aa r X i v : . [ m a t h . G R ] F e b CORRECTION TO: REDUCTIONS TO SIMPLE FUSION SYSTEMS
BOB OLIVER
Abstract.
We fill in a gap in the proof of the main theorem in our earlier paper [Ol].At the same time, we prove a slightly stronger version of the theorem needed for anotherpaper.
The main theorem in our earlier paper [Ol] stated (very roughly) that if
E ≤ F are satu-rated fusion systems such that E is normal in F and satisfies certain additional conditions,then there is a sequence of saturated fusion subsystems E = F E F E · · · E F m = F ,each normal in the following system and normal in F , such that F i has p -power index orindex prime to p in F i +1 for each i . We refer to [Ol, Theorem 2.3], or to Theorems 5 and6 below, for the precise statement.The theorem was proven by an inductive argument, where we assume that F i +1 hasalready been constructed with certain properties before constructing F i . This inductiveargument requires that E be normal in F i for each i , a property that was not justified in[Ol]. The missing details are not hard to fill in, but we think it’s best to do so formally,especially since the theorem has been applied by various people, either directly as in [HL],or indirectly via Lemma 2.22 in [AO].Most of the notation and terminology used in [Ol] will be assumed here; we refer tothat paper for their definitions. As one exception, since the details of the definition ofnormal fusion subsystems play an important role here, we begin by recalling them. Definition 1.
Let
E ≤ F be saturated fusion systems over finite p -groups T ≤ S . Thesubsystem E is weakly normal in F if • T is strongly closed in F (in particular, T E S ), and • (strong invariance condition) for each P ≤ Q ≤ T , each ϕ ∈ Hom E ( P, Q ) , and each ψ ∈ Hom F ( Q, T ) , ψϕ ( ψ | P ) − ∈ Hom E ( ψ ( P ) , ψ ( Q )) .The subsystem E is normal in F ( E E F ) if it is weakly normal and • (extension condition) each α ∈ Aut E ( T ) extends to some α ∈ Aut F ( T C S ( T )) suchthat [ α, C S ( T )] ≤ Z ( T ) .This is different than the definition of a normal fusion subsystem used in [Ol], but thetwo definitions are equivalent by [AKO, Proposition I.6.4]. This one has the advantagethat it simplifies the proof of point (a) in the following proposition. Proposition 2.
Let
E ≤ F ≤ F be saturated fusion systems over T ≤ S ≤ S . Thenthe following hold. (a) If E is weakly normal in F , then E is weakly normal in F . (b) If E E F and E = O p ′ ( E ) , then E E F . Mathematics Subject Classification.
Primary 20E25. Secondary 20D20, 20D05, 20D25, 20D45.
Key words and phrases.
Fusion systems, Sylow subgroups, finite simple groups, generalized Fittingsubgroup, p -solvable groups.B. Oliver is partially supported by UMR 7539 of the CNRS. (c) If E E F and F has p -power index in F [AKO, Definition I.7.3] , then E E F .Proof. If T is strongly closed in F , then it is also strongly closed in F . If the stronginvariance condition holds for E ≤ F , then it also holds for
