Eliminating components in Quillen's Conjecture
aa r X i v : . [ m a t h . G R ] J a n ELIMINATING COMPONENTS IN QUILLEN’S CONJECTURE
KEVIN IV ´AN PITERMAN*DEPARTAMENTO DE MATEM ´ATICAIMAS-CONICET, FCEYNUNIVERSIDAD DE BUENOS AIRESBUENOS AIRES ARGENTINAE-MAIL:
STEPHEN D. SMITHDEPARTMENT OF MATHEMATICSUNIVERSITY OF ILLINOIS AT CHICAGOCHICAGO ILLINOIS USA(HOME: 728 WISCONSIN, OAK PARK IL 60304 USA)E-MAIL:
Abstract.
We generalize an earlier result of Segev, which shows that some component in a mini-mal counterexample to Quillen’s conjecture must admit an outer automorphism. We show in factthat every component must admit an outer automorphism. Thus we transform his restriction-result on components to an elimination-result: namely one which excludes any component whichdoes not admit an outer automorphism. Indeed we show that the outer automorphisms ad-mitted must include p -outers: that is, outer automorphisms of order divisible by p . This givesstronger, concrete eliminations: for example if p is odd, it eliminates sporadic and alternat-ing components—thus reducing to Lie-type components (and typically forcing p -outers of fieldtype). For p = 2, we obtain similar but less restrictive results. We also provide some tools tohelp eliminate suitable components that do admit p -outers in a minimal counterexample. Contents
1. Introduction 22. Notation and preliminaries 53. Discussion: Segev’s method, via a Mayer-Vietoris argument 74. The generalized method, leading to elimination results 155. Some more concrete consequences of the generalized method 236. Some properties of the image poset in the generalized method 267. Elimination of components of sporadic type HS (for the prime 2) 318. Elimination of certain alternating components (for the prime 2) 33References 36THIS PAPER IS DEDICATED TO THE MEMORY OF JAN SAXL
Mathematics Subject Classification.
Key words and phrases. p -subgroups, Quillen’s conjecture, posets, finite groups.*Supported by a CONICET postdoctoral fellowship and grants PIP 11220170100357, PICT 2017-2997, andUBACYT 20020160100081BA. . Introduction
This paper began life as a sequel to Segev’s article “Quillen’s conjecture and the kernel oncomponents” [Seg96]; and expanded from there. In brief summary:One consequence of Segev’s main result—essentially the contrapositive of the final remarkon Out G ( L ) in Theorem 1.4 below—can be stated in the form: Corollary 1.1 (Segev) . If G is a minimal counterexample to Quillen’s conjecture, then G inducesouter automorphisms on some component. We call this kind of result on a counterexample to Quillen’s conjecture a restriction result, sinceit restricts at least one component to have a certain Property P G . (Here, P G would be that thecomponent admits an outer automorphism induced in G .)By contrast, a consequence of our extension of Segev’s work can be stated in the related form: Corollary 1.2. If G is a minimal counterexample to Quillen’s conjecture, then G induces outerautomorphisms on all components. We call this kind of result an elimination result, since it establishes that every component in a mini-mal counterexample to Quillen’s conjecture must have a certain Property P G —so that a componentwith not- P G (here, one having no outers in G ) is eliminated from a minimal counterexample.In fact, we can sharpen Corollary 1.2, to eliminate components with only p ′ -outers; the followingis essentially the contrapositive of the final statement about Out G ( L ) in Theorem 1.6 below: Corollary 1.3. If G is a minimal counterexample to Quillen’s conjecture, then G induces outerautomorphisms of order p on all components. As a preview of our elimination-applications: Assume p is odd. Then Corollary 1.3 eliminatesalternating and sporadic components L , since they have Out( L ) a 2-group. So by the Classificationof Finite Simple Groups (CFSG), all the components are of Lie-type. These have Out( L ) givenby diagonal \ field \ graph automorphisms. But we can have diagonal \ graph automorphisms for onlysome p ; so “mostly” we get just field automorphisms.In the remainder of this Introduction, we further expand on the above summary:We begin with some background on Quillen’s Conjecture: Let G be a finite group and p aprime dividing its order. In [Qui78], Quillen introduced the poset A p ( G ) of nontrivial elementaryabelian p -subgroups and studied its homotopical and topological properties via its order com-plex. He showed that if G has a nontrivial normal p -subgroup, then A p ( G ) is contractible; andconjectured the converse: that if A p ( G ) is contractible, then G should have a nontrivial nor-mal p -subgroup. Contrapositively, if the largest normal p -subgroup of G is trivial, then A p ( G )should not be contractible. This is the well-known Quillen Conjecture , which we abbreviateby (QC). The Conjecture remains open in general; but there have been important advances, suchas [AK90, AS93, Pit20, Qui78, Seg96] (see [Smi11, Ch.8] for a fuller historical discussion).In this article, we restrict attention to the following stronger homology-version of (QC); recallthat O p ( G ) denotes the largest normal p -subgroup of G :(H-QC) If O p ( G ) = 1, then ˜ H ∗ ( A p ( G ) , Q ) = 0.Here, ˜ H ∗ ( X, Q ) denotes the rational homology of the finite poset X , which is the homology of itsorder complex. We will work with rational homology, so in general we will drop the coefficientnotation and write ˜ H ∗ ( X ) for ˜ H ∗ ( X, Q ).We next indicate some existing (H-QC)-results, as background to Segev’s work: Quillen es-tablished (H-QC) for solvable groups, and groups of p -rank at most 2. Later, various authorsextended (H-QC) to p -solvable groups (see [Smi11, 8.2.12]). In this direction, in [AS93, 1.6] nd [Pit20] (cf. Theorem 2.4 below), it is proved that a minimal counterexample G to (H-QC)has O p ′ ( G ) = 1. It then follows that the generalized Fitting subgroup F ∗ ( G ) is the direct productof the components of G —which are simple, and of order divisible by p (see Lemma 2.2 below). Thecase where G itself is simple was included in the work of Aschbacher and Kleidman in [AK90]—who established (H-QC) for almost-simple groups (i.e., where F ∗ ( G ) is simple). SubsequentlyAschbacher and Smith in [AS93] obtained a result for fairly general