δ, π ) be a linking system associated to a saturated fusion system F over a finite p -group S . Then (a) δ is injective on all morphism sets; and (b) all morphisms in L are monomorphisms and epimorphisms in the categorical sense.Conditions for the existence of restrictions and extensions of morphisms are as follows: (c) For every morphism ψ ∈ Mor L ( P, Q ) , and every P , Q ∈ Ob( L ) such that P ≤ P , Q ≤ Q , and π ( ψ )( P ) ≤ Q , there is a unique morphism ψ | P ,Q ∈ Mor L ( P , Q ) (the“restriction” of ψ ) such that ψ ◦ ι P ,P = ι Q ,Q ◦ ψ | P ,Q . CARLES BROTO, JESPER M. MØLLER, BOB OLIVER, AND ALBERT RUIZ (d)
Let
P, Q, P , Q ∈ Ob( L ) and ψ ∈ Mor L ( P, Q ) be such that P E P , Q ≤ Q , and for each g ∈ P there is h ∈ Q such that ψ ◦ δ P ( g ) = δ Q,Q ∗ ( h ) ◦ ψ ( Q ∗ = hQh − ). Then there is aunique morphism ψ ∈ Mor L ( P , Q ) such that ψ | P,Q = ψ .Proof. See points (c), (f), (b), and (e), respectively, in [O2, Proposition 4]. (cid:3)
Lemma 1.8.
Let G be a finite group with S ∈ Syl p ( G ) , and set F = F S ( G ) . (a) For each Q ≤ S , Q is fully normalized (fully centralized) if and only if N S ( Q ) ∈ Syl p ( N G ( Q )) ( C S ( Q ) ∈ Syl p ( C G ( Q )) ). If this holds, then N F ( Q ) = F N S ( Q ) ( N G ( Q )) ( C F ( Q ) = F C S ( Q ) ( C G ( Q )) ). (b) In all cases, O p ( G ) ≤ O p ( F ) and O p ( Z ( G )) ≤ Z ( F ) .Proof. Point (a) is shown in [AKO, Proposition I.5.4]. In particular, when Q = O p ( G ), wehave N F ( Q ) = F S ( G ) = F and hence Q ≤ O p ( F ). Similarly, when Q = O p ( Z ( G )), it saysthat C F ( Q ) = F S ( G ) = F and hence that Q ≤ Z ( F ). (cid:3) We note here the existence and uniqueness of linking systems shown by Chermak, Oliver,and Glauberman-Lynd. Two linking systems ( L , δ , π ) and ( L , δ , π ) associated to thesame fusion system F are isomorphic if there is an isomorphism of categories ρ : L ∼ = −−→ L such that ρ ◦ δ = δ and π ◦ ρ = π . Theorem 1.9 ([Ch, O3, GLn]) . Let F be a saturated fusion system over a finite p -group S ,and let H be a set of subgroups of S such that F cr ⊆ H ⊆ F q , and such that H is closed under F -conjugacy and overgroups. Then up to isomorphism, there is a unique linking system L H associated to F with object set H .Proof. The existence and uniqueness of a centric linking system associated to F was shownby Chermak. See [Ch, Main theorem] and [O3, Theorem A] for two versions of his originalproof, and [GLn, Theorem 1.2] for the changes to the proof in [O3] needed to make itindependent of the classification of finite simple groups.More generally, if F cr ⊆ H ⊆ F c , the uniqueness of an H -linking system follows by thesame obstruction theory (shown to vanish in [O3, Theorem 3.4] and [GLn, Theorem 1.1]) asthat used in the centric case (by the same argument as in the proof of [BLO2, Proposition3.1]). For arbitrary H ⊆ F q containing F cr , the existence and uniqueness now follows from[AKO, Proposition III.4.8], applied with H ⊇ H ∩ F c in the role of b H ⊇ H . (cid:3) Normal fusion and linking subsystems.
Let F be a fusion system over a finite p -group S . A fusion subsystem of F is a subcategory E ⊆ F which is itself a fusion systemover a subgroup T ≤ S (in particular, Ob( E ) = S ( T )). We write E ≤ F when E is a fusionsubsystem, and also sometimes say that E ≤ F is a pair of fusion systems over T ≤ S . Definition 1.10.
Let F be a fusion system over a finite p -group S . (a) Let R be another finite p -group and let α : S −→ R be an isomorphism. We denote by α F the fusion system over R with morphism sets Hom α F ( P, Q ) = α ◦ Hom F ( α − ( P ) , α − ( Q )) ◦ α − for each pair of subgroups P, Q ≤ R . (b) Let E be another fusion system over a finite p -group T . We say that E and F areisomorphic fusion systems if there is an isomorphism α : S −→ T such that E = α F . AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 7
A more general concept of morphism between fusion systems is given in [AKO, DefinitionII.2.2].Consider now the following definition from [AKO, Definition I.6.1].
Definition 1.11.
Fix a saturated fusion system F over a finite p -group S . (a) A fusion subsystem
E ≤ F over T E S is weakly normal if T is strongly closed in F and the following conditions hold: • (invariance condition) α E = E for each α ∈ Aut F ( T ) , and • (Frattini condition) for each P ≤ T and each ϕ ∈ Hom F ( P, T ) , there are α ∈ Aut F ( T ) and ϕ ∈ Hom E ( P, T ) such that ϕ = α ◦ ϕ . (b) A fusion subsystem
E ≤ F over T E S is normal ( E E F ) if E is weakly normal in F and • (Extension condition) each α ∈ Aut E ( T ) extends to α ∈ Aut F ( T C S ( T )) such that [ α, C S ( T )] ≤ Z ( T ) . (c) A saturated fusion system F over a finite p -group S is simple if it contains no propernontrivial normal fusion subsystem. It will be convenient to say that “ E E F is a normal pair of fusion systems over T E S ”to mean that F is a fusion system over S and E E F is a normal subsystem over T .Note that if F is a saturated fusion system over a finite p -subgroup S , and P ≤ S , then P E F if and only if F P ( P ) E F [A3, (7.9)].Note also that what are called “normal fusion subsystems” in [AOV, Definition 1.18] arewhat we are calling “weakly normal” subsystems here.When ( L , δ, π ) is a linking system associated to the fusion system F over S , and F ≤ F is a fusion subsystem over S ≤ S , then a linking subsystem associated to F is a linkingsystem ( L , δ , π ) associated to F , where L is a subcategory of L and T Ob( L ) ( S ) δ −−−−−→ L π −−−−−→ F are the restrictions of δ and π . In this situation, we write L ≤ L , and sometimes say that L ≤ L is a pair of linking systems. Note in particular the special case where S = S and F = F but Ob( L ) ⊆ Ob( L ): a pair of linking systems with different objects associated tothe same fusion system. Definition 1.12.
Fix a pair of saturated fusion systems
E ≤ F over finite p -groups T E S such that E E F , and let M ≤ L be a pair of associated linking systems. Then, M is normalin L ( M E L ) if: (a) Ob( L ) = { P ≤ S | P ∩ T ∈ Ob( M ) } , and (b) for all γ ∈ Aut L ( T ) and ψ ∈ Mor( M ) , γψγ − ∈ Mor( M ) .If M E L , then we define L / M = Aut L ( T ) / Aut M ( T ) . Notice that not every normal pair of fusion systems has an associated normal pair oflinking systems.Definition 1.12 differs from Definition 1.27 in [AOV] in that there is no “Frattini condition”in the definition we give here. We have omitted it since it follows from the Frattini conditionfor normal fusion subsystems, as shown in the next lemma.
CARLES BROTO, JESPER M. MØLLER, BOB OLIVER, AND ALBERT RUIZ
Lemma 1.13. If M E L is a normal pair of linking systems associated to fusion systems E E F over finite p -groups T E S , then for all P, Q ∈ Ob( M ) and all ψ ∈ Mor L ( P, Q ) ,there are morphisms γ ∈ Aut L ( T ) and ψ ∈ Mor M ( γ ( P ) , Q ) such that ψ = ψ ◦ γ | P,γ ( P ) .Proof. Let ψ ∈ Mor L ( P, Q ) be as above, and assume first that P is fully centralized in F .Then π ( ψ ) ∈ Hom F ( P, Q ), and by the Frattini condition on E E F , there are α ∈ Aut F ( T )and ϕ ∈ Hom E ( α ( P ) , Q ) such that π ( ψ ) = ϕ ◦ α | P,α ( P ) . Choose lifts e ϕ ∈ Mor M ( α ( P ) , Q ) of ϕ and e α ∈ Aut L ( T ) of α . By axiom (A2) for the linking system L , there is z ∈ C S ( P ) suchthat e ϕ ◦ e α | P,α ( P ) = ψ ◦ δ P ( z ) | P,P . Set γ = e α ◦ δ S ( z ) − ∈ Aut L ( T ), and then ψ = e ϕ ◦ γ | P,γ ( P ) .If P is not fully centralized in F , then choose P ∗ ∈ P F that is fully centralized, andfix ω ∈ Iso L ( P ∗ , P ). Then P ∗ ≤ T since T is strongly closed in F , and so P ∗ ∈ Ob( M ).We just showed that there are morphisms γ , γ ∈ Aut L ( T ), ψ ∈ Mor M ( γ ( P ∗ ) , Q ), and ψ ∈ Mor M ( γ ( P ∗ ) , P ) such that ψ ◦ ω = ψ ◦ γ | P ∗ ,γ ( P ∗ ) and ω = ψ ◦ γ | P ∗ ,γ ( P ∗ ) . Also, ψ is an isomorphism since ω is an isomorphism. Set γ = γ γ − ∈ Aut L ( T ). Then ψ = ( ψ ◦ ω ) ◦ ω − = ψ ◦ γ | γ ( P ∗ ) ,γ ( P ∗ ) ◦ ψ − = ψ ◦ ( γψ γ − ) − ◦ γ | P,γ ( P ) where γψ γ − ∈ Iso M ( γ ( P ∗ ) , γ ( P )) by Definition 1.12(b). (cid:3) Automorphism groups and tameness
The aim of this section is to introduce the concept of tameness for fusion systems. Thiswas originally defined in [AOV] and it is the main subject of this article.Before doing so, we must define automorphism and outer automorphism groups of fusionand linking systems.
Definition 2.1.
Fix a saturated fusion system F over a finite p -group S . Then (a) Aut( F ) = { α ∈ Aut( S ) | α F = F } : the group of automorphisms of S that send F toitself; (b) Out( F ) = Aut( F ) / Aut F ( S ) is the group of outer automorphisms of F ; and (c) for each α ∈ Aut( F ) , we let c α : F −→ F denote the functor that sends an object P to α ( P ) and a morphism ϕ ∈ Hom F ( P, Q ) to αϕα − ∈ Hom F ( α ( P ) , α ( Q )) . Now that we have defined automorphisms, we can define characteristic subsystems:
Definition 2.2.