E ≤ F . This proves (a).Point (b) was shown by David Craven in [Cr, Corollary 8.19].Under the hypotheses of (c), E is weakly normal in F by (a), and it remains to provethat each α ∈ Aut E ( T ) extends to α ∈ Aut F ( T C S ( T )) such that [ α, C S ( T )] ≤ Z ( T ) .This clearly holds for α ∈ Inn ( T ) , and since Inn ( T ) ∈ Syl p ( Aut E ( T )) , it suffices to show itfor α ∈ Aut E ( T ) of order prime to p . For such α , since E E F , there is b α ∈ Aut F ( T C S ( T )) such that b α | T = α and [ b α, C S ( T )] ≤ Z ( T ) , and b α restricts to α ∈ Aut F ( T C S ( T )) since [ b α, C S ( T )] ≤ T . Upon replacing α by α k for some appropriate k ≡ (mod | α | ), we canarrange that α has order prime to p (and still α | T = α and [ α, C S ( T )] ≤ Z ( T ) ). But then α ∈ O p ( Aut F ( T C S ( T ))) ≤ Aut F ( T C S ( T )) since F has p -power index in F , provingthe extension condition for E ≤ F . (cid:3) We refer to [Cr, Example 8.18] for an example of saturated fusion systems
E ≤ F ≤ b F where E E b F , and E is weakly normal but not normal in F . Proposition 3 (Compare with [Ol, Proposition 1.8]) . Let E E F be saturated fusion sys-tems over finite p -groups T E S . Let χ : Aut F ( T ) −→ ∆ be a surjective homomorphism,for some ∆ = 1 of order prime to p , such that Aut E ( T ) ≤ Ker ( χ ) . Then there is a uniqueproper normal subsystem F E F over S such that Aut F ( S ) = (cid:8) α ∈ Aut F ( S ) (cid:12)(cid:12) α | T ∈ Ker ( χ ) (cid:9) , (1) and Aut F ( T ) = Ker ( χ ) and E E F .Proof. This was shown in [Ol, Proposition 1.8], except for the statements that Aut F ( T ) = Ker ( χ ) and E is normal in F . Since E is weakly normal in F by Proposition 2(a),normality will follow one we have checked the extension condition.Set H ∗ = { P ∈ F c | P ∩ T ∈ E c } . Thus T C S ( T ) ∈ H ∗ . In the proof of [Ol, Proposition1.8], we construct a map b χ : Mor ( F | H ∗ ) −−−−−→ ∆ with the property that for each P ∈ H ∗ such that P ≥ T , and each β ∈ Aut F ( P ) , wehave b χ ( β ) = χ ( β | T ) and Aut F ( P ) = Aut F ( P ) ∩ b χ − (1) . (Note that β | T ∈ Aut F ( T ) since T is strongly closed in F .) So (1) holds, andAut F ( T C S ( T )) = { α ∈ Aut F ( T C S ( T )) | α | T ∈ Ker ( χ ) } . (2)Since each β ∈ Aut F ( T ) extends to some β ∈ Aut F ( T C S ( T )) by the extension axiom[AKO, Proposition I.2.5] applied to F , (2) shows that Aut F ( T ) ≤ Ker ( χ ) . Similarly,each γ ∈ Ker ( χ ) extends to some γ ∈ Aut F ( T C S ( T )) by the extension axiom for F , and γ, γ ∈ Mor ( F ) by (2) again. Thus Aut F ( T ) = Ker ( χ ) .For each α ∈ Aut E ( T ) , the extension condition for E E F implies that there exists α ∈ Aut F ( T C S ( T )) extending α and with [ α, C S ( T )] ≤ Z ( T ) . Then α ∈ Aut F ( T C S ( T )) by (2) and since α | T = α ∈ Aut E ( T ) . So the extension condition holds for E ≤ F , provingthat E E F . (cid:3) Definition 4.
Let E E F be saturated fusion systems over finite p -groups T E S , anddefine C S ( E ) = { x ∈ S | C F ( x ) ≥ E } . We say that E is centric in F if C S ( E ) ⊆ T . ORRECTION TO: REDUCTIONS TO SIMPLE FUSION SYSTEMS 3
By a theorem of Aschbacher (see Notation 6.1 and (6.7.1) in [As]), for each such E E F , C S ( E ) is a subgroup of S , and C F ( C S ( E )) contains E . Thus each morphism in E extendsto a morphism in F between subgroups containing C S ( E ) that is the identity on C S ( E ) .For each saturated fusion system F over a finite p -group S , we setAut ( F ) = { β ∈ Aut ( S ) | β F = F } : the group of “fusion preserving” automorphisms of S . (This group was denoted Aut ( S, F ) in [Ol].) For β ∈ Aut ( F ) , let c β be the automorphism of the category F that sends P to β ( P ) and sends ϕ ∈ Hom F ( P, Q ) to βϕβ − ∈ Hom F ( β ( P ) , β ( Q )) .The next theorem contains most of Theorems 1.14 and 2.3 in [Ol], together with someadditional information about automorphisms of the systems. Theorem 5.