G —showing that (H-QC)holds for p > G which are unitary groups.Along the way, in [AS93, 1.7] it is shown that every component in a minimal counterexampleto (H-QC) must fail the QD-property (“Quillen dimension”, defined at [AS93, p.474])—notice thisis an elimination result, ruling out components with QD.In [Seg96], Segev worked with the kernel on components of G . This subgroup of G is the kernelof the conjugation action of G on its set of components, namely: H := \ L N G ( L ) . Here the intersection runs over all the components L of G , and N G ( L ) denotes the normalizerof L in G . Segev established (H-QC) under certain conditions on the kernel H on components,when O p ′ ( G ) = 1. As indicated in the previous paragraph, F ∗ ( G ) is the product of the com-ponents. In particular, his main result (stated as Theorem 1.4 below) gives (H-QC) under thehypothesis that H = F ∗ ( G )—which implies that the kernel H on components induces only inner automorphisms on each component: Theorem 1.4 (Segev, [Seg96, Thms 2 & 3]) . Suppose that O p ′ ( G ) = 1 . Let H := T L N G ( L ) be thekernel on components. If for each component L of G , the inclusion map A p ( L ) ⊆ A p (Aut H ( L )) isnot the zero map in homology, then G satisfies (H-QC).In particular, if H = F ∗ ( G ) , then G satisfies (H-QC).Indeed if Out H ( L ) or Out G ( L ) is a p ′ -group for every component L of G , then G has (H-QC). Here Aut G ( L ) = N G ( L ) /C G ( L ), with C G ( L ) the centralizer of L in G ; also Out G ( L ) = Aut G ( L ) /L .We have Aut H ( L ) ≤ Aut G ( L ) and Out H ( L ) ≤ Out G ( L ).We will next indicate some ways in which we can weaken Segev’s hypotheses. Note that Segev’stheorem requires a common behavior in all the components of G ; and also requires O p ′ ( G ) = 1.In this article, we will show that we can instead focus on the behavior of a single component.Therefore, we will convert Segev’s restriction result into an elimination result—as we had indicatedearlier, when we contrasted Corollaries 1.1 and 1.2.We will also drop the requirement that O p ′ ( G ) = 1. Instead, we will often work under one of thefollowing inductive hypotheses (H1) or (H L ( p ))—which will at least limit some aspects of behaviorof the components. Hypothesis (H1) below is motivated by previous results [AS93, Pit20], with aneye to a minimal counterexample to (H-QC): (H1) Proper subgroups and proper central quotients of G satisfy (H-QC).And indeed in the context of a counterexample of minimal order to (H-QC), we do have theabove hypothesis (H1). Hence, results proved under (H1) will also hold under the minimal-counterexample variant of (H1): namely that (H-QC) holds for all groups H such that | H | < | G | .For our other inductive hypothesis, let L be a component of G , and L , . . . , L t its G -orbit; define:(H L ( p )) p divides the order of L , and C G ( L . . . L t ) satisfies (H-QC). Recall that O p ′ ( G ) denotes the largest normal p ′ -subgroup of G . Our convention is that that “simple” means non-abelian simple. Recall that a proper central quotient of G is a quotient by a nontrivial subgroup Z ≤ Z ( G ), where Z ( G ) is thecenter of G . n fact, later Remark 2.5 gives an implication-relation between our two hypotheses: Namely un-der (H1), G satisfies either (H-QC), or (H L ( p )) for every component L of G . So within the contextof proving (H-QC), the hypothesis (H L ( p )) is weaker than (H1). Hence in proving various results,e.g. our main (H-QC)-result Theorem 1.6 below (and Corollary 5.1 which leads to it), we will beable to assume as hypothesis “either (H1), or (H L ( p )) for some L ”—and then it suffices to givethe proof just under (H L ( p )) for that L .Now we turn our attention away from hypotheses, towards sharpening Segev’s conclusions—thatis, to obtaining elimination-results. Namely in Theorem 1.6 below, we will see that, under (H1), ifthere is some component L of G such that no elementary abelian p -subgroup of G induces outerautomorphisms on L , then G satisfies (H-QC).We then immediately obtain many of our particular elimination-results as consequences: Recallwe had mentioned that Corollary 1.3 (and hence also Corollary 1.2) follows via the contrapositiveof the final remark on Out G ( L ) in Theorem 1.6.First assume p is odd. Then using Corollary 1.3, we can eliminate alternating and sporadiccomponents from a minimal counterexample to (H-QC): For (using e.g. [GLS98, Secs 5.2,5.3]),if L = A n is the alternating group on n letters, then Out( L ) = C for n = 6, and Out( A ) = C × C ;while if L is a sporadic group, then Out( L ) ≤ C . Therefore, if p is odd and L is a componentof G of alternating or sporadic type, Out G ( L ) is a p ′ -group. (This gives case (1) below.)Now assume p = 2. Using Corollary 1.3 as above, we can also eliminate Suzuki (including F )and Ree components: since their outer automorphisms are 2 ′ -groups (using for example [GLS98,Ch 4]). (This gives case (4) below.)Furthermore we will also able, in the final two sections of the paper, to obtain elimination-consequences from various other results (such as Proposition 6.9), which give us some tools towardsuitable conditions on the outer automorphisms of a simple group (and on the fixed points of thoseautomorphisms)—in order to then guarantee the hypotheses of Theorem 1.6, and hence estab-lish (H-QC) for G . Such results allow us to also eliminate, for example, at least some alternatingand sporadic components (see cases (2) and (3) below) from a minimal counterexample to (H-QC),in the present subcase p = 2 as well.We summarize our various concrete elimination-consequences of the above types in: Corollary 1.5.
Assume (H1). Then (H-QC) holds if p and a component L of G satisfy one of:(1) p is odd, and L is alternating or sporadic; or:We have p = 2 , with L given by one of:(2) L is sporadic HS .(3) L is alternating A or A .(4) L has one of the Lie types Sz( q ) , or F ( q ) , or Ree( q ) . Note furthermore that any simple L with Out( L ) = 1 is similarly eliminated (for any p ) byCorollary 1.2. For example, the reader can consult [GLS98, Table 5.3] for the 14 sporadic L with Out( L ) = 1.We conclude our Introduction with the promised statement of our main general theorem leadingto elimination results: Theorem 1.6.