Fix a saturated fusion system F over a finite p -group S . A fusion subsystem E ≤ F over T E S is characteristic if E is normal in F and c α ( E ) = E for all α ∈ Aut( F ) .Likewise, a subgroup P of S is characteristic in F if P E F and α ( P ) = P for all α ∈ Aut( F ) ; equivalently, if F P ( P ) is a characteristic subsystem of F . For example, when F is a saturated fusion system over a finite p -group S , then thesubsystems O p ( F ) and O p ′ ( F ) (see Definition 3.6(b,c)) and the subgroups O p ( F ) and Z ( F )are all characteristic in F . Lemma 2.3.
Let E E F be a normal pair of fusion systems over finite p -groups T E S .Then (a) if D E E is characteristic in E , then D E F ; and (b) O p ( E ) ≤ O p ( F ) . AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 9
Proof.
Point (a) is shown in [A3, 7.4]. Since O p ( E ) is characteristic in E , O p ( E ) E F by (a),and so O p ( E ) ≤ O p ( F ), proving (b). (cid:3) The following definitions of automorphism groups are taken from [AOV, Definition 1.13& Lemma 1.14].
Definition 2.4.
Let F be a fusion system over a finite p -group S and let L be an associatedlinking system. (a) For each pair of objects P ≤ Q in L , we call ι P,Q def = δ P,Q (1) ∈ Mor L ( P, Q ) the inclusion in L of P in Q . For each P in L , we call δ P ( P ) ≤ Aut L ( P ) the distinguished subgroup of Aut L ( P ) . (b) Let
Aut( L ) be the group of automorphisms of the category L that send inclusions toinclusions and distinguished subgroups to distinguished subgroups. (c) For γ ∈ Aut L ( S ) , let c γ ∈ Aut( L ) be the automorphism which sends an object P to c γ ( P ) def = π ( γ )( P ) , and sends ψ ∈ Mor L ( P, Q ) to c γ ( ψ ) def = γ | Q, γ Q ◦ ψ ◦ ( γ | P, γ P ) − . Set Out( L ) = Aut( L ) (cid:14) { c γ | γ ∈ Aut L ( S ) } . The notation in Definitions 2.1 and 2.4 is slightly different from that used in [AOV] and[AKO], as described in the following table:Notation used here Aut( F ) Out( F ) Aut( L ) Out( L )Used in [AOV, AKO] Aut( S, F ) Out( S, F ) Aut I typ ( L ) Out typ ( L )By [AOV, Lemma 1.14], the above definition of Out( L ) is equivalent to Out typ ( L ) in[BLO2], and by [BLO2, Lemma 8.2], both are equivalent to Out typ ( L ) in [BLO1]. So by[BLO1, Theorem 4.5(a)], Out( L cS ( G )) ∼ = Out( BG ∧ p ): the group of homotopy classes of selfhomotopy equivalences of the space BG ∧ p .The next result shows how an automorphism of a linking system automatically preservesthe structure functors. For use in the next section, we state this for certain full subcategoriesof a linking system that need not themselves be linking systems because their objects mightnot be closed under overgroups. (Compare with Proposition 6 in [O2].) Proposition 2.5.
Let ( L , δ, π ) be a linking system associated to a fusion system F over afinite p -group S , let L ⊆ L be a full subcategory such that Ob( L ) ⊇ F cr , and let Aut( L ) be the group of automorphisms of the category L that send inclusions to inclusions anddistinguished subgroups to distinguished subgroups. Fix α ∈ Aut( L ) , and let β ∈ Aut( S ) be such that α ( δ S ( g )) = δ S ( β ( g )) for all g ∈ S . Then β ∈ Aut( F ) , α ( P ) = β ( P ) for each P ∈ Ob( L ) , and the following diagram of functors T Ob( L ) ( S ) δ / / T ( β ) (cid:15) (cid:15) L π / / α (cid:15) (cid:15) F c β (cid:15) (cid:15) T Ob( L ) ( S ) δ / / L π / / F (2.1) commutes. Let e µ L : Aut( L ) −→ Aut( F ) denote the homomorphism that sends α ∈ Aut( L ) to β as defined above.Proof. Clearly, α ( S ) = S , and hence α S sends δ S ( S ) ∈ Syl p (Aut L ( S )) to itself. Thus β iswell defined. Since α sends inclusions to inclusions, it commutes with restrictions. So for P, Q ∈ Ob( L ) and g ∈ T S ( P, Q ) (the transporter set), we have α ( δ P,Q ( g )) = δ α ( P ) ,α ( Q ) ( β ( g )) (2.2)since α ( δ S ( g )) = δ S ( β ( g )). In particular, the left-hand square in (2.1) commutes.When Q = P , (2.2) says that δ α ( P ) ( β ( P )) = α P ( δ P ( P )), and α P ( δ P ( P )) = δ α ( P ) ( α ( P ))since α sends distinguished subgroups to distinguished subgroups. So α ( P ) = β ( P ) since δ α ( P ) is a monomorphism (Proposition 1.7(a)).Fix P, Q ∈ Ob( L ) and ψ ∈ Mor L ( P, Q ), and set ϕ = π ( ψ ) ∈ Hom F ( P, Q ). For each g ∈ P ,consider the following three squares: P ψ / / δ P ( g ) (cid:15) (cid:15) Q δ Q ( ϕ ( g )) (cid:15) (cid:15) P ψ / / Q α ( P ) α ( ψ ) / / δ α ( P ) ( β ( g )) (cid:15) (cid:15) α ( Q ) δ α ( Q ) ( β ( ϕ ( g ))) (cid:15) (cid:15) α ( P ) α ( ψ ) / / α ( Q ) α ( P ) α ( ψ ) / / δ α ( P ) ( β ( g )) (cid:15) (cid:15) α ( Q ) δ α ( Q ) ( π ( α ( ψ ))( β ( g ))) (cid:15) (cid:15) α ( P ) α ( ψ ) / / α ( Q ) .The first and third of these squares commute by axiom (C) in Definition 1.6, and the secondcommutes since it is the image under α of the first. Since morphisms in L are epimorphismsand δ α ( Q ) is injective (Proposition 1.7(a,b)), this implies β ( ϕ ( g )) = π ( α ( ψ ))( β ( g )). Thus π ( α ( ψ )) = βϕβ − = c β ( π ( ψ )), proving that the right-hand square in (2.1) commutes.In particular, since π is surjective on morphism sets (axiom (A2) in Definition 1.6), βϕβ − ∈ Hom F ( β ( P ) , β ( Q )) for each P, Q ∈ Ob( L ) and each ϕ ∈ Hom F ( P, Q ). SinceOb( L ) includes all subgroups which are F -centric and F -radical, all morphisms in F arecomposites of restrictions of morphisms between objects of L by Theorem 1.3. Hence β F ≤ F with equality since F is a finite category, and so β ∈ Aut( F ). (cid:3) We next define a homomorphism κ G that connects the automorphism group of a group tothat of its linking system. We refer to [AOV, § κ G and the proofthat it is well defined. Definition 2.6.
Let G be a finite group and choose S ∈ Syl p ( G ) . Let κ G : Out( G ) −−−→ Out( L cS ( G )) denote the homomorphism that sends the class of α ∈ Aut( G ) such that α ( S ) = S to theclass of the automorphism of L cS ( G ) induced by α . We are now ready to define tameness.
Definition 2.7.
Let F be a saturated fusion system over a finite p -group S . Then (a) F is tamely realized by a finite group G if F ∼ = F S ∗ ( G ) for some S ∗ ∈ Syl p ( G ) and thehomomorphism κ G : Out( G ) −→ Out( L cS ∗ ( G )) is split surjective; and (b) F is tame if it is tamely realized by some finite group. Products of fusion and linking systems
In this section, we first define centric linking systems associated to products of two or morefusion systems (Lemma 3.5). This is followed by a description of the group of automorphismsof such a product linking systems that leave the factors invariant up to permutation, as wellas conditions on the fusion systems that guarantee that these are the only automorphismsof the linking system (Proposition 3.8(a,b)). As a consequence, we show that a product oftame fusion systems that satisfy these same conditions is always tame (Proposition 3.8(c)).
AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 11
Recall that if F and F are fusion systems over finite p -groups S and S , then F × F isthe fusion system over S × S generated by all morphisms ϕ × ϕ ∈ Hom( P × P , Q × Q )for ϕ i ∈ Hom F i ( P i , Q i ) ( i = 1 , F and F are both saturated, then so is F × F [AKO, Theorem I.6.6].The following notation and hypotheses will be used throughout the section. Hypotheses 3.1.
Let F , . . . , F k be saturated fusion systems over finite p -groups S , . . . , S k (some k ≥ ), and set S = S × · · · × S k and F = F × · · · × F k . For each i , let pr i : S −→ S i be the projection. For each P ≤ S , we write P i = pr i ( P ) (for ≤ i ≤ k ) and b P = P × · · · × P k ≤ S . Thus b P ≥ P for each P . We first check which subgroups are centric in a product of fusion systems.
Lemma 3.2.
Assume Hypotheses 3.1. Then a subgroup P ≤ S is F -centric if and only if P i is F -centric for all i and Z ( P ) × · · · × Z ( P k ) ≤ P .Proof. For each P ≤ S , we have C S ( P ) = C S ( P ) × · · · × C S k ( P k ) = C S ( b P ). Hence • P is fully centralized in F if and only if P i is fully centralized in F i for each i , and • C S ( P ) ≤ P if and only if C S i ( P i ) ≤ P i for each i and Z ( b P ) ≤ P .The result now follows since P is F -centric if and only if P is fully centralized in F and C S ( P ) ≤ P . (cid:3) Note also that under Hypotheses 3.1, if P ≤ S is F -centric and F -radical, then P = b P = P × · · · × P k (see, e.g., [AOV, Lemma 3.1]). But that will not be needed here.The following easy consequence of the Krull-Remak-Schmidt theorem will be needed.Recall that a group is indecomposable if it is not the direct product of two of its propersubgroups. Proposition 3.3.
Assume G , . . . , G k are finite, indecomposable groups, and set G = G ×· · · × G k . Then the following hold for each α ∈ Aut( G ) . (a) There is σ ∈ Σ k such that α ( G i Z ( G )) = G σ ( i ) Z ( G ) for each ≤ i ≤ k . (b) If (cid:0) | Z ( G ) | , | G/ [ G, G ] | (cid:1) = 1 , then there is σ ∈ Σ k such that α ( G i ) = G σ ( i ) for each i .Proof. An automorphism β ∈ Aut( G ) is normal if β commutes with all inner automorphismsof G ; equivalently, if [ β, G ] ≤ Z ( G ). In particular, if (cid:0) | Z ( G ) | , | G/ [ G, G ] | (cid:1) = 1, then the onlynormal automorphism of G is the identity.By the Krull-Remak-Schmidt theorem in the form stated in [Sz1, Theorem 2.4.8] (andapplied with Ω = 1 or Ω = Inn( G )), any two direct product decompositions of G withindecomposable factors have the same number of factors, and there is always a normalautomorphism of G that sends one to the other up to a permutation of the factors . Points(a) and (b) follow immediately from this, applied to the decompositions of G as the productof the G i and of the α ( G i ). (cid:3) One immediate consequence of Proposition 3.3 is the following description of Out( G ) when G is a product of simple groups. Proposition 3.4.