Let E E F be saturated fusion systems over finite p -groups T E S . Assumethat Aut F ( T ) / Aut E ( T ) is p -solvable (equivalently, that Out F ( T ) is p -solvable). (a) In all cases, there is a sequence F ≤ F ≤ F ≤ · · · ≤ F m = F of saturated fusion subsystems (for some m ≥ ) such that for each ≤ i < m , F i is normal of p -power index orindex prime to p in F i +1 and E E F i E F ; (3) and such that F is a fusion system over T C S ( T ) and Aut F ( T ) = Aut E ( T ) . (b) If E is centric in F , then there is a sequence of subsystems satisfying (3) such that F = E .In either case, the subsystems can be chosen so that for each ≤ j ≤ m , and each β ∈ Aut ( F j ) with c β ( E ) = E , we have c β ( F i ) = F i for all ≤ i < j .Proof. We outline here the proof as given in [Ol]: enough to explain how Propositions2(c) and 3 are used to prove that E E F i for each i , and explain why the last statement istrue. We refer frequently to the following transitivity result for normality (see [As, 7.4]):If F E F E F are saturated fusion systems over finite p -groups S E S E S such that c α ( F ) = F for each α ∈ Aut F ( S ) , then F E F . (4) (a) Set G = Aut F ( T ) and G = Aut E ( T ) . Since G/G is p -solvable, there are subgroups G E G E · · · E G m = G (some m ≥ ) such that for each ≤ i < m , either G i = O p ( G i +1 ) G (hence G i +1 /G i is a p -group), or G i = O p ′ ( G i +1 ) G (hence G i +1 /G i hasorder prime to p ). In particular, the G i are all normal in G since G is. For each i , set S i = N G i S ( T ) def = { x ∈ S | c x ∈ G i } . Thus S i E S and Aut S i ( T ) = G i ∩ Aut S ( T ) ∈ Syl p ( G i ) for each i , S m = S , and S = T C S ( T ) . We will construct successively subsystems F = F m ≥ F m − ≥ · · · ≥ F in F such that for each ≤ i ≤ m − , F i is a fusion system over S i , Aut F i ( T ) = G i , and the conditions on F i in (3) all hold.Assume, for some ≤ i < m , that F i +1 E F has been constructed satisfying theseconditions. Thus Aut F i +1 ( T ) = G i +1 . If G i +1 /G i has order prime to p , then by Proposition3, applied with G i +1 /G i in the role of ∆ , there is a unique saturated subsystem F i E F i +1 of index prime to p over S i +1 = S i such that Aut F i ( S i +1 ) = { α ∈ Aut F i +1 ( S i +1 ) | α | T ∈ G i } ,and also E E F i and Aut F i ( T ) = G i .If G i +1 /G i is a p -group, then the argument in the proof of [Ol, Theorem 1.14] showsthat there is a unique F i E F i +1 over S i of p -power index such that Aut F i ( T ) = G i and E ≤ F i . Also, E E F i by Proposition 2(c). Since Aut F i ( T ) has p -power index in BOB OLIVER
Aut F i +1 ( T ) , we haveAut F i ( T ) ≥ O p ( Aut F i +1 ( T )) Aut S i ( T ) ≤ O p ( G i +1 ) G i = G i , where the first inclusion is an equality since Aut S i ( T ) ∈ Syl p ( Aut F i ( T )) and the secondis an equality since Aut S i ( T ) ∈ Syl p ( G i ) . Thus Aut F i ( T ) = G i . (b) By (a) and (4), it suffices to prove this when S = T C S ( T ) . By [Ol, Corollary 2.2], S/T = T C S ( T ) /T is abelian.Set H = { P ≤ S | P ≥ C S ( T ) } , and let F ∗ ⊆ F be the full subcategory with Ob ( F ∗ ) = H . Define χ : Mor ( F ∗ ) −−−−−−→ Aut F /T ( S/T ) by sending ϕ ∈ Hom F ( P, Q ) to the induced automorphism of S/T = P T /T = QT /T ∼ = P/ ( P ∩ T ) ∼ = Q/ ( Q ∩ T ) . Here,