Let L be a component of G and L , . . . , L t its G -orbit; and set H := T i N G ( L i ) .Suppose that G satisfies (H1) or (H L ( p ) ).If A p ( L ) ⊆ A p (cid:0) Aut H ( L ) (cid:1) is not the zero map in homology, then G satisfies (H-QC).In particular, if we have H = L . . . L t C G ( L . . . L t ) , then G satisfies (H-QC).Indeed, if Out H ( L ) or Out G ( L ) is a p ′ -group, then (H-QC) holds. ote that the “In particular” part of Theorem 1.6 applies for some component of G (in contrastto each component, in Segev’s Theorem 1.4); so that its contrapositive forces nontrivial Out G ( L )for all L in a counterexample to (H-QC). Hence we do indeed have an elimination result.In our Theorem 1.6, we in fact work with the local kernel H := T ti =1 N G ( L i ), where L , . . . , L t isthe G -orbit of the component L . And for this local kernel H , rather than requiring H = F ∗ ( G ) asin the “In particular” part of Segev’s Theorem 1.4, we instead ask for the inner-only condition juston the orbit of L , namely H = L . . . L t C G ( L . . . L t )—which holds for example if Out H ( L ) = 1.The behavior of components in other G -orbits is hidden in C G ( L . . . L t ); and for this centralizer,we will be able to exploit the inductive hypotheses (H1) or (H L ( p )). Moreover, the originalrequirement O p ′ ( G ) = 1 in Segev’s theorems is relaxed to just the divisibility of the order of thecomponent L by p .Our Theorem 1.6 will in fact follow as a special case of the more general and technical Theo-rem 4.1—which has still-more-flexible hypotheses.2. Notation and preliminaries
In this section, we establish some notation and recall some fundamental constructions on finitegroups that will be used throughout this article. For more details on the assertions on finiteposets and their homotopy properties in relation with their order complexes, we refer the readerto [Qui78]. For results on finite groups we refer to [Asc00]. We will follow the conventions of [GL83]for finite simple groups; the reader should be aware that there may be minor notational variationsfrom other standard sources.All the posets and simplicial complex considered here are finite. If X is a finite poset, then K ( X )denotes its order complex. Recall that the simplices of K ( X ) are the non-empty chains of X .We study the homotopy properties of X by means of its order complex. If f : X → Y is anorder-preserving map between finite posets, then f induces a simplicial map f : K ( X ) → K ( Y ).If f, g : X → Y are two order-preserving maps between finite posets and f ≤ g (i.e. f ( x ) ≤ g ( x )for all x ∈ X ), then the induced simplicial maps f, g : K ( X ) → K ( Y ) are homotopic. Write X ≃ Y for finite posets X, Y if their order complexes K ( X ) and K ( Y ) are homotopy equivalent.We denote by X ∗ Y the join of the posets X and Y . The underlying set of this join is thedisjoint union of X and Y , and the order is given as follows. We keep the given order in X and Y ,and we put x < y for x ∈ X and y ∈ Y . It can be shown that K ( X ∗ Y ) = K ( X ) ∗ K ( Y ), where thelatter join is the join of simplicial complexes. Moreover, its geometric realization coincides withthe classical join of topological spaces. That is, if K, L are simplicial complexes and | K | denotesthe geometric realization of K , then we have a homeomorphism | K ∗ L | ≡ | K | ∗ | L | . For moredetails on these join-properties, see [Qui78]. If f : X → Y and g : X → Y are order-preservingmaps, then we have an induced map f ∗ g : X ∗ X → Y ∗ Y defined by ( f ∗ g )( x ) = f ( x ) ∈ Y if x ∈ X , and ( f ∗ g )( x ) = g ( x ) ∈ Y if x ∈ X .Below we recall a generalized version of Quillen’s fiber lemma (cf. [Qui78, Prop. 1.6]. See also[BWW05]). Recall that an n -equivalence is a continuous function f : X → Y such that f inducesisomorphisms in the homotopy groups f ∗ : π i ( X ) → π i ( Y ) with i < n , and an epimorphism in the n -th homotopy group. By the Hurewicz theorem, an n -equivalence also induces isomorphisms inthe homology groups of degree < n , and an epimorphism in degree n . The topological space X is n -connected if its homotopy groups of degree at most n are trivial (and hence its homology groupsof degree at most n also vanish). By convention, ( − − Proposition 2.1 (Quillen’s fiber lemma) . Let f : X → Y be a map between finite posets. As-sume n ≥ . Suppose that for all y ∈ Y , f − ( Y ≤ y ) ∗ Y >y (resp. f − ( Y ≥ y ) ∗ Y Suppose that O p ( G ) = 1 = O p ′ ( G ) . Then F ( G ) = 1 and F ∗ ( G ) = E ( G ) is thedirect product of the components of G , which are all simple and of order divisible by p . That is, wehave F ∗ ( G ) = L . . . L n ; and each L i is a simple component of G . Moreover, the self-centralizingproperty gives C G (cid:0) F ∗ ( G ) (cid:1) = Z (cid:0) F ( G ) (cid:1) = 1 , so we have a natural inclusion: F ∗ ( G ) ≤ G ≤ Aut( F ∗ ( G )) , and so G/F ∗ ( G ) ≤ Out (cid:0) F ∗ ( G ) (cid:1) . ecall the outer automorphism group of G is Out( G ) = Aut( G ) / Inn( G ), where Inn( G ) = G/Z ( G )is the group of inner automorphisms of G . Recall for H ≤ G that Aut G ( H ) = N G ( H ) /C G ( H ) isthe group of automorphisms of H induced by G , and Out G ( H ) = N G ( H ) / ( HC G ( H )) is the groupof outer automorphisms of H induced by G . The subgroup HC G ( H ) is the subgroup of G whoseelements induce inner automorphisms on H . We will say that a subgroup K ≤ G induces (or actsby) outer automorphisms on H , if K normalizes H , and K contains no inner automorphism of H .That is, K induces outer automorphisms on H if and only if K ∩ ( HC G ( H )) = 1.In the following lemma, we relate the subgroups of inner automorphisms for sets of commutingsubgroups of a given group. The proof is a straightforward use of the Dedekind modular law. Lemma 2.3. Let A, B ≤ G such that [ A, B ] = 1 . Then (cid:0) AC G ( A ) (cid:1) ∩ (cid:0) BC G ( B ) (cid:1) = ABC G ( AB ) .Indeed if A , . . . , A r ≤ G commute pairwise, then T i (cid:0) A i C G ( A i ) (cid:1) = ( A . . . A r ) C G ( A . . . A r ) . In Proposition 1.6 of [AS93] it is shown that: If G satisfies (H1) with p odd, and does not containcomponents of type L (8), U (8) or Sz(32) for p = 3 , , O p ′ ( G ) = 1implies (H-QC) for G . Inspired by this result, in [Pit20, Thm 1] it is shown that the restrictionson p and on the components are unnecessary. We recall this generalization below: Theorem 2.4. Suppose that G satisfies (H1). If further O p ′ ( G ) = 1 , then G satisfies (H-QC). In particular this result leads to an implication relation between our inductive hypotheses: Remark 2.5. Suppose that G satisfies (H1). If O p ′ ( G ) = 1, then G has (H-QC) by Theorem 2.4.Otherwise O p ′ ( G ) = 1: and then it follows that every component L of G has order divisible by p .Moreover, writing L , . . . , L t for the G -orbit of L , since L is nonabelian, we have C G ( L . . . L t ) isa proper subgroup of G —and hence it satisfies (H-QC) by (H1). That is, we obtain (H L ( p )). So:Under (H1), G satisfies either (H-QC), or (H L ( p )) for every component L .So in effect, our later proofs under (H1) can often proceed just under (H L ( p )). ♦ To finish this section, we recall the definition (for a subgroup H ≤ G ) of the inflation N G ( H )of H , consisting of the elements of A p ( G ) which intersect H nontrivially. This poset had beenused earlier in e.g. [Seg96, SW94], and more recently in [Pit20]. Definition 2.6. For H ≤ G and B ⊆ A p ( G ), set: N B ( H ) := { B ∈ B : B ∩ H = 1 } . If B = A p ( K ) for some subgroup K ≤ G , then we also write N B ( H ) = N K ( H ).We recall some special features of this subposet: Remark 2.7. We have N G ( H ) ≃ A p ( H )—using the retraction E r ( E ) = E ∩ H , withhomotopy-inverse given by the inclusion of A p ( H ) in N G ( H ). So we regard the subposet N G ( H )as the inflation of A p ( H ) in G .Moreover, if we consider an E ∈ A p ( G ) − N G ( H ) (that is, with E ∩ H = 1), then r restricts toa homotopy equivalence N G ( H ) >E ≃ A p (cid:0) C H ( E ) (cid:1) , with homotopy-inverse given by A AE . Sowhen we consider the fibers of the inclusion i : N G ( H ) ⊆ A p ( G ), for E ∈ A p ( G ) − N G ( H ) we get: i − ( A p ( G ) ≥ E ) = N G ( H ) >E ≃ A p (cid:0) C H ( E ) (cid:1) . This relation connects Quillen’s fiber lemma (Proposition 2.1) with the study of centralizers. ♦ Discussion: Segev’s method, via a Mayer-Vietoris argument The proof of our main technical result Theorem 4.1 (which leads to our more concrete elimination-results such as Theorems 1.6 and Corollary 5.2) uses a generalization of Segev’s method in [Seg96].In this section, we outline some fundamental aspects of Segev’s proofs; and along the way, weindicate where we can make our extensions. e begin the section with some general background for Segev’s theorems:We are pursuing results which establish (H-QC). So:We assume O p ( G ) = 1;and we need to obtain: (Goal) ˜ H ∗ (cid:0) A p ( G ) (cid:1) = 0.Segev’s arguments primarily involve: H := the kernel of the G -conjugation action on its components. Remark 3.1 (Some consequences of O p ′ ( G ) = 1) . Segev works under the further assumptionthat O p ′ ( G ) = 1; and using this along with O p ( G ) = 1, he gives some standard consequencesat pp955–956 of [Seg96]. We summarize a number of such consequences (cf. our 2.2) as:(i) F ∗ ( G ) = E ( G ) is a direct product of simple components L , . . . , L n , with G faithful on F ∗ ( G );(ii) The components of G have order divisible by p .Segev’s arguments seem to frequently depend on these properties—but very often only implicitly .For our analysis of his logic with an eye to generalization, we will always try to be much moreexplicit. For example, we mention that using (ii) above we get:(iii) A p ( L i ) = ∅ for all i , and hence A p ( H ) = ∅ .This fact seems to be fundamental—though unstated—throughout [Seg96]. ♦ A further point which seems at best implicit in [Seg96] is the handling of the case n = 1—namelywhere there is just a single component L : Note using Remark 3.1(i) above that then F ∗ ( G ) = L ,so that G is almost-simple. Now the well-known result of Aschbacher and Kleidman in [AK90]gives (H-QC) for G almost-simple; so that result could be quoted to finish when n = 1. But infact, Segev only explicitly quotes [AK90] on p956: and there, only to show (for any n ) that thehypothesis of Theorem 3 in [Seg96] satisfies the hypothesis of Theorem 2 there. Thus it seems likelythat a different argument was intended to cover the case n = 1: For notice that this condition inparticular gives a special case of trivial G -conjugation action on components—that is, H = G . Sowe will include the reduction to n ≥ H < G provides the foundationfor the “generic-context” in our logic-analysis, namely: A Mayer-Vietoris sequence, based on nontrivial action on components: Before starting on the proofs, we develop an outline for the overall approach followed in them.The more specific hypotheses of Theorems 1 and 2 in [Seg96] involve restrictions related to thehomology of A p ( H ). And the proofs there seem to begin by assuming that the case H < G musthold. So for completeness, when we later outline the logic in those proofs, we will at those pointsadd a treatment of the seemingly-omitted case where H = G —that is, where G normalizes allthe components. Indeed, we will treat the slightly more general situation where A p ( H ) = A p ( G ):for since (Goal) only involves A p ( G ) (via its homology), we may as well assume in that situationthat H = G . In brief summary: We will verify there that in the trivial-action case, the hypothesesof Theorems 1 and 2 essentially already contain the conclusion (H-QC); so that no further proofis then actually required.Hence, in the remainder of our introductory Mayer-Vietoris discussion, i.e. through Remark 3.4:We assume temporarily that we have already reduced to A p ( H ) ( A p ( G );and hence also to H < G —namely nontrivial action on components. In this situation, we want toset up a general context for the main arguments, within this “generic” case of nontrivial action. In fact, an unpublished result of Thompson (in a preprint on “separating sets”, from July 1991) shows that acounterexample G to (H-QC) under the hypotheses of Aschbacher-Smith [AS93] should have H < G —indeed G/H should be nonsolvable. But we won’t in this paper follow that direction: since we are instead focusing on thekernel H of the action, and outer automorphisms that it induces on components. n order to work toward the nonzero homology for A p ( G ) in (Goal), we will proceed by de-composing that homology over certain other p -subgroup posets related to H . In particular, wewill make use of the inflation Y := N G ( H ) of A p ( H ), and the complement Z := A p ( G ) − A p ( H )to A p ( H ); that is, we will work with the decomposition A p ( G ) = Y ∪ Z . We set Y := Y ∩ Z for the overlap. We also recall by Remark 2.7 that Y is homotopy equivalent to A p ( H ), via theretraction r ( E ) = E ∩ H ; we then further set V := r ( Y ), and let b : V ⊆ A p ( H ) denote theinclusion. We will need: Remark 3.2 (Some consequences of reducing to nontrivial action A p ( H ) ( A p ( G )) . Recall weare assuming here that we have in fact reduced to A p ( H ) ( A p ( G ). Let’s examine the effect ofthis assumption on the terms of the above decomposition A p ( G ) = Y ∪ Z .Notice first that Y = N G ( H ) is nonempty—for it contains A p ( H ), which we saw at Re-mark 3.1(iii) is nonempty.Next note that our assumed reduction provides us with some A ∈ A p ( G ) −A p ( H ) = Z , so that Z also is nonempty. In fact, we get A = ( A ∩ H ) × B , where nontrivial B ∈ A p ( G ) has B ∩ H = 1—sothat B ∈ Z − Y . And then our decomposition Y ∪ Z is nontrivial —that is, we really do get two independent nonempty parts. Indeed we can add here, again using A p ( H ) = ∅ , that any E ∈ Z must centralize some F ∈ A p ( H )—giving a member EF of Y , since EF ∩ H ≥ F > 1; so thatalso Y is nonempty, and hence V is nonempty. We summarize these observations in:(i) The decomposition A p ( G ) = Y ∪ Z is nontrivial; with Y, Z, Y , V nonempty.Note also that from the assumed reduction to H < G , we immediately get:(ii) We have n ≥ p ≥ G .These observations (and indeed some other consequences of H < G ) will be important at variouspoints, as our analysis below continues. ♦ In view of our assumed reduction, we now see via Remark 3.2(i) that our fundamental decom-position A p ( G ) = Y ∪ Z is nontrivial. And in using the decomposition, we can roughly regard theinflation Y as the neighborhood of A p ( H ), and Y = Y ∩ Z as the boundary of that neighborhood.In this viewpoint, the condition (A) below exploits a local-requirement related to H : namely thatthe boundary Y should (perhaps not surprisingly) be “thinner” than the full neighborhood Y , inthe sense of not being surjective in homology; and then from this restriction, we will obtain the(perhaps surprising) global-consequence of nonzero homology for A p ( G ). To implement this plan,we will use the natural homological context for our decomposition: Remark 3.3 (The Mayer-Vietoris sequence for the decomposition A p ( G ) = Y ∪ Z ) . The Mayer-Vietoris exact sequence for our decomposition A p ( G ) = Y ∪ Z takes the form: . . . → ˜ H k +1 (cid:0) A p ( G ) (cid:1) → ˜ H k ( Y ) α → ˜ H k ( Y ) ⊕ ˜ H k ( Z ) β → ˜ H k (cid:0) A p ( G ) (cid:1) → . . . Recall via our assumed reduction to A p ( H ) ( A p ( G ) and Remark 3.2(i) that Y, Z = ∅ . ♦ Note that if we had A p ( H ) = A p ( G ) in Remark 3.3 above, we would get Z = ∅ ; this then leadsto Y = A p ( H ) = A p ( G ), and hence we get the trivial decomposition A p ( G ) = Y ∪ Z = A p ( G ) ∪∅ —in which case, the sequence would degenerate to just an isomorphism of ˜ H ∗ (cid:0) A p ( G ) (cid:1) with itself.(This why our development has emphasized the need to obtain nontriviality for the decomposition;the issue of nontriviality is seemingly omitted, when the sequence is introduced at p958 of [Seg96].)Recall Y = ∅ by Remark 3.2(i). In the context of the Mayer-Vietoris sequence in Remark 3.3,our approach to (Goal) will be to now establish, for the inclusion a : Y ⊆ Y , that:(A) The induced map in homology a ∗ : ˜ H ∗ ( Y ) → ˜ H ∗ ( Y ) is not surjective.For since a ∗ gives the Y -coordinate of the value of α in the sequence, we see under (A) that then α also is non-surjective: that is, has proper image. But by exactness, im( α ) = ker( β ) is then alsoproper, and this proper kernel gives β the desired nonzero image in ˜ H ∗ (cid:0) A p ( G ) (cid:1) . o now prove (A), we usually proceed somewhat indirectly—via examining further related mapsin homology. As we continue our analysis below, we will build up successive homological terms,and maps among them; these are collected in our “omnibus” commutative diagram, in later Re-mark 3.10—as the culmination of our analysis of the overall logic for proving Theorems 1 and 2.The first few such “related maps” are reasonably self-evident:Recall by Remark 2.7 that Y is homotopy equivalent to A p ( H ), via the retraction r ( E ) = E ∩ H ;so condition (A) is equivalent to:(A ′ ) The composition map ˜ H ∗ ( Y ) a ∗ → ˜ H ∗ ( Y ) r ∗ → ˜ H ∗ (cid:0) A p ( H ) (cid:1) is not surjective.Now recall by Remark 3.2(i) that V = r ( Y ) = ∅ . We can formulate a condition analogous to (A):(B) The induced map in homology b ∗ : ˜ H ∗ ( V ) → ˜ H ∗ ( A p ( H )) is not surjective.And we see under (B) that non-surjectivity of b ∗ gives non-surjectivity also for the compositiongiven by b ∗ ◦ ( r | Y ) ∗ —which (cf. the diagram in Remark 3.10) is the composition r ∗ ◦ a ∗ , so thatwe in fact get (A ′ ). We summarize these deductions in: Remark 3.4 (Some initial implication-relations) . So far we have obtained the logical implications:(B) ⇒ (A ′ ) ⇔ (A) ⇒ (Goal) .In particular, we see that one route toward (Goal) is via establishing (B). ♦ To now prove (B), we will need suitable further hypotheses on the homology of A p ( H ). Someof these will lead us to formulate more conditions and implications—in particular extending, byan additional term implying (B), the chain in Remark 3.4 to that in later Remark 3.9. But first,in the proof of Theorem 1 in [Seg96] below, we will instead fairly directly obtain (B) itself: An outline of the proof of Segev’s Theorem 1: For Theorem 1, the “suitable further hypothesis” is that the inclusion: i : D p ( H ) ⊆ A p ( H )should induce a map i ∗ in homology which is non-surjective: where D p ( H ) is the “diagonal sub-poset” of H defined on [Seg96, p956]. We use the equivalent definition that D p ( H ) is given bythe A ∈ A p ( H ) such that there exist components L , . . . , L t , t ≥ 2, with C A ( L · · · L t ) = C A ( L i )for all i .The proof of Theorem 1 in [Seg96] begins with the decomposition A p ( G ) = Y ∪ Z , as in thesetup for Remark 3.2 above; thus in effect, it assumes that the case Z = ∅ must hold. So, in orderto cover the seemingly-omitted case Z = ∅ , we give here our promised explicit treatment of the(equivalent) case A p ( H ) = A p ( G ) of trivial action. Note that in any case for A p ( H ), the hypothesisin Theorem 1 of non-surjectivity of the map i ∗ implies that its codomain-space ˜ H ∗ (cid:0) A p ( H ) (cid:1) mustbe nonzero. So when A p ( H ) = A p ( G ), we in fact have the nonzero homology of A p ( G ) requiredfor (Goal). Thus for this case, the hypothesis already includes the conclusion (H-QC) of the result.That is, Theorem 1 becomes essentially a tautology here: so that this subcase doesn’t really require further proof. However, the overall proof of Theorem 1 seems incomplete, without some mentionof this “not-required” fact.Thus we have explicitly reduced to A p ( H ) ( A p ( G ) (and hence nontrivial action H < G ).Consequently by Remark 3.2(i): we do indeed have nontriviality of the decomposition A p ( G ),including Z = ∅ ; and thus we may use the generic-context of the Mayer-Vietoris sequence given inearlier Remark 3.3. We mention that this reduction to H < G using i ∗ is valid, even when the domain-poset D p ( H ) for i is empty—so that reduced homology in formal dimension − D p ( H ) = ∅ gives n = 1; and then in that situation, where G is almost-simple, finishingvia (H-QC) using [AK90] as we had mentioned earlier. ithin that Mayer-Vietoris context, and the corresponding terms appearing in the left half ofthe summary-diagram in Remark 3.10, the crucial step in the proof is that:We get an inclusion j : V ⊆ D p ( H ).We see then that the non-surjectivity hypothesis on i ∗ now gives non-surjectivity also for the com-position i ∗ ◦ j ∗ —which is the map b ∗ : that is, we get (B)—and hence (Goal) using the implicationsin Remark 3.4.We mention that the proof of this inclusion j on p959 of [Seg96] begins by assuming that thecase V = ∅ holds. The seemingly-omitted case V = ∅ can in fact be covered by invoking ourreduction to A p ( H ) ( A p ( G ), as we saw in Remark 3.2(ii).For later comparison, we also indicate here the rest of the proof of that inclusion j : As justindicated, we can indeed assume that V = ∅ ; so consider any B ∈ V . Then we have B = A ∩ H for some A ∈ Y , where A contains some a H . Since H is the kernel on components, a must haveat least one nontrivial orbit O (of size p ≥ 2) on components; and then the nontrivial elementsof B ≤ C G ( a ) must exhibit isomorphic actions on all members L of O —which gives us the condition(namely C B ( O ) = C B ( L ) for all L ∈ O ) defining B ∈ D p ( H ).We turn now to arguments which will involve further conditions in order to lead to (B): An outline of the proof of Segev’s Theorem 2: These further conditions arise in the proof of Theorem 2 in [Seg96], which we stated earlier asTheorem 1.4; in fact we will indicate a slight extension of Segev’s argument, which we will use inour results later in this paper.We recall that for Theorem 2, the “suitable further hypothesis on H ” is that for each compo-nent L i of G , the inclusion A p ( L i ) ⊆ A p (cid:0) Aut H ( L i ) (cid:1) should induce a nonzero map in homology.