Assume G = G × · · · × G k , where G , . . . , G k are finite indecomposablegroups and (cid:0) | Z ( G ) | , | G/ [ G, G ] | (cid:1) = 1 . Let Γ be the group of all γ ∈ Σ k such that G γ ( i ) ∼ = G i for each i . Then there is an isomorphism Φ G : (cid:0) Aut( G ) × · · · × Aut( G k ) (cid:1) ⋊ Γ ∼ = −−−−−−→ Aut( G ) with the property that Φ G ( β , . . . , β k ) = β × · · · × β k for each k -tuple of automorphisms β i ∈ Aut( G i ) . Also, Φ G sends Q ki =1 Inn( G i ) isomorphically to Inn( G ) , and hence induces anisomorphism Φ G : (cid:0) Out( G ) × · · · × Out( G k ) (cid:1) ⋊ Γ ∼ = −−−−−−→ Out( G ) . To define Φ G more precisely, fix isomorphisms λ ij : G i −→ G j for each ≤ i < j ≤ k suchthat G i ∼ = G j , chosen so that λ iℓ = λ jℓ λ ij whenever G i ∼ = G j ∼ = G ℓ , and set λ ji = λ − ij . Also,set λ ii = Id G i for each i . Then Φ G can be chosen so that for each γ ∈ Γ , Φ G ( γ )( g , . . . , g k ) = (cid:0) λ γ − (1) , ( g γ − (1) ) , . . . , λ γ − ( k ) ,k ( g γ − ( k ) ) (cid:1) . Proof.
Without loss of generality, we can assume that G i = G j and λ ij = Id G i for each i, j such that G i ∼ = G j . Thus Φ G ( γ )( g , . . . , g k ) = ( g γ − (1) , . . . , g γ − ( k ) ) for each γ ∈ Γ. ThenΦ G is clearly an injective homomorphism, and it factors through a homomorphism Φ G asabove since Inn( G ) = Φ G (cid:0)Q ki =1 Inn( G i ) (cid:1) . Each automorphism of G permutes the factors byProposition 3.3(b), and hence Φ G is surjective. (cid:3) In the next proposition, we describe one way to construct linking systems associated toproducts of fusion systems.
Proposition 3.5.
Assume Hypotheses 3.1. For each ≤ i ≤ k , let L i be a centric linkingsystem associated to F i , with structure functors δ i and π i . Let L be the category whose objectsare the F -centric subgroups of S , and where for each P, Q ∈ Ob( L ) , Mor L ( P, Q ) = (cid:8) ( ϕ , . . . , ϕ k ) ∈ Q ki =1 Mor L i ( P i , Q i ) (cid:12)(cid:12) ( π ( ϕ ) , . . . , π k ( ϕ k ))( P ) ≤ Q (cid:9) . Define T Ob( L ) ( S ) δ −−−−−→ L π −−−−−→ F by setting, for all P, Q ∈ F c = Ob( L ) , δ P,Q ( g ) = (cid:0) ( δ ) P ,Q ( g ) , . . . , ( δ k ) P k ,Q k ( g k ) (cid:1) all g = ( g , . . . , g k ) ∈ T S ( P, Q ) π P,Q ( ϕ ) = (cid:0) π ( ϕ ) , . . . , π k ( ϕ k ) (cid:1) all ϕ = ( ϕ , . . . , ϕ k ) ∈ Mor L ( P, Q ). Then the following hold: (a)
The functors δ and π make L into a centric linking system associated to F . (b) Define ξ L : L × · · · × L k −→ L by setting ξ L ( P , . . . , P k ) = P × · · · × P k and ξ L ( ϕ , . . . , ϕ k ) = ( ϕ , . . . , ϕ k ) . Then ξ L is an isomorphism of categories from L ×· · ·×L k to the full subcategory b L ⊆ L whose objects are those P ∈ F c such that P = b P ; equivalently, the products P × · · · × P k for P i ∈ F ci . Also, the following square commutes T Ob( L ) ( S ) × · · · × T Ob( L k ) ( S k ) η ∼ = / / δ ×···× δ k (cid:15) (cid:15) T Ob( b L ) ( S ) δ (cid:15) (cid:15) L × · · · × L k ξ L / / L where η is the natural isomorphism that sends ( P , . . . , P k ) to Q ki =1 P i . (c) Let ρ i : L i −→ L be the functor that sends P i ∈ Ob( L i ) to its product with the S j for j = i , and sends ϕ i ∈ Mor( L i ) to its product with Id S j for j = i . Then ρ i is injectiveon objects and on morphism sets. If α ∈ Aut( L ) is such that α S ( δ S ( S i )) = δ S ( S i ) foreach i , then α ( ρ i ( L i )) = ρ i ( L i ) for each i . AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 13
Proof.
By Lemma 3.2, for each P ∈ Ob( L ) = F c , pr i ( P ) is F i -centric for each i . So thedefinitions of Mor L ( P, Q ) and δ make sense. (a) Axiom (A1) is clear. Fix
P, Q ∈ Ob( L ) and set P i = pr i ( P ) and Q i = pr i ( Q ); then C S ( P ) = Z ( P ) = Z ( P ) × · · · × Z ( P k ). So by axiom (A2) for the L i , for ϕ, ϕ ′ ∈ Mor L ( P, Q ), π ( ϕ ) = π ( ϕ ′ ) if and only if ϕ ′ = ϕ ◦ δ P ( z ) for some z ∈ Z ( P ). For each P, Q ∈ Ob( L ), each ϕ ∈ Hom F ( P, Q ) is the restriction of some morphism Q ki =1 ϕ i ∈ Hom F ( b P , b Q ) (see [AKO,Theorem I.6.6]), and hence the surjectivity of π on morphism sets follows from that of the π i . The rest of (A2) (the effect of ϕ and π on objects) is clear. Likewise, axioms (B) and(C) for L follow immediately from the corresponding axioms for the L i . Thus L is a centriclinking system with structure functors δ and π . (b) Both statements ( ξ L is an isomorphism of categories and the diagram commutes) areimmediate from the definitions and since P × · · · × P k is F -centric if P i is F i -centric for each i (Lemma 3.2). (c) Let α ∈ Aut( L ) be such that α ( δ S ( S i )) = δ S ( S i ) ≤ Aut L ( S ) for each i . We must showthat α ( ρ i ( L i )) = ρ i ( L i ) for each i . Fix some i , let S i ≤ S be the product of the S j for j = i ,and identify S i × S i with S . Thus ρ i ( P i ) = P i × S i and ρ i ( ϕ i ) = ( ϕ i × Id S i ) for P i ∈ Ob( L i )and ϕ i ∈ Mor( L i ).Set β = e µ L ( α ) ∈ Aut( F ), as defined in Proposition 2.5. By that proposition, α ( P ) = β ( P )for P ∈ Ob( L ), and π ◦ α = c β ◦ π as functors from L to F . By assumption, β ( S i ) = S i and β ( S i ) = S i . So for each P i ∈ Ob( L i ), we have α ( ρ i ( P i )) = β ( P i × S i ) = P ∗ i × S i for some P ∗ i ≤ S i . Thus α permutes the objects in ρ i ( L i ).Now fix a morphism ϕ i ∈ Mor L i ( P i , Q i ). Since π ◦ α = c β ◦ π , we have π ( α ( ρ i ( ϕ i ))) = c β ( π ( ϕ i × Id S i )) = c β ( π ( ϕ i ) × Id S i ) = π ( ψ ) × Id S i = π ( ρ i ( ψ ))for some ψ ∈ Mor( L i ). So by axiom (A2) in Definition 1.6, α ( ρ i ( ϕ i )) = ρ i ( ψ ) δ S ( z, z ′ ) forsome z ∈ Z ( P i ) and z ′ ∈ Z ( S i ). Since z ′ has p -power order, this shows that z ′ = 1 and α ( ρ i ( ϕ i )) ∈ Mor( ρ i ( L i )) if ϕ i is an automorphism of order prime to p . Also, α ( ρ i ( δ S i ( g i ))) = α ( δ S ( g i )) = δ S ( β ( g i )) ∈ Mor( ρ i ( L i ))for all g ∈ S i since β ( g i ) ∈ β ( S i ) = S i .By [AOV, Theorem 1.12], each morphism in L i is a composite of restrictions of elementsof Aut L i ( P ) for fully normalized subgroups P ∈ F cri . Also, when P is fully normalized, δ P ( N S ( P )) ∈ Syl p (Aut L i ( P )) (see [AKO, Proposition III.4.2(c)]), and so Aut L i ( P ) is gen-erated by δ P ( N S ( P )) and elements of order prime to P . Thus each morphism in L i is acomposite of restrictions of automorphisms of order prime to p and elements of δ i ( S i ), andso α ( ρ i ( L i )) = ρ i ( L i ). (cid:3) As one example, if G , . . . , G k are finite groups, S i ∈ Syl p ( G i ), and L i = L cS i ( G i ), then itis an easy exercise to show that the linking system L defined in Proposition 3.5 is the centriclinking system of G × · · · × G k .We next recall the definition of reduced fusion systems. Definition 3.6.
Let F be a saturated fusion system over a finite p -group S . (a) Set hyp ( F ) = (cid:8) g − α ( g ) (cid:12)(cid:12) g ∈ P ≤ S, α ∈ O p (Aut F ( P )) (cid:9) (the hyperfocal subgroup of F ). (b) A saturated fusion subsystem
E ≤ F over T ≤ S has p -power index if T ≥ hyp ( F ) , and Aut E ( P ) ≥ O p (Aut F ( P )) for all P ≤ T . The smallest normal subsystem of p -powerindex is denoted O p ( F ) . (c) A saturated fusion subsystem
E ≤ F over T ≤ S has index prime to p if T = S and Aut E ( P ) ≥ O p ′ (Aut F ( P )) for all P ≤ T . The smallest normal subsystem of index primeto p is denoted O p ′ ( F ) . (d) The fusion system F is reduced if O p ( F ) = 1 and O p ( F ) = F = O p ′ ( F ) . For the existence of the minimal subsystems O p ( F ) and O p ′ ( F ), see, e.g., Theorems I.7.4and I.7.7 in [AKO].Recall also that a saturated fusion system is indecomposable if it is not the direct productof two proper fusion subsystems.The subcategory b L ⊆ L defined in Proposition 3.5(b) is not a linking system, since Ob( b L )is not closed under overgroups. However,Ob( b L ) = (cid:8) P = P × · · · × P k (cid:12)(cid:12) P i ≤ S i , P ∈ F c (cid:9) does include all subgroups of S that are F -centric and F -radical: this is shown in [AOV,Lemma 3.1] when k = 2 and follows in the general case by iteration. So Proposition 2.5applies to the automorphism groupAut( b L ) = (cid:8) α ∈ Aut cat ( b L ) (cid:12)(cid:12) α ( δ P,S (1)) = δ α ( P ) ,S (1) , α ( δ P ( P )) = δ α ( P ) ( α ( P )) ∀ P ∈ Ob( b L ) (cid:9) . Lemma 3.7.