P T = QT = S since P, Q ∈ H and S = T C S ( T ) , and ϕ ( P ∩ T ) ≤ Q ∩ T since T is strongly closed. Thus each ϕ ∈ Mor ( F ∗ ) factors through some ϕ ∈ Aut ( S/T ) . In thenotation of Craven [Cr, Definition 5.5], ϕ ∈ Aut F T ( S/T ) , and so ϕ ∈ Aut F /T ( S/T ) by[Cr, Theorem 5.14]. See also [As, Theorem 12.5] for a different proof that ϕ ∈ Mor ( F /T ) .We now apply [Ol, Lemma 1.6], whose hypotheses (i)–(v) are shown to hold in the proofof [Ol, Theorem 2.3]. By that lemma, F def = h χ − (1) i is a saturated fusion subsystem over S normal of index prime to p in F such that Aut F ( S ) = Ker ( χ | Aut F ( S ) ) .By Proposition 2(a), E is weakly normal in F . If α ∈ Aut E ( T ) , then since E E F , α extends to α ∈ Aut F ( S ) such that [ α, C S ( T )] ≤ Z ( T ) . Since S = T C S ( T ) , this impliesthat χ ( α ) = 1 , and hence that α ∈ Aut F ( S ) . So the extension condition holds, and E E F .The construction of F E F of p -power index such that E E F and has index primeto p follows from exactly the same argument as used in [Ol], except that E is normal in F by Proposition 2(c). (a,b) It remains to prove the last statement (invariance under automorphisms), andshow that F i E F for all i (not only for i = m − ). To see this, choose ≤ j ≤ m and β ∈ Aut ( F j ) ≤ Aut ( S j ) such that c β ( E ) = E . Then β ( T ) = T and β Aut E ( T ) = Aut E ( T ) .In (b), we have c β ( F i ) = F i for ≤ i < j by the uniqueness of choices of subsystemsat each stage. In (a), β Aut F i ( T ) = Aut F i ( T ) for each ≤ i < j by construction of G i = Aut F i ( T ) , and hence by the uniqueness of the choices (depending only on the G i ),we have c β ( F i ) = F i .In particular, if ≤ i < m is such that F i +1 E F , this says that c β ( F i ) = F i for each β ∈ Aut F ( S i +1 ) ≤ Aut ( F i +1 ) , and together with (4), it implies that F i E F . It nowfollows inductively that F i E F for each i . (cid:3) For each saturated fusion system F over a finite p -group S , set F ∞ = F for anysequence F E F E · · · E F m = F of saturated subsystems such that O p ( F ) = O p ′ ( F ) = F , and such that F i E F and F i has index prime to p or p -power index in F i +1 for each ≤ i < m . By [Ol, Lemma 1.13], F ∞ is independent of the choice of the F i . Theorem 6 ([Ol, Theorem 2.3]) . Let E E F be saturated fusion systems over finite p -groups T E S such that E is centric in F . Assume either (a) Aut F ( T ) / Aut E ( T ) is p -solvable; or ORRECTION TO: REDUCTIONS TO SIMPLE FUSION SYSTEMS 5 (b) Out ( E ) def = Aut ( E ) / Aut E ( T ) is p -solvable.Then F ∞ = E ∞ .Proof. Since Aut F ( T ) ≤ Aut ( E ) (since E E F ), we have Aut F ( T ) / Aut E ( T ) ≤ Out ( E ) .So (b) implies (a). It thus suffices to prove the theorem when (a) holds, and this followsimmediately from Theorem 5(b) and [Ol, Lemma 1.13]. (cid:3) References [As] 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, CambridgeUniv. Press (2011)[AO] M. Aschbacher & B. Oliver, Fusion systems, Bull. Amer. Math. Soc. 53 (2016), 555–615[Cr] D. Craven, The theory of fusion systems, an algebraic approach, Cambridge Studies in AdvancedMathematics 131, Cambridge Univ. Press (2011)[HL] E. Henke & J. Lynd, Fusion systems with Benson-Solomon components, arXiv:1806.01938 [Ol] B. Oliver, Reductions to simple fusion systems, Bull. London Math. Soc. 48 (2016), 923–934
Université Sorbonne Paris Nord, LAGA, UMR 7539 du CNRS, 99, Av. J.-B. Clément,93430 Villetaneuse, France.
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