(The faithful action of L i here implicitly uses simplicity of L i in Remark 3.1(i).)Segev’s proof of Theorem 2 in fact involves the join X of factors which are given by the aboveposets A p (cid:0) Aut H ( L i ) (cid:1) ; along with some other associated constructions. In our generalization ofhis method in Section 4, we will want to instead choose X as the join of certain variants on thosefactors. So in order to focus on some of the issues involved, before we begin the main proof ofTheorem 2, we will first spend a few paragraphs sketching various aspects involved in working withsuch joins. We will give an overall context a little more general than in [Seg96]; though we willstill indicate the features given by Segev at several different points in that paper. Preliminary discussion: Constructions related to the join-poset X . In order to exploit the posetsin the hypothesis of Theorem 2, Segev on p957 of [Seg96] defines their join: Definition 3.5 (The fundamental join X ) . We set: X := X ∗ · · · ∗ X n ; where X i := A p (cid:0) Aut H ( L i ) (cid:1) .This X provides the codomain for the crucial poset map ψ : A p ( H ) → X indicated below. ♦ Various related structures are general; i.e. they do not yet involve our hypothesis on the X i :There are several direct-product groups in the background (cf. p956 of [Seg96]). Using Re-mark 3.1(i), we have E ( G ) = L × · · · × L n ≤ H , with H acting faithfully on E ( G ). Welet π i : H → Aut H ( L i ) be the natural projection; and let π := ( π , . . . , π n ) denote the “co-ordinate” map into the formal direct product J = J × . . . × J n , where J i := Aut H ( L i ). Fromfaithfulness, it is standard that π maps H to an isomorphic subgroup π ( H ) ≤ J . That is: though H itself need not admit a direct-product decomposition, it is in effect embedded in the abstractly-given formal direct product J of the individual groups Aut H ( L i ) = J i .It is also standard that a direct product of groups leads to certain analogous join-relationshipsat the level of A p -posets: First A p (cid:0) E ( G ) (cid:1) is homotopy equivalent to the join B of the individ-ual A p ( L i ). Similarly A p ( J ) is homotopy equivalent to the join X of the individual A p ( J i ) = X i ; nd we write ψ for this standard homotopy equivalence (compare [Qui78, Prop 2.6]). We will de-scribe ψ more precisely at later Definition 3.7. The restriction of this standard ψ to A p (cid:0) π ( H ) (cid:1) is es-sentially the map which Segev denotes by “ ψ : A p ( H ) → X ” —and we will similarly write ψ ratherthan ψ | A p ( H ) = ψ ◦ π for this restriction. Indeed within the join X , we get a relationship analogouswith that in the previous paragraph: namely ψ maps A p ( H ) to ψ (cid:0) A p ( H ) (cid:1) ⊆ X = X ∗ · · · ∗ X n .That is, though A p ( H ) itself need not admit a join-decomposition, it has a natural image in theabstractly-given formal join X of the A p (cid:0) Aut H ( L i ) (cid:1) = X i .Finally we recall the standard fact (cf. our (2.1), or [Seg96, (2.2)]) that the reduced homologyof a join such as X is the tensor product of the reduced homology of the factors X i .With these features in place, we now do apply our specific hypothesis on components:Namely the homology of the join A p (cid:0) E ( G ) (cid:1) is the tensor product of the homology of thefactors A p ( L i ); and our hypothesis says that the “inclusions” A p ( L i ) ⊆ A p (cid:0) Aut H ( L i ) (cid:1) = X i arenonzero in homology. Hence, the join gives an inclusion map A p ( L ) ∗ . . . ∗ A p ( L n ) = B ֒ → X which is also nonzero in homology. Then the composition: ψ | A p ( E ( G )) : A p (cid:0) E ( G ) (cid:1) →A p ( L ) ∗ . . . ∗ A p ( L n ) ֒ → X is not the zero map in homology, since ψ | A p ( E ( G )) is a homotopy equivalence with its image.Since ψ | A p ( E ( G )) = ψ ◦ e for e : A (cid:0) E ( G ) (cid:1) ⊆ A p ( H ), we see that ψ also is nonzero in homology.Thus we have established: Remark 3.6 ( A p ( H )-form of hypothesis for Theorem 2) . For the join X of the X i = A p (Aut H ( L i )):Under the hypothesis of Theorem 2, ψ : A p ( H ) → X induces a nonzero map in homology. ♦ This A p ( H )-form of the hypothesis appears as (4.2)(3) in [Seg96]; and also as (D) in our mainanalysis for Theorem 2 below. And it is the form that is actually used in the proof of Theorem 2.We complete our preliminary discussion with several further standard features related to X .We will need for certain later arguments the precise definition of the above poset isomorphism ψ : Definition 3.7 (The map ψ : A p ( H ) → X ) . First define for E ∈ A p ( H ): i E := the largest index i , such that π i ( E ) = 1 (i.e. E (cid:2) C H ( L i )).Then (cf. [Seg96, p960]) we select only the corresponding projection π i E for application to E : thatis, we define ψ ( E ) := π i E ( E ); this image of course lies in A p (Aut H ( L i E )) = X i E ⊆ X . ♦ We emphasize in particular that the term X i E of the join X which contains ψ ( E ) is uniquelydetermined: since for any other i = i E , we do not use the projection π i ( E ) ∈ X i . Also: It isstandard that the particular homotopy equivalence ψ depends on the ordering of the factors of thedirect product; but of course any chosen ordering does give such a homotopy equivalence.Next (cf. [Seg96, p957]) the order complex K := K ( X ) of the join X determines some usefulstandard subcomplexes, as follows: Definition 3.8 ( K and ˆ K ) . For a fixed i , let X ˆ i denote the maximal sub-join determined bythe X j for j = i , and then take: K := S ni =1 K ( X ˆ i ) .Notice that K = ∅ if and only if n ≥ c in K are “sub-maximal”, in the sense that c determines at least one index i c , such that the members of c do notlie in X i c .Finally (for any n ≥ 1) we indicate a standard contractible (cf. [Seg96, p958]) subcomplex ˆ K of K = K ( X ): We first fix some arbitrary vertex v i ∈ X i for each i . (We have X i = ∅ byRemark 3.1(iii).) Then we set: ˆ K := S ni =1 St K ( v i ) , here St denotes the topological star. Notice in particular that when n ≥ K = ∅ as noted above), we get K ( X ˆ i ) ⊆ St K ( X i ) ( v i ) ∗ K ( X ˆ i ) = St K ( v i ) ⊆ ˆ K , so that we have aninclusion f : K ⊆ ˆ K . Note then that for c : K ⊆ K , we have c = d ◦ f , where d : ˆ K ⊆ K . Herethe contractibility of ˆ K guarantees that d is zero in homology; so that also c is zero in homology. ♦ We note that K = K ( X ), K , and c are all needed in our summary diagram in later Remark 3.10.With the above preliminary