Assume Hypotheses 3.1. Let L , . . . , L k be centric linking systems associated to F , . . . , F k , respectively, and let L be the centric linking system associated to F defined as inProposition 3.5. Let b L ⊆ L be the full subcategory with objects the subgroups P ×· · ·× P k ≤ S for P i ∈ F ci = Ob( L i ) , and let ξ L : Q ki =1 L i ∼ = −−→ b L ≤ L be as in Proposition 3.5(b). Then (a) ξ L induces a homomorphism c ξ : Aut( L ) × · · · × Aut( L k ) −−−−−−→ Aut( b L ) that sends ( α , . . . , α k ) to ξ L ( α × · · · × α k ) ξ − L ; (b) each b α ∈ Aut( b L ) has a unique extension E L ( b α ) to an automorphism of L , in this waydefining an injective homomorphism E L : Aut( b L ) −→ Aut( L ) ; and (c) if O p ′ ( F i ) is reduced and indecomposable for each i , then E L is an isomorphism.Proof. (a) This formula clearly defines a homomorphism to the group Aut cat ( b L ) of allautomorphisms of b L as a category. That ξ L ( α × · · · × α k ) ξ − L (for α i ∈ Aut( L i )) sendsinclusions in b L to inclusions and sends distinguished subgroups to distinguished subgroupsfollows from the commutativity of the square in Proposition 3.5(b). (b) We will show that each b α ∈ Aut( b L ) extends to some α ∈ Aut( L ). By the definition inProposition 3.5, each morphism in L is a restriction of a morphism in b L , and hence thereis at most one such extension. So upon setting E L ( b α ) = α , we get a well defined injectivehomomorphism from Aut( b L ) to Aut( L ).Fix b α ∈ Aut( b L ), and let β = e µ L ( b α ) ∈ Aut( F ) be the automorphism of Proposition 2.5.Thus π ◦ b α = c β ◦ π , and b α ( P ) = β ( P ) for all P ∈ Ob( b L ). By the definition in Proposition3.5, for all P, Q ∈ Ob( L ),Mor L ( P, Q ) = (cid:8) ψ ∈ Mor b L ( b P , b Q ) (cid:12)(cid:12) π ( ψ )( P ) ≤ Q (cid:9) (3.1) AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 15 (where b P and b Q are as in Hypotheses 3.1). Also, β ( b P ) = [ β ( P ) for each P ∈ Ob( L ), since b P and [ β ( P ) are the unique minimal objects of b L containing P and β ( P ), and since β permutesthe objects of b L and of L . (We are not assuming here that β permutes the factors S i .)For all P, Q ∈ Ob( L ) and ψ ∈ Mor L ( P, Q ) ⊆ Mor b L ( b P , b Q ), we have π ( b α ( ψ ))( β ( P )) = c β ( π ( ψ ))( β ( P )) = β ( π ( ψ )( P )) ≤ β ( Q ) :the first equality since π ◦ b α = c β ◦ π and the inequality since π ( ψ )( P ) ≤ Q by (3.1). So b α ( ψ ) ∈ Mor L ( β ( P ) , β ( Q )) by (3.1) again. We can thus define α ∈ Aut( L ) extending b α bysetting α ( P ) = β ( P ) for all P , and letting α P,Q be the restriction of b α b P , b Q for P, Q ∈ Ob( L ). (c) Now assume that O p ′ ( F i ) is reduced and indecomposable for each i . By [AOV, Propo-sition 3.4], O p ′ ( F ) = Q ki =1 O p ′ ( F i ) and is reduced, and by [AOV, Proposition 3.6], O p ′ ( F )has a unique factorization as a product of indecomposable fusion systems.Fix α ∈ Aut( L ), and set β = e µ L ( α ) ∈ Aut( F ) (Proposition 2.5). Then c β sends O p ′ ( F )to itself, and by the uniqueness of its factorization, there is γ ∈ Σ k such that c β ( O p ′ ( F i )) = O p ′ ( F γ ( i ) ) for each 1 ≤ i ≤ k . In particular, β ( S i ) = S γ ( i ) for each i . So for each object P = P × · · · × P k in b L , β ( P ) = Q ki =1 β ( P i ) is also an object in b L . Hence α ( b L ) = b L and α = E L ( α | b L ). Since α ∈ Aut( L ) was arbitrary, E L is onto. (cid:3) We are now ready to prove our main results concerning the automorphism group of aproduct of linking systems.
Proposition 3.8.
Assume Hypotheses 3.1. Let L i be a centric linking system associated to F i for each ≤ i ≤ k , and let L be the centric linking system associated to F defined as inProposition 3.5. Set Γ = (cid:8) σ ∈ Σ I (cid:12)(cid:12) F σ ( i ) ∼ = F i for each 1 ≤ i ≤ k (cid:9) . (a) There is an injective homomorphism Φ L : (cid:0) Aut( L ) × · · · × Aut( L k ) (cid:1) ⋊ Γ −−−−−−→ Aut( L ) with the property that for each ( α , . . . , α k ) ∈ Q ki =1 Aut( L i ) , ( P , . . . , P k ) ∈ Q ki =1 Ob( L i ) ,and ( ϕ , . . . , ϕ k ) ∈ Q ki =1 Mor( L i ) , we have Φ L (cid:0) α , . . . , α k (cid:1)(cid:0) P × · · · × P k (cid:1) = α ( P i ) × · · · × α k ( P k )Φ L (cid:0) α , . . . , α k (cid:1)(cid:0) ϕ , . . . , ϕ k (cid:1) = (cid:0) α ( ϕ i ) , . . . , α k ( ϕ k ) (cid:1) (3.2) for α i ∈ Aut( L i ) . Furthermore, Φ L (cid:0)Q ki =1 Aut( L i ) (cid:1) = Aut ( L ) def = (cid:8) α ∈ Aut( L ) (cid:12)(cid:12) α S ( δ S ( S i )) = δ S ( S i ) for each 1 ≤ i ≤ k (cid:9) . (b) If O p ′ ( F i ) is reduced and indecomposable for each i , then Φ L is an isomorphism, andinduces an isomorphism Φ L : (cid:0) Out( L ) × · · · × Out( L k ) (cid:1) ⋊ Γ ∼ = −−−−−−→ Out( L ) . (c) Assume, for each i , that O p ′ ( F i ) is reduced and indecomposable, and F i is tame. Then F is tame. If G , . . . , G k are finite groups such that O p ′ ( G i ) = 1 and F i is tamelyrealized by G i for each i , and such that F i ∼ = F j implies G i ∼ = G j , then F is tamelyrealized by the product G × · · · × G k .Proof. Let b L ⊆ L be the full subcategory defined in Proposition 3.5(b). Thus Ob( b L ) is theset of all P ∈ Ob( L ) = F c such that P = b P . Without loss of generality, for each pair of indices i, j such that F i ∼ = F j , we can assumethat F i = F j and S i = S j . Then L i ∼ = L j by Theorem 1.9, and so we can also assume that L i = L j . (a) Define Φ L to be the compositeΦ L : (cid:0) Aut( L ) × · · · × Aut( L k ) (cid:1) ⋊ Γ b c ξ −−−−−→ Aut( b L ) E L −−−−−→ Aut( L )where E L is the homomorphism of Lemma 3.7(b), where the restriction of b c ξ to Q ki =1 Aut( L i )is the homomorphism c ξ of Lemma 3.7(a), and where b c ξ ( γ )( ϕ , . . . , ϕ k ) = ( ϕ γ − (1) , . . . , ϕ γ − ( k ) ) (3.3)for γ ∈ Γ and ϕ i ∈ Mor( L i ). Then (3.2) holds by the definition of c ξ . One easily checksusing (3.2) and (3.3) that Φ L is an injective homomorphism.It remains to check that Φ L (cid:0)Q ki =1 Aut( L i ) (cid:1) = Aut ( L ): the subgroup of those α ∈ Aut( L )such that α S ( δ S ( S i )) = δ S ( S i ) for each i . The inclusion of the first group in Aut ( L ) is clear.By Proposition 3.5(c), there are embeddings of categories ρ i : L i −→ L sending P i ≤ S i toits product with the S j for all j = i , and α ( ρ i ( L i )) = ρ i ( L i ) for each α ∈ Aut ( L ). We canthus define Ψ L ,i : Aut ( L ) −→ Aut( L i ) by sending α to ρ − i αρ i . Set Ψ L = (Ψ L , , . . . , Ψ L ,k );then Φ L ◦ Ψ L is the inclusion of Aut ( L ) into Aut( L ), and so Aut ( L ) ≤ Im(Φ L ). (b) Fix α ∈ Aut( L ), and set β = e µ L ( α ) ∈ Aut( F ) (see Proposition 2.5). Then β restrictsto an automorphism β of O p ′ ( F ) = O p ′ ( F ) × · · · × O p ′ ( F k ) (see [AOV, Proposition 3.4]). Ifeach O p ′ ( F i ) is reduced and indecomposable, then by [AOV, Proposition 3.6], c β permutesthe factors O p ′ ( F i ).Let γ ∈ Σ k be such that c β ( O p ′ ( F i )) = O p ′ ( F γ ( i ) ) for each i , and hence also β ( S i ) = S γ ( i ) and α S ( δ S ( S i )) = δ S ( S γ ( i ) ). Then c β (Aut F ( S i )) = Aut F ( S γ ( i ) ) for each i , and sinceAut F ( S i ) = Aut F i ( S i ) and F i = h O p ′ ( F i ) , Aut F i ( S i ) i , it follows that c β ( F i ) = F γ ( i ) for each i and hence that γ ∈ Γ. Then Φ L ( γ ) − ◦ α ∈ Aut ( L ), and since Aut ( L ) ≤ Im(Φ L ) by (a),this shows that Aut( L ) ≤ Im(Φ L ) and hence that Φ L is onto.Since Aut L ( S ) = Aut L ( S ) × · · · × Aut L k ( S k ), Φ L induces an isomorphism of quotientgroups Φ L : (cid:0) Out( L ) × · · · × Out( L k ) (cid:1) ⋊ Γ ∼ = −−−−−−→ Out( L ) . (c) Assume now that for each i , O p ′ ( F i ) is reduced and indecomposable, and F i is tame.Let G , . . . , G k be such that O p ′ ( G i ) = 1 and F i is tamely realized by G i for each i , and suchthat F i ∼ = F j implies G i ∼ = G j . Thus Z ( G i ) = O p ( G i ) = 1 for each i by Lemma 1.8(b) andsince O p ′ ( F i ) is reduced (and since normal p -subgroups of F i are also normal in O p ′ ( F i )).Without loss of generality, we can assume that G i = G j whenever G i ∼ = G j and also (by theuniqueness of linking systems again) that F i = F S i ( G i ) and L i = L cS i ( G i ) for each i . Notethat each G i is indecomposable: since O p ′ ( G i ) = 1, a nontrivial factorization of G i wouldinduce nontrivial factorizations of F i and O p ′ ( F i ).Set κ i = κ G i : Out( G i ) −→ Out( L i ) for short. Fix splittings s i : Out( L i ) −→ Out( G i ) forall i , chosen so that s i = s j if G i = G j . AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 17
Consider the following diagram (cid:0)
Out( L ) × · · · × Out( L k ) (cid:1) ⋊ Γ ( s ,...,s k ) ⋊ Id Γ (cid:15) (cid:15) Φ L ∼ = / / Id (cid:28) (cid:28) Out( L ) s (cid:15) (cid:15) (cid:0) Out( G ) × · · · × Out( G k ) (cid:1) ⋊ Γ ( κ ,...,κ k ) ⋊ Id Γ (cid:15) (cid:15) Φ G ∼ = / / Out( G ) κ G (cid:15) (cid:15) (cid:0) Out( L ) × · · · × Out( L k ) (cid:1) ⋊ Γ Φ L ∼ = / / Out( L )where Φ G is the isomorphism of Proposition 3.4 (as defined when taking λ ij = Id G i foreach i < j such that G i = G j ), where s is defined to make the top square commute, andwhere the commutativity of the bottom square is immediate from the definitions. Thus κ G ◦ s = Id Out( L ) , so κ G is split surjective, and F is tamely realized by G . (cid:3) Components of groups and of fusion systems
In this section, we set up some tools that will be used later when proving inductively thatall realizable fusion systems are tame. The starting point for the inductive procedure is thefollowing theorem, which summarizes the main results in [OR]. The statement in point (c)that certain subsystems are exotic depends on the classification of finite simple groups.
Theorem 4.1.
Fix a prime p and a known finite simple group G such that p | | G | . Fix S ∈ Syl p ( G ) , and set F = F S ( G ) . Then either (a) S E F ; or (b) p = 3 and G ∼ = G ( q ) for some q ≡ ± (mod ), in which case | O ( F ) | = 3 , and O ′ ( F ) < F is realized by SL ± ( q ) ; or (c) p ≥ , G is one of the classical groups PSL ± n ( q ) , PSp n ( q ) , Ω n +1 ( q ) , or P Ω ± n +2 ( q ) where n ≥ and q , ± (mod p ), in which case O p ′ ( F ) is simple and exotic; or (d) O p ′ ( F ) is simple, and it is realized by a known finite simple group G ∗ .Moreover, in case (c) , there is a subsystem F E F of index at most in F with the propertythat for each saturated fusion system E over S such that O p ′ ( E ) = O p ′ ( F ) , E is realizable ifand only if it contains F .Proof. See [OR, Theorem A]. A more precise statement of the theorem is given in [OR,Theorem 4.8]. (cid:3)
Some of the results in this section need the hypothesis that a certain fusion or linkingsubsystem be centric , which we now define.
Definition 4.2.
Let E E F be a normal pair of saturated fusion systems over finite p -groups T E S . (a) Let C S ( E ) denote the unique largest subgroup X ≤ C S ( T ) such that C F ( X ) ≥ E . Sucha largest subgroup exists by [A3, 6.7] , or (via a different proof ) by [He, Theorem 1(a)] . (b) The subsystem E is centric in F if C S ( E ) ≤ T ; i.e., if there is no x ∈ C S ( T ) r T suchthat C F ( x ) ≥ E . (c) If M E L are linking systems associated to E E F , then M is centric in L if C Aut L ( T ) ( M ) ≤ Aut M ( T ) ; equivalently, if for each ψ ∈ Aut L ( T ) r Aut M ( T ) , c ψ = Id M . For pairs of linking systems, this is the definition used in [AOV] (Definition 1.27). Theterm “centric fusion subsystem” was not used in [O6], but the condition in Definition 4.2(b)appears in Proposition 2.1 and Theorem 2.3 of that paper (and the term is used in [O6c]).In the next lemma, we look at the relation between normal centric fusion subsystems andnormal centric linking subsystems.
Lemma 4.3.
Let E E F be a normal pair of saturated fusion systems over finite p -groups T E S with associated linking systems M E L , and set C S ( M ) = (cid:8) x ∈ S (cid:12)(cid:12) c δ T ( x ) = Id M (cid:9) E S. Then (a) C Aut L ( T ) ( M ) = δ T ( C S ( M )) and Z ( E ) Z ( F ) ≤ C S ( M ) ≤ C S ( E ) ; (b) M is centric in L if and only if C S ( M ) ≤ T , and this holds if E is centric in F ; (c) if T = S , then C S ( M ) = C S ( E ) and hence M is centric in L if and only if E is centricin F ; and (d) the conjugation action of Aut L ( T ) on C S ( M ) ∼ = δ T ( C S ( M )) E Aut L ( T ) induces an ac-tion of the quotient group L / M = Aut L ( T ) / Aut M ( T ) on C S ( M ) , and C C S ( M ) ( L / M ) = Z ( F ) .If, in addition, p is odd, then (e) C S ( M ) = C S ( E ) and C S ( E ) Z ( T ) = C S ( T ) .Proof. Throughout the proof, “axiom (–)” always refers to the definition of a linking system(Definition 1.6).We first claim that ∀ x ∈ S, δ T ( x ) ∈ Aut M ( T ) ⇐⇒ x ∈ T. (4.1)The implication “ ⇐ =” is clear. To see the converse, fix x ∈ S such that δ T ( x ) ∈ Aut M ( T ).Then c x ∈ Aut E ( T ), and c x ∈ Inn( T ) ∈ Syl p (Aut E ( T )) since it has p -power order. Thus thereis t ∈ T such that xt − ∈ C S ( T ), δ T ( xt − ) = δ T ( x ) δ T ( t ) − ∈ Aut M ( T ), and so xt − ∈ Z ( T )by axiom (A2) applied to M and since δ T is injective (Proposition 1.7(a)). Hence x ∈ T .We next claim that C S ( E ) ∩ T = Z ( E ) . (4.2)The inclusion Z ( E ) ≤ C S ( E ) ∩ T is immediate from the definitions. If x ∈ C S ( E ) ∩ T ≤ Z ( T ),then since E ≤ C F ( x ), each ϕ ∈ Hom E ( h x i , S ) extends to a morphism in F that sends x toitself, and hence ϕ ( x ) = x . Thus x E = { x } , so x ∈ Z ( E ) by [AKO, Lemma I.4.2], finishingthe proof of (4.2). (a) Fix α ∈ C Aut L ( T ) ( M ). By axiom (C) for the linking system L , for all g ∈ T , αδ T ( g ) α − = δ T ( π ( α )( g )), so δ T ( g ) = δ T ( π ( α )( g )) since c α = Id M , and g = π ( α )( g ) since δ T is injective(see Proposition 1.7(a)). So π ( α ) = Id T , and α = δ T ( x ) for some x ∈ C S ( T ) by axiom (A2).Then x ∈ C S ( M ) by definition, and thus C Aut L ( T ) ( M ) = δ T ( C S ( M )).If x ∈ Z ( E ) = C S ( E ) ∩ T , then for each P, Q ≤ T and ψ ∈ Mor M ( P, Q ), π ( ψ ) extendsto some ϕ ∈ Hom E ( P h x i , Q h x i ), and ϕ = π ( b ψ ) for some b ψ ∈ Mor M ( P h x i , Q h x i ). Set ψ ′ = b ψ | P,Q ∈ Mor M ( P, Q ). Then ψ ′ δ P ( x ) = δ Q ( x ) ψ ′ since b ψδ P h x i ( x ) = δ Q h x i ( x ) b ψ by axiom(C). Also, π ( ψ ′ ) = π ( ψ ), so if P is fully centralized in E , then ψ ′ = ψδ P ( y ) for some AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 19 y ∈ C T ( P ). Then [ x, y ] = 1 since x ∈ Z ( T ), so c δ T ( x ) ( δ P ( y )) = δ P ( y ), and thus c δ T ( x ) ( ψ ) = ψ .This holds for all ψ ∈ Mor( M ) whose domain is fully centralized, and hence for all morphismsin M . So x ∈ C T ( M ) ≤ C S ( M ).Thus Z ( E ) ≤ C S ( M ). By a similar argument but working in L and F instead of M and E , we also have Z ( F ) ≤ C S ( L ) ≤ C S ( M ), and so Z ( E ) Z ( F ) ≤ C S ( M ).It remains to show that C S ( M ) ≤ C S ( E ). Fix x ∈ C S ( M ); we must show that E ≤ C F ( x ).Fix P, Q ≤ T and ϕ ∈ Hom E ( P, Q ), and choose ψ ∈ Mor M ( P, Q ) such that π ( ψ ) = ϕ . Set P = P h x i and Q = Q h x i . Then ψδ P ( x ) = δ Q ( x ) ψ since c δ T ( x ) = Id M , and ψδ P ( g ) = δ Q ( ϕ ( g )) ψ for all g ∈ P by axiom (C) applied to M . So for each g ∈ P , there is h ∈ Q such that ψδ P ( g ) = δ Q ( h ) ψ in Mor L ( P, Q ), and by Proposition 1.7(d), ψ extends to aunique morphism ψ ∈ Mor L ( P , Q ). Set ϕ = π ( ψ ) ∈ Hom F ( P , Q ). By axiom (C) again(but applied to L ) we have ψδ P ( x ) = δ Q ( ϕ ( x )) ψ , and after restriction to P and Q thisgives ψδ P ( x ) = δ Q ( ϕ ( x )) ψ . Hence δ Q ( ϕ ( x )) ψ = δ Q ( x ) ψ , so δ Q ( ϕ ( x )) = δ Q ( x ) since ψ isan epimorphism in the categorical sense (see Proposition 1.7(b)), and ϕ ( x ) = x by theinjectivity of δ Q . Thus each morphism in E extends to a morphism in F that sends x toitself, and hence C F ( x ) ≥ E and x ∈ C S ( E ). (b) By (a) and Definition 4.2(c), M is centric in L if and only if δ T ( C S ( M )) ≤ Aut M ( T ),and this holds exactly when C S ( M ) ≤ T by (4.1). Since C S ( M ) ≤ C S ( E ) by (a), this is thecase whenever E is centric in F . (c) If T = S , then C S ( E ) = Z ( E ) = C S ( M ) by (a) and (4.2). So by (b), M is centric in L if and only if E is centric in F . (d) By (a), δ T ( C S ( M )) is the kernel of the homomorphism from Aut L ( T ) to Aut( M )induced by conjugation. So δ T ( C S ( M )) E Aut L ( T ), and Aut L ( T ) acts by conjugation on C S ( M ) ∼ = δ T ( C S ( M )). Since this subgroup centralizes M by definition, Aut M ( T ) actstrivially, and hence the action of Aut L ( T ) factors through an action of the quotient group L / M .By Definitions 1.2(g) and 4.2(a), Z ( F ) = C S ( F ). So by (a) applied when M = L , C Aut L ( S ) ( L ) = δ S ( Z ( F )) . (4.3)Thus for each x ∈ Z ( F ), δ S ( x ) acts trivially on L and hence δ T ( x ) acts trivially on Aut L ( T ).So Z ( F ) ≤ C C S ( M ) (Aut L ( T )) = C C S ( M ) ( L / M ), and it remains to prove the opposite inclu-sion.Fix x ∈ C S ( M ) such that δ T ( x ) ∈ Z (Aut L ( T )). In particular, δ T ( x ) ∈ Z ( δ T ( S )), so x ∈ Z ( S ) by the injectivity of δ T (Proposition 1.7(a)), and c δ T ( x ) ( P ) = x P = P for each P ∈ Ob( L ). We must show that δ T ( x ) acts trivially on L . Fix P, Q ≤ S and ψ ∈ Mor L ( P, Q ),and set P = P ∩ T , Q = Q ∩ T , and ψ = ψ | P,Q (see Proposition 1.7(c)). By the Frattinicondition on a normal linking subsystem (Lemma 1.13), ψ is the composite of the restrictionof a morphism γ ∈ Aut L ( T ) followed by some χ ∈ Mor( M ). Since δ T ( x ) commutes with γ by assumption and commutes with χ by (a) ( Z ( F ) ≤ C S ( M )), we have( ψδ P ( x )) | P ,Q = ψ δ P ( x ) = δ Q ( x ) ψ = ( δ Q ( x ) ψ ) | P ,Q . Then ψδ P ( x ) = δ Q ( x ) ψ by the uniqueness of extensions in a linking system (Proposition1.7(d)), and hence c δ T ( x ) ( ψ ) = ψ .Thus c δ S ( x ) is the identity on all objects and morphisms in L . So x ∈ Z ( F ) by (4.3) andthe injectivity of δ S (Proposition 1.7(a)). (e) Assume now that p is odd. Then the natural homomorphism µ M : Out( M ) −→ Out( E )induced by e µ M (see Proposition 2.5 and [AKO, § III.4.3] or [AOV, § x ∈ C S ( T ), and let ξ = [ c δ T ( x ) ] ∈ Out( M ) be the class of c δ T ( x ) ∈ Aut( M ). Then µ M ( ξ ) = 1 since δ T ( x ) centralizes T , and ξ = 1 since µ M is an isomorphism. Thus c δ T ( x ) = c γ for some γ ∈ Aut M ( T ). Also, γ has p -power order, so γ ∈ δ T ( T ). Let t ∈ T be suchthat γ = δ T ( t ); then t ∈ Z ( T ) since c γ is the identity on T , and c δ T ( xt − ) = Id M . So xt − ∈ C S ( M ), and x ∈ C S ( M ) Z ( T ). Since x ∈ C S ( T ) was arbitrary, this finishes the proofthat C S ( T ) = C S ( M ) Z ( T ).Since C S ( M ) ≤ C S ( E ) ≤ C S ( T ), this proves that C S ( M ) Z ( T ) = C S ( E ) Z ( T ). Also, C S ( M ) ∩ Z ( T ) = C S ( E ) ∩ Z ( T ) by (a) and (4.2), so C S ( M ) = C S ( E ). (cid:3) Let Comp( G ) denote the set of components of a finite group G ; i.e., the set of subnormalsubgroups of G that are quasisimple. (Recall that a subgroup H of G is subnormal, denoted H E E G , if there is a sequence H = H E H E · · · E H k = G with each subgroup normalin the following one, and H is quasisimple if H is perfect and H/Z ( H ) is simple.) Thecomponents of G commute with each other pairwise (see [A1, §
31] or [AKO, Lemma A.12]).In particular, when O q ( G ) = 1 for all primes q , they are all simple groups, and the subgroup E ( G ) def = h Comp( G ) i is their direct product.Similarly, the components of a saturated fusion system F over a finite p -group S are itssubnormal fusion subsystems C E E F that are quasisimple (i.e., O p ( C ) = C and C /Z ( C ) issimple). The set of components of F will be denoted Comp( F ), and E ( F ) = h Comp( F ) i .When O p ( F ) = 1, the subsystem E ( F ) is the direct product of the components, all of whichare simple. See [A3, Chapter 6] for more details.We need to understand the role played by the components of G and of F S ( G ) whendetermining automorphisms of the linking system L cS ( G ) and tameness. The next propositionis a first step towards doing that. Proposition 4.4.
Let G be a finite group such that O p ( G ) = O p ′ ( G ) = 1 , let L , . . . , L k beits components, and set L = L × · · · × L k E G . Fix S ∈ Syl p ( G ) , set F = F S ( G ) , andassume that O p ( F ) = 1 . For each i , set T i = S ∩ L i ∈ Syl p ( L i ) and E i = O p ′ ( F T i ( L i )) , andset T = T × · · · × T k = S ∩ L ∈ Syl p ( L ) and E = O p ′ ( F T ( L )) . Then (a) Comp( F ) = {E , . . . , E k } , the E i are all simple, and E = E × · · · × E k ; (b) there is a unique minimal normal fusion subsystem E ∗ E F containing E that is realiz-able, and E = O p ′ ( E ∗ ) ; (c) E ∗ = E ∗ × · · · × E ∗ k , where for each ≤ i ≤ k , we have E ∗ i ≥ E , E i = O p ′ ( E ∗ i ) , and E ∗ i = F T i ( H i ) for some finite simple group H i with T i ∈ Syl p ( H i ) ; and (d) E ∗ and E are both centric and characteristic in F .Proof. (a) For each 1 ≤ i ≤ k , we have E i = O p ′ ( F T i ( L i )) E F T i ( L i ) E F T ( L ) E F S ( G ) = F :the first normality relation by [AKO, Theorem I.7.7] and the other two by [AKO, PropositionI.6.2] and since L i E L E G . Thus E i E E F for each i . Also, since L = L × · · · × L k E G and T = T × · · · × T k , we have E = O p ′ ( F T ( L )) = O p ′ (cid:0) F T ( L ) × · · · × F T k ( L k ) (cid:1) = O p ′ ( F T ( L )) × · · · × O p ′ ( F T k ( L k )) = E × · · · × E k , AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 21 where the second equality holds by [AOV, Proposition 3.4], and so E is the direct productof the E i .Since O p ( G ) = O p ′ ( G ) = 1, the components L i of G are simple, and all have ordera multiple of p . Since O p ( F ) = 1, we have O p ( F T ( L )) = 1 by Lemma 2.3(b), and so O p ( F T i ( L i )) = 1 for each i . Hence E i = O p ′ ( F T i ( L i )) is simple for each i by Theorem 4.1.We just saw that each E i is subnormal in F , and so it is a component of F .Assume D E E F is another component. By [A3, 9.8.2] and since O p ( F ) = 1, F containsthe direct product of D and the E i . In particular, D ≤ C F ( T ), where C F ( T ) is the fusionsystem of C G ( T ) by Lemma 1.8. (Note that T is fully centralized since it is normal in S .)Conjugation by each element of G permutes the subgroups L i , and since L i ∩ T = T i =1 for each i , each element of C G ( T ) normalizes each of the L i . So conjugation induceshomomorphisms C G ( T ) c −−−−−−→ Aut( L ) × · · · × Aut( L k ) c −−−−−−→ Out( L ) × · · · × Out( L k )where Ker( c ) = C G ( L ) and Ker( c c ) = C L ( T ) C G ( L ). Also, C G ( L ) = Z ( L ) is abeliansince L = E ( G ) and O q ( G ) = 1 for all primes q (see, e.g., [AKO, Theorem A.13(c)]), and C L ( T ) ≤ N L ( T ) is p -solvable since T ∈ Syl p ( L ). Finally, Out( L i ) is solvable for each i bythe Schreier conjecture (see [GLS3, Theorem 7.1.1]), and so C G ( T ) is p -solvable.Thus C F ( T ) = F C S ( T ) ( C G ( T )) is solvable in the sense of [AKO, Definition II.12.1], andits saturated fusion subsystems are all solvable by [AKO, Lemma II.12.8]. So D is solvable,and hence constrained by [AKO, Lemma II.12.5(b)] (see also [AKO, Definition I.4.8]). Butthen O p ( D ) = 1, and since D E E F and O p ( F ) = 1, this contradicts Lemma 2.3(b). Weconclude that E , . . . , E k are the only components of F . (b,c) By points (c) and (d) in Theorem 4.1, for each 1 ≤ i ≤ k , there is a fusion subsystem E ∗ i E F T i ( L i ) over T i , realized by a finite group H i with T i ∈ Syl p ( H i ), which is the uniqueminimal realizable fusion subsystem of F T i ( L i ) containing E i . Assume each H i is chosen sothat | H i | is minimal among orders of groups realizing E ∗ i . Then O p ( H i ) = O p ′ ( H i ) = 1, andso the components of H i are simple. By (a), applied with H i in the role of G , the fusionsystem of each component of H i contains a component of E ∗ i E E F and hence a componentof F . Since E i is the only component of F contained in E ∗ i , H i has only one componentand that must be O p ′ ( H i ). But F T i ( O p ′ ( H i )) = F T i ( H i ) by the minimality assumption on E ∗ i = F T i ( H i ), and so H i = O p ′ ( H i ) and is simple by the minimality of | H i | .Set E ∗ = E ∗ × · · · × E ∗ k ≤ F . Thus E ∗ = F T ( H ) where H = H × · · · × H k , and E i = O p ′ ( F T i ( H i )) since E i = O p ′ ( F T i ( L i )) and E i ≤ F T i ( H i ) ≤ F T i ( L i ). Also, E = O p ′ ( E ∗ )and E ∗ E F T ( L ). Each α ∈ Aut F ( T ) permutes the factors T i in T and F T i ( L i ) in F T ( L ),and hence permutes the factors E ∗ i in E ∗ . Thus E ∗ E F T ( L ) E F and c α ( E ∗ ) = E ∗ for each α ∈ Aut F ( T ), so E ∗ E F by [A3, 7.4]. It remains to show that each normal realizable fusionsubsystem of F containing E also contains E ∗ .Assume that K is a finite group with subgroup U ∈ Syl p ( K ) such that T ≤ U E S and E ≤ F U ( K ) E F . As usual, we can assume that O p ′ ( K ) = 1, and O p ( K ) = 1 by Lemma1.8(b) and since O p ( F U ( K )) = 1. Let K . . . , K m E E K be the components of K , and set U i = U ∩ K i . The K i are simple since O p ( K ) = O p ′ ( K ) = 1, and by (a), the subsystems O p ′ ( F U i ( K i )) for 1 ≤ i ≤ m are the components of F U ( K ). Each component of F U ( K )is subnormal in F (hence a component) since F U ( K ) E F , and each component of F iscontained in and hence a component of F U ( K ) by assumption. Thus m = k , and we canassume that the indices are chosen so that U i = T i and O p ′ ( F T i ( K i )) = E i for each i . For each i , F T i ( K i ) and F T i ( H i ) are both realizable fusion systems over T i where O p ′ ( F T i ( K i )) = O p ′ ( F T i ( H i )), and so F T i ( K i ) ≥ F T i ( H i ) by Theorem 4.1 (the last statement) and the mini-mality assumption. So F U ( K ) contains E ∗ = F T ( H ), and thus E ∗ is minimal among normalrealizable subsystems. (d) To see that E ∗ is centric in F , we must show that C S ( E ∗ ) ≤ T , where C S ( E ∗ ) ≤ C S ( T ) isthe unique largest subgroup X ≤ C S ( T ) such that C F ( X ) ≥ E ∗ (see Definition 4.2(a,b)). By[A3, Theorem 4] (see also [He, Theorem 2]), there is a normal fusion subsystem C F ( E ∗ ) E F over C S ( E ∗ ). Also, C S ( E ∗ ) ∩ T = Z ( E ∗ ) ≤ O p ( E ∗ ) = 1, and O p ( C F ( E ∗ )) ≤ O p ( F ) = 1 byLemma 2.3(b). So if C S ( E ∗ ) = 1, then each minimal subnormal fusion subsystem of C F ( E ∗ )is simple, and hence a component of F not contained in E . Since this contradicts (a), weconclude that C S ( E ∗ ) = 1 and hence that E ∗ is centric in F . By a similar argument, E isalso centric in F .Finally, E is characteristic in F since it is the product of the components of F , and so E ∗ is characteristic in F by the uniqueness property in (b) and since E ∗ E F . (cid:3) Tameness of realizable fusion systems
We are now ready to show that all realizable fusion systems are tame. This has alreadybeen shown in earlier papers for fusion systems of simple groups (see Proposition 5.2). When F is the fusion system of an arbitrary finite group G , we will show that it is tame via a seriesof reductions based on an examination of the components of G . Throughout this section,we assume the classification of finite simple groups.We first restrict attention to tameness of fusion systems of finite simple groups. This wasshown in most cases in earlier papers, and will be summarized below, but there were twocases whose proofs assumed earlier results that were in error: Lemma 5.1.
Let ( G, p ) be one of the pairs (He , or ( Co , , choose S ∈ Syl p ( G ) , and set F = F S ( G ) and L = L cS ( G ) . Then Out( L ) = 1 , and so F is tamely realized by G .Proof. Since p is odd, Out( F ) ∼ = Out( L ) in both cases. The simplest proof of this is givenin [O1, Theorem C] (and the sporadic groups are handled in Proposition 4.4 of that paper).A more general result is shown in [O3, Theorem C] and [GLn, Theorem 1.1].When G = He and p = 3, the argument in [O7, p. 139] claimed (wrongly) that F issimple, but did not actually use this. Since S is extraspecial of order 27 and exponent 3 andOut G ( S ) ∼ = D , we have D ∼ = Out G ( S ) ≤ Aut( F ) / Inn( S ) ≤ N Out( S ) (Out G ( S )) ∼ = SD . Elements in N Aut( S ) (Aut G ( S )) r Aut G ( S ) exchange subgroups of S of order 9 with non-isomorphic automizers, and hence do not normalize F . So Aut( F ) = Aut G ( S ), and Out( F ) =1. When G = Co and p = 5, the proof that Out( F ) = 1 in [O7, p. 138] used the incorrectclaim that F has a normal subsystem of index 2. So we replace that argument with thefollowing one. By [Cu, Theorem 5.1] and the correction in [W, p. 145], S contains aunique abelian subgroup Q ∼ = E of index 5, and N G ( Q ) /Q ∼ = Aut G ( Q ) ∼ = C × Σ . Set H = N G ( Q ). By [O7, Lemma 1.2(b)] and since C H ( Q ) = Q , we have | Out( F ) | ≤ | Out( H ) | .By [OV, Lemma 1.2], there is an exact sequence0 −−−→ H ( H/Q ; Q ) −−−−−→ Out( H ) −−−−−→ N Out( Q ) (Out H ( Q )) / Out H ( Q ) , AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 23 and by [Be2, p. 110], there is a 5-term exact sequence for the homology of
H/Q as anextension of C by Σ that begins with0 −−−→ H (Σ ; H ( C ; Q )) −−−−−→ H ( H/Q ; Q ) −−−−−→ H (Σ ; H (Σ ; H ( C ; Q )) . Since H ( C ; Q ) = H ( C ; Q ) = 0 ( C acts on Q ∼ = C × C × C via multiplication byscalars), this proves that H ( H/Q ; Q ) = 0. Also, N Out( Q ) (Out H ( Q )) / Out H ( Q ) ∼ = N GL (5) ( C × Σ ) / ( C × Σ )is trivial since GL (5) ∼ = C × PSL (5) and GO (5) ∼ = Σ is a maximal subgroup of PSL (5)(see, e.g., [GLS3, Theorem 6.5.3]). So Out( F ) = Out( H ) = 1. (cid:3) We now summarize what we need to know here about tameness of fusion systems of finitesimple groups.
Proposition 5.2.
Fix a known simple group G , choose S ∈ Syl p ( G ) , and assume that S F S ( G ) . Then F S ( G ) is tamely realized by some known simple group G ∗ .Proof. Set F = F S ( G ) and L = L cS ( G ) for short. Note that G is nonabelian since S F S ( G ).Assume first that G ∼ = A n for some n ≥
5. By [AOV, Proposition 4.8], if p = 2 and n ≥ p is odd and p ≤ n ≡ , p ), then κ G is an isomorphism. If p is odd and p < n ≡ k (mod p ) where 2 ≤ k ≤ p −
1, then F is still tamely realized by A n : Out( L ) = 1since F is isomorphic to the fusion system of Σ n and also that of Σ n − k . If p = 2 and n = 6 , F is tamely realized by A ∼ = PSL (9) (and κ A is an isomorphism). In all other cases, S is abelian and hence S E F .If G is of Lie type in defining characteristic p , or if p = 2 and G ∼ = F (2) ′ , then by [BMO2,Theorems A and D], κ G is an isomorphism except when p = 2 and G ∼ = SL (2). In thisexceptional case, F is tamely realized by A again.If G is of Lie type in defining characteristic q for some prime q = p , then by [BMO2,Theorem B], F is tamely realized by some other simple group G ∗ of Lie type. See also Tables0.1–0.3 in [BMO2] for a list of which groups of Lie type do tamely realize their fusion system,and when they do not, which other groups they can be replaced by.If G is a sporadic simple group (and S F ), then by [O7, Theorem A] and Lemma5.1, κ G is an isomorphism except when ( G, p ) is one of the pairs ( M ,
2) or (He , G, p ) = (He , | Out( G ) | = 2 and Out( L ) = 1, so F is stilltamely realized by G . If ( G, p ) = ( M , F is the unique simple fusion system over SD , and is tamely realized by G ∗ = PSU (5) (and κ G ∗ is an isomorphism) by [AOV,Proposition 4.4]. (cid:3) The statements in the next proposition are very similar to results proven in [AOV], butexcept for part (a) are not stated there explicitly. Their proof consists mostly of repeatingthose arguments. Since many of the results referred to in [AOV, §
2] require consideringlinking systems that are not centric, they depend in a crucial way on [AOV, Lemma 1.17].This lemma states that Out( L ) ∼ = Out( L ) whenever L ≤ L are linking systems associatedto the same fusion system F and Ob( L ) ⊆ Ob( L ) are both Aut( F )-invariant. Proposition 5.3.
Let F be a saturated fusion system over a finite p -group S . (a) If F /Z ( F ) is tame, then F is tame. (b) Assume F E F is a characteristic subsystem over S E S , with the property that F cr ⊆ F c . If F is tame, then F is also tame. (c) If F E F is a characteristic subsystem of index prime to p , and F is tame, then F istame. (d) If F E F is a characteristic subsystem of p -power index, F is tame, and Z ( F ) = 1 ,then F is tame.Proof. (a) See [AOV, Proposition 2.18]. (b)
Set H = S ( S ) ∩ F c , and let H be the set of P ≤ S such that P ∩ S ∈ H . For each P ∈ H , P ∩ S ∈ F c by assumption, and hence P ∈ F c . So by [AOV, Lemma 1.30], there isa normal pair of linking systems L E L associated to F E F with object sets H and H .By Lemma 4.3(a), for α ∈ Aut L ( S ), c α = Id L only if α ∈ δ S ( C S ( F )) ≤ δ S ( C S ( S )).Since S ∈ ( F ) cr ⊆ F c , we have C S ( S ) ≤ S , and hence α ∈ Aut L ( S ). Thus L is centricin L (Definition 4.2(c)).If in addition, F is tamely realized by the finite group G , then by the uniqueness oflinking systems, L ∼ = L H S ( G ). By construction, the sets of objects H and H are invariantunder the actions of Aut( F ) and Aut( F ), respectively, and L is Aut( L )-invariant. Allhypotheses in [AOV, Proposition 2.16] are thus satisfied, and so F is tame. (c) This is a special case of (b). (d)
Assume F has p -power index in F . By [5a2, Proposition 3.8(b)], a subgroup P ≤ S is F -quasicentric if and only if it is F -quasicentric. Let H be the set of all P ≤ S suchthat P ∩ S is F -quasicentric; then H ⊆ F q since overgroups of quasicentric subgroupsare quasicentric. By Theorem 1.9, there is a unique linking system L associated to F withOb( L ) = H , and by [5a2, Theorem 4.4], there is a unique linking system L ≤ L associatedto F with Ob( L ) = ( F ) q . Then L E L : the condition on objects (Definition 1.12(a))holds by construction, and the invariance condition (1.12(b)) holds by the uniqueness of L .By Lemma 4.3(d), there is an action of L / L on C S ( L ) such that C C S ( F ) ( L / L ) = Z ( F ),where Z ( F ) = 1 by assumption. Also, L / L = Aut L ( S ) / Aut L ( S ) is a p -group since F has p -power index in F , so C S ( L ) = 1. Hence C Aut L ( S ) ( L ) = 1 by Lemma 4.3(a), and so L is centric in L .If in addition, F is characteristic in F , then each α ∈ Aut( L ) induces an element β = e µ L ( α ) ∈ Aut( F ) (Proposition 2.5), and c β ( F ) = F by assumption. Hence α ( L ) = L bythe uniqueness of the linking systems in [5a2, Theorems 4.4 & 5.5]. Also, by construction,Ob( L ) and Ob( L ) are invariant under the actions of Aut( F ) and Aut( F ).Assume F is tamely realized by G , and set Z = Z ( F ). Thus Z ≥ Z ( G ) by Lemma1.8(b). By Theorem 1.9 (the uniqueness of centric linking systems), L ∼ = L cS ( G ). If Z >Z ( G ), set H = C G ( Z ): then H still realizes F and Z ( H ) = Z . Each γ ∈ N Aut( G ) ( S )acts on F and hence on Z ( F ), and thus restricts to an automorphism of H . For each g ∈ N G ( S ), c g ∈ Aut G ( S ) = Aut F ( S ) centralizes Z = Z ( F ), and so g ∈ H . The splitsurjective homomorphism κ G : Out( G ) −→ Out( L ) thus factors through Out( H ), and κ H is also split surjective. Upon replacing G by H , we can arrange that Z ( G ) = Z ( F ).We have now shown that the hypotheses of [AOV, Proposition 2.16] all hold. So F istamely realized by some G such that G E G and G/G ∼ = L / L . (cid:3) We are now ready to prove Theorem A.
Theorem 5.4.
Assume the classification of finite simple groups. Then for each prime p ,every realizable fusion system over a finite p -group is tame. AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 25
Proof.
Let
N T be the class of all finite groups G such that F S ( G ) is not tame for S ∈ Syl p ( G ). We must show that N T = ∅ . Assume otherwise, and fix G ∈ N T such that | G | is minimal. In particular, O p ′ ( G ) = 1. Choose S ∈ Syl p ( G ), and set F = F S ( G ).We show in Step 1 that O p ( G ) = 1 = O p ( F ), and in Step 2 that each component of G isnormal in G . Then, in Step 3, we show that such a G cannot exist. Step 1:
Assume O p ( F ) = 1, and set Q = O p ( F ) for short. Then F = N F ( Q ) is the fusionsystem of N G ( Q ) by Lemma 1.8(a), so G = N G ( Q ) (i.e., Q E G ) by the minimality of | G | .Set G = QC G ( Q ), S = QC S ( Q ), and F = F S ( G ). Then Z ( F ) ≥ Z ( G ) ≥ Z ( Q ) = 1(see Lemma 1.8(b)), so F /Z ( F ) is the fusion system of G /Z ( F ) and hence tame by theinduction assumption. So F is tame by Proposition 5.3(a).Now, G = QC G ( Q ) is normal in G = N G ( Q ) since Q and C G ( Q ) are both normal, so F E F by [AKO, Proposition I.6.2]. By [AKO, Proposition I.5.4], applied with Inn( Q ) inthe role of K , and since S = N Inn( Q ) S ( Q ) ∈ Syl p ( G ), we have F = N Inn( Q ) F ( Q ): the fusionsystem over S where for P , P ≤ S and ϕ ∈ Hom F ( P , P ), ϕ ∈ Mor( N Inn( Q ) F ( Q )) if andonly if it extends to some ϕ ∈ Hom F ( P Q, P Q ) such that ϕ ( Q ) = Q and ϕ | Q ∈ Inn( Q ).From this description, it is clear that c α ( F ) = F (see Definition 2.1(c)) for each α ∈ Aut( F ),and hence that F is characteristic in F .For each P ∈ ( F ) cr , P ≥ Q by [AKO, Proposition I.4.5] and since Q E F , and so C S ( P ) ≤ C S ( Q ) ≤ S and hence C S ( P ) = C S ( P ) ≤ P . If R ∈ P F , then R ∈ ( F ) cr since Q E F , so C S ( R ) ≤ R by a similar argument, and we conclude that P ∈ F c . So ( F ) cr ⊆ F c ,and by Proposition 5.3(b), F is tame since F is tame. Since this contradicts our assumptionthat G ∈ N T , we conclude that O p ( F ) = 1, and hence that O p ( G ) = 1 by Lemma 1.8(b). Step 2:
Let L , . . . , L k be the components of G , and set G = T ki =1 N G ( L i ) E G . Set S = S ∩ G ∈ Syl p ( G ) and F = F S ( G ) E F . For each 1 ≤ i ≤ k , set T i = S ∩ L i ∈ Syl p ( L i ) and Z i = Z ( S ) ∩ L i = C T i ( S ).Since O p ( G ) = 1 = O p ′ ( G ) by Step 1, each L i is a simple group with p | | L i | . So T i = 1,and (since T i E S ) Z i = C T i ( S ) = 1. For each P ∈ F c and each Q ∈ P F , Q ∈ F c since F E F , so Q ≥ Z ( S ), and hence Q ≥ Z i for each i . Thus for each x ∈ C S ( Q ), x centralizeseach Z i and hence normalizes each L i (recall that each element of G permutes the L i ). So C S ( Q ) = C S ( Q ) ≤ Q . This proves that P ∈ F c , and hence that F c ⊆ F c .Assume α ∈ Aut( F ) ≤ Aut( S ). Since Comp( F ) is the set of all O p ′ ( F T i ( L i )) by Propo-sition 4.4(a), there is τ ∈ Σ k such that α ( T i ) = T τ ( i ) for each i . Also, α ( S ) = S since S = T ki =1 N S ( T i ), so α ( Z ( S )) = Z ( S ), and α ( Z i ) = Z τ ( i ) for each i .Fix P, Q ∈ F c and ϕ ∈ Hom F ( P, Q ), and let g ∈ G and h ∈ G be such that ϕ = c g | P and c α ( ϕ ) = c h | α ( P ) . Thus P, α ( P ) ≥ Z ( S ), and g Z i ≤ S ∩ L i = T i for each i since g ∈ G .So for each i , h Z i = αϕα − ( Z i ) = α ( g ( Z τ − ( i ) )) ≤ α ( T τ − ( i ) ) = T i , Since c h permutes the L i , this shows that it normalizes each of them, and hence that h ∈ G and c α ( ϕ ) ∈ Mor( F ). Thus c α (Mor( F c )) ⊆ Mor( F ), and c α ( F ) ≤ F by Theorem 1.3(Alperin’s fusion theorem).Thus F is characteristic in F and F c ⊆ F c . So F is not tame by Proposition 5.3(b) andsince F is not tame. Hence G ∈ N T , so G = G by the minimality assumption, and allcomponents of G are normal in G . Step 3:
Again let L , . . . , L k be the components of G , and set T i = S ∩ L i ∈ Syl p ( L i ).By Step 2, L i E G and T i E S for each i , and by Step 1, O p ( F ) = O p ( G ) = O p ′ ( G ) = 1. Set L = L · · · L k . Thus the L i are simple, and L = E ( G ) is the direct product of the L i . Set E i = O p ′ ( F T i ( L i )) where T i ∈ Syl p ( L i ); T = T × · · · × T k ∈ Syl p ( L ); and E = E × · · · × E k = O p ′ ( F T ( L )) . (5.1)Let E ∗ E F T ( L ) be the unique minimal realizable normal subsystem containing E ofProposition 4.4(b,c). By the same proposition, E ∗ = E ∗ × · · · × E ∗ k , where each factor E ∗ i isrealized by a finite simple group. Hence each E ∗ i is tame by Proposition 5.2, and so E ∗ isalso tame by Proposition 3.8(c).By Proposition 4.4(d), E ∗ is centric and characteristic in F . Since G = LN G ( T ) andhence G/L ∼ = N G ( T ) /N L ( T ) by the Frattini argument, the group Aut F ( T ) / Aut L ( T ) =Aut G ( T ) / Aut L ( T ) is a quotient group of G/L ∼ = Out G ( L ), and this is a subgroup of Q ki =1 Out( L i ). Since Out( L i ) is solvable for each i (see, e.g., [GLS3, Theorem 7.1.1]), thegroup Aut F ( T ) / Aut L ( T ) is solvable. Also, Aut L ( T ) / Aut E ∗ ( T ) has order prime to p since O p ′ ( F T ( L )) ≤ E ∗ ≤ F T ( L ), and so Aut F ( T ) / Aut E ∗ ( T ) is p -solvable.The hypotheses of [O6c, Theorem 5(b)] thus hold for the pair E ∗ E F . By that theorem,there is a sequence E ∗ = F E F E · · · E F m = F of saturated fusion subsystems, for some m ≥
0, such that(i) for each 0 ≤ i < m , F i is normal of p -power index or index prime to p in F i +1 and E ∗ E F i E F ; and(ii) for each 1 ≤ j ≤ m and each α ∈ Aut( F j ) with c α ( E ∗ ) = E ∗ , we have c α ( F i ) = F i forall 0 ≤ i < j .Recall that Comp( F ) = {E , . . . , E k } by Proposition 4.4(a). For each 0 ≤ i ≤ m − F i ) ⊆ Comp( F ) since F i E E F , and the opposite inclusion holds since E j E E E F ∗ E F i for each j . Hence E is characteristic in F i . For each α ∈ Aut( F i ), c α ( E ) = E since E is characteristic, hence c α ( E ∗ ) = E ∗ since E ∗ is the unique minimal realizable subsystemcontaining E , and hence c α ( E j ) = E j for all 0 < j < i by condition (ii) above.In particular, this shows that F i is characteristic in F i +1 for each i . Also, Z ( F i ) ≤ O p ( F i ) = 1 for each i by Lemma 2.3(b) and since O p ( F ) = 1 and F i E E F . So byProposition 5.3(c,d), and since F i has index prime to p or p -power index in F i +1 and Z ( F i ) =1, F i +1 is tame if F i is. We already showed that E ∗ = F is tame. Hence F = F m is alsotame, thus contradicting our assumption that G ∈ N T . (cid:3) References [AOV] K. Andersen, B. Oliver, & J. Ventura, Reduced, tame, and exotic fusion systems, Proc. LondonMath. Soc. 105 (2012), 87–152[A1] M. Aschbacher, Finite group theory, Cambridge Univ. Press (1986)[A3] M. Aschbacher, The generalized Fitting subsystem of a fusion system, Memoirs Amer. Math. Soc.209 (2011), nr. 986[AKO] M. Aschbacher, R. Kessar, & B. Oliver, Fusion systems in algebra and topology, Cambridge Univ.Press (2011)[AO] M. Aschbacher & B. Oliver, Fusion systems, Bull. Amer. Math. Soc. 53 (2016), 555–615[Be2] D. Benson, Representations and cohomology II: cohomology of groups and modules, Cambridge Univ.Press (1991)
AMENESS OF FUSION SYSTEMS OF FINITE GROUPS 27 [Be3] D. Benson, Cohomology of sporadic groups, finite loop spaces, and the Dickson invariants, Geometryand cohomology in group theory,
London Math. Soc. Lecture notes ser. , Cambridge Univ. Press(1998), 10–23.[5a2] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Extensions of p -local finite groups, Trans.Amer. Math. Soc. 359 (2007), 3791–3858[BLO1] C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of p -completed classifying spaces of finitegroups, Invent. math. 151 (2003), 611–664[BLO2] C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16(2003), 779–856[BMO2] C. Broto, J. Møller, & B. Oliver, Automorphisms of fusion systems of finite simple groups of Lietype, Memoirs Amer. Math. Soc. 1267 (2020), 1–117[Ch] A. Chermak, Fusion systems and localities, Acta Math. 211 (2013), 47–139[Cr] D. Craven, The theory of fusion systems, Cambridge Univ. Press (2011)[Cu] R. Curtis, On subgroups of · Co , J. Algebra 85 (1983), 144–165 Departament de Matem`atiques, Edifici Cc, Universitat Aut`onoma de Barcelona, 08193Cerdanyola del Vall`es (Barcelona), Spain.Centre de Recerca Matem`atica, Edifici Cc, Campus de Bellaterra, 08193 Cerdanyoladel Vall`es (Barcelona), Spain.
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Email address : [email protected] Departament de Matem`atiques, Edifici Cc, Universitat Aut`onoma de Barcelona, 08193Cerdanyola del Vall`es (Barcelona), Spain.Centre de Recerca Matem`atica, Edifici Cc, Campus de Bellaterra, 08193 Cerdanyoladel Vall`es (Barcelona), Spain.
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