discussion of constructions related to X completed, we begin themain proof of Theorem 2: The main logical analysis for the proof of Theorem 2 . The proof of Theorem 2 in [Seg96] beginson p960 by setting up for a reduction to the hypothesis of Theorem 1 (and hence in effect tocondition (B) of Remark 3.4). We will instead indicate a slight generalization of the argument,which leads more directly to (B). By either route, the argument depends in several ways (as was thecase for Theorem 1) on having made the reduction to the nontrivial-action case A p ( H ) ( A p ( G ).So again we give the earlier-promised explicit discussion of the seemingly-omitted trivial-actioncase A p ( H ) = A p ( G ). Here, we can simply use the A p ( H )-form of the hypothesis for Theorem 2that we derived in Remark 3.6: Note that in any case for A p ( H ), nonzero-ness of the inducedmap ψ ∗ there forces nonzero-ness of its domain-space ˜ H ∗ (cid:0) A p ( H ) (cid:1) . So when A p ( H ) = A p ( G ), wein fact have the nonzero homology of A p ( G ) required for (Goal). Thus for this case, the hypothesisalready includes the conclusion (H-QC) of the result. That is, Theorem 2 becomes essentially atautology here: so that this case doesn’t really require proof. However, again the overall proof ofTheorem 2 seems incomplete, without some mention of this “not-required” fact.We also now recall from Remark 3.2(ii) that our reduction here to A p ( H ) ( A p ( G ) also reducesus to the case of n ≥ n = 1 at p957 in [Seg96], where n ≥ n ≥ K = ∅ , whichis seemingly omitted on p958.Since we have reduced to the nontrivial-action case A p ( H ) ( A p ( G ), then just as for Theorem 1,by Remark 3.2(i) we again get nontriviality of the decomposition A p ( G ) = Y ∪ Z ; and we can indeeduse the generic-context of the Mayer-Vietoris sequence as in Remark 3.3.This time, we will investigate the homology of the right-hand terms in the summary-diagramin Remark 3.10—namely X and K , constructed in Definitions 3.5 and 3.8. We reproduce, ascondition (D) below, the A p ( H )-form of the hypothesis for Theorem 2, which we had obtainedin Remark 3.6 above. Our discussion of ˆ K in Definition 3.8 corresponds roughly to Segev’sat p958: in particular we observed there that the inclusion c : K ⊆ K ( X ) is zero in homology,and we reproduce this as (E) below. At this point, Segev can finish by establishing the furthercontainment (C):(C) ψ (cid:0) K ( V ) (cid:1) ⊆ K .(D) The induced map ψ ∗ : ˜ H ∗ (cid:0) A p ( H ) (cid:1) → ˜ H ∗ ( X ) is nonzero.(E) The induced map c ∗ : ˜ H ∗ ( K ) → ˜ H ∗ ( X ) is zero.For note then that we get c ∗ ◦ ( ψ | V ) ∗ = 0 using (C + E); so by commutativity in the diagram inRemark 3.10 below, we get ψ ∗ ◦ b ∗ = 0—and then nonzero-ness of ψ ∗ in (D) forces non-surjectivityof b ∗ , as required for (B). Thus by the implications in Remark 3.4, we get (Goal), completing theproof of Theorem 2. Remark 3.9 (Summary of logical implications) . The previous paragraph in particular exhibits afurther implication-relation among our various conditions—so that we can now write:(C + D + E) ⇒ (B) ⇒ (A ′ ) ⇔ (A) ⇒ (Goal) ; xtending our earlier sequence in Remark 3.4. ♦ The maps involved in these relations are indicated in: Remark 3.10 (Overall commutative diagram) . We have:˜ H ∗ ( Y ) a ∗ (cid:15) (cid:15) ( r | Y ) ∗ / / ˜ H ∗ ( V ) b ∗ (cid:15) (cid:15) ( ψ | V ) ∗ / / ˜ H ∗ ( K ) =0 c ∗ (cid:15) (cid:15) ˜ H ∗ ( Y ) r ∗ ∼ = / / ˜ H ∗ ( A p ( H )) ψ ∗ =0 / / ˜ H ∗ ( K )where in the right half, we have included the desired properties of the maps given in (D,E). ♦ To our outline above of the proof of Theorem 2, we now add some comments on certain details:Segev’s proof does not explicitly use the inclusion ψ (cid:0) K ( V ) (cid:1) ⊆ K that we indicated in (C);instead he establishes in (4.3) of [Seg96] just the terminal-segment ψ ( K (cid:0) D p ( H ) (cid:1) ) ⊆ K of thatinclusion. His proof there begins by assuming that the case D p ( H ) = ∅ must hold; so we indicatean explicit treatment to eliminate the seemingly-omitted case where D p ( H ) = ∅ : Namely via ourearlier reduction to nontrivial-action H < G , we may (as noted in Remark 3.2(ii)) assume thatwe have n ≥ L , L .We had also seen using Remark 3.1(ii) that components have order divisible by p ; so taking a i oforder p in each L i , we have A := h a a i ∈ A p ( H ). But then 1 = C A ( L L ) = C A ( L i ) for each i ,giving the condition defining A ∈ D p ( H ). That is, we may indeed assume that D p ( H ) = ∅ .Segev then finishes the proof as follows: In the final commutative-diagram argument aboveusing (C + E), he in effect begins at D p ( H ) rather than V —and obtains instead non-surjectivityfor i ∗ induced by the inclusion i : D p ( H ) ⊆ A p ( H )—that is, the hypothesis for Theorem 1. Thushe obtains (H-QC) in Theorem 2 as corollary of his Theorem 1. (So the use of condition (B) forthe proof of Theorem 2 remains implicit–i.e. via the proof of Theorem 1.)By contrast, our extension above of his argument obtains the inclusion ψ (cid:0) K ( V ) (cid:1) ⊆ K in (C) byapplying his inclusion ψ ( K (cid:0) D p ( H ) (cid:1) ) ⊆ K in Theorem 2 after his earlier inclusion j : V ⊆ D p ( H )from the proof Theorem 1. Thus we are quoting only part of the argument for Theorem 1, asopposed to Segev’s quoting the result . To see that our use of V and j from that argument inTheorem 1 is in fact valid for Theorem 2, we need to recall that that earlier argument depended onhaving V = ∅ , via the reduction to H < G —and we did in fact we also obtain H < G independentlyfor the proof of Theorem 2. And note furthermore that the inclusion-argument for j in Theorem 1depended only on the definition of D p ( H )—and not on the specific restriction on D p ( H ) in thehypothesis of Theorem 1. So we can indeed use that inclusion also in the proof of Theorem 2.We conclude the section with: Further remarks on generalizing Segev’s methods In our generalizations starting in the next section, we will proceed via essentially the samesequence of implications as indicated in Remark 3.9 above. However, in contrast with Segev, wewill now mainly take H to be the “local” version of the kernel on components—that is, the kernelof the permutation action on the G -orbit of a single component. So we will make appropriateadjustments to the constructions involved; notably to the factors in the join X .The diagonal subposet D p ( H ) will not usually be involved; instead, we will want to establishthe inclusion in (C) “directly”; that is, without proceeding as in [Seg96] via the inclusion of suchan intermediate poset. So to get (Goal), we will look for hypotheses on our H , which lead to asuitable poset X —allowing us to prove (D). Conditions (C) and (E) will then follow easily fromthe naturality of our construction of X (and hence K ). his sequence (C + D + E) of implications that was used for Theorem 2 provides one route—which we include in summarizing below several possible routes to (Goal) suggested by our analysis.Namely we might proceed via any of the following approaches: • Prove (A) (or (A ′ )): e.g. by showing that a ∗ : ˜ H ∗ ( Y ) → ˜ H ∗ ( Y ) is the zero map,while ˜ H ∗ ( Y ) = 0; or more generally, showing just that a ∗ is not surjective. • Prove (B): e.g. showing b ∗ : ˜ H ∗ ( V ) → ˜ H ∗ (cid:0) A p ( H ) (cid:1) is the zero map, while ˜ H ∗ (cid:0) A p ( H ) (cid:1) = 0;or more generally, showing just that b ∗ is not surjective. • Prove the following sequence: – (C): showing that inclusion—based on a natural construction of X (and hence K ); – (D): showing that ψ ∗ = 0—based on hypotheses on H , giving the construction of X ; – (E): showing that c ∗ = 0—again based on natural constructions related to X .In proving our Theorem 4.1 below, we will in fact follow the sequence (C + D + E): In particular,we will prove (C) and (E) via an appropriate new choice of X and K as mentioned above. Theconstruction of these posets is inspired by the join-construction of the corresponding posets inSegev’s work, which we described in Definitions 3.5 and 3.8; we will need comparatively slightvariations on the mechanics of those earlier join-constructions. Then we will obtain (D)—byusing a component-related technical hypothesis on H —which essentially defines our new X as ananalogous join: where now the old X i = A p (cid:0) Aut H ( L i ) (cid:1) are replaced by subposets that we call A i —which this time are defined using essentially the A p (cid:0) Aut C i ( H ) ( L i ) (cid:1) , for certain sub groups C i ( H )of H . Finally, Theorem 4.1 can be regarded as an analogue of the “core” part of Theorem 2in [Seg96]: Thus rather than assuming hypotheses on the individual L i , we will instead assumethat a suitable mapping ψ H , from A p ( H ) itself to the new X , is nonzero in homology. That is, wedesign the construction of X so that we will “automatically” get condition (D). Thus for this initialresult of ours, condition (D) is the key point. (In later Corollary 5.1, we will give an analogue moreclosely following the original hypotheses of Theorem 2 in [Seg96]—that is, where we do assumehypotheses on the individual components L i .) Remark 3.11 (Good behavior for ψ in (D)) . Our setup of implications-analysis in Remark 3.9already involves a number of different maps; and we will want to be clear about the differentbehaviors that we will want to be establishing for them. So we quickly review this aspect in ourearlier conditions:(A,A ′ ,B): We seek a non-surjective map.(D): We seek a nonzero map ψ ∗ .(E): We seek a zero map c ∗ .In fact we will mainly be focusing on (D), so that our “usual” expectation will then be a nonzeromap. So we will sometimes use the informal expression good behavior , to express the desiredoutcome of ψ ∗ = 0 in that situation. ♦ The generalized method, leading to elimination results From this section on, we begin to provide alternative versions of Segev’s results in [Seg96].While Segev’s Theorem 2 (which we stated as Theorem 1.4 in our Introduction) requires a commonbehavior for all the components of G , we show that in general we can focus on the behavior ofa single component L —if we further assume a suitable inductive hypothesis. Indeed starting inthe following Section 5, we will use inductive hypotheses such as (H1) mentioned earlier in ourIntroduction.However for our fundamental result Theorem 4.1 in this section, we use a more technical kindof inductive hypothesis—somewhat more directly akin to that for Theorem 2 of [Seg96], (whichwe discussed in the previous Section 3). The resemblance is in fact closest to the A p ( H )-formdeduced in Remark 3.6 from the original hypothesis of Theorem 2; we had also stated that form s condition (D) there. Recall that (D) required nonzero-ness in homology of the map inducedfrom the poset map ψ , which takes A p ( H ) into the formal join X of the A p (cid:0) Aut H ( L i ) (cid:1) for thecomponents L i . Similarly, when we now choose as “ H ” the kernel on the G -conjugates L i of oursingle component L , our hypothesis in 4.1 will require nonzero-ness in homology of a map ψ H defined on A p ( H )—into a corresponding new choice of X . This time the join X will arise fromfactors A i which, though still based on the A p (Aut H ( L i )) as in Theorem 2 above, are constructedin a somewhat more complicated inductive way—a way we can exploit, in verifying the hypothesisfor applications.We emphasize one further feature of working with just a single component L : Note that theproduct N := L · · · L t of the G -conjugates of L may well be proper in F ∗ ( G ), so that we can’tassume faithful action on N . For example, in the discussion of constructions after Definition 3.5,we were able to use faithfulness to embed H in the formal direct product of the Aut H ( L i ); butthis time such an embedding holds only for the faithful quotient H/C H ( N ). For this reason, ournew join X in the statement of 4.1 also includes a term A p (cid:0) C H ( N ) (cid:1) ; which later we will similarlycall A .After the proof of Theorem 4.1, the section proceeds to Proposition 4.9—which gives one possibleapproach to verifying the hypotheses of 4.1 (though we won’t actually be quoting Proposition 4.9in the later sections of this paper). That approach more explicitly involves the filtration of H given by certain centralizer-subgroups C i ( H ) below, and the building-up of the map ψ H via suc-cessive approximations ψ i . Finally, we will close the section with Theorem 4.10, which provides ageneralization of Theorem 1 of [Seg96].Thus we begin the work of the section by stating our main technical result below: Theorem 4.1. Let L ≤ G be a component of order divisible by p , with { L , . . . , L t } its G -orbitunder conjugation. Set H := T i N G ( L i ) and N := L . . . L t , with posets A i as in Definition 4.3,and poset map ψ H as in Definition 4.5. Assume also that: ψ H : A p ( H ) → X := A p (cid:0) C H ( N ) (cid:1) ∗ A ∗ . . . ∗ A t satisfies ( ψ H ) ∗ = 0 in homology.(Indeed, it suffices if there exists a subposet B ⊆ A p ( H ) with ( ψ H | B ) ∗ = 0 .)Then G satisfies (H-QC).Hence such an L is eliminated from a counterexample to (H-QC). We temporarily postpone the proof, while we provide various relevant background details. Preliminaries: construction of posets and maps for Theorem 4.1 We begin with a few further comments relevant to hypotheses:The results in [Seg96] assume the hypothesis O p ′ ( G ) = 1. However, in our Theorem 4.1 here,this can be relaxed to only requiring that some components L of G has order divisible by p ; as wewill emphasize in the proof below. In particular, in paralleling arguments from [Seg96], we mustbe careful about uses there of consequences of O p ′ ( G ) = 1 in Remark 3.1 (especially since theseare sometimes not explicit in [Seg96]). In the case of 3.1(i), properties related to faithful actioncan be handled by careful treatment of centralizers which may now be nontrivial. For example,our component L might not now be simple—but since we assume O p ( G ) = 1 in results on (H-QC),we will at least have the following partial replacement for 3.1(i):(i) L Z ( L ) is a p ′ -group.On the other hand, 3.1(ii) is now directly assumed in our hypothesis. So we will call that:(ii) L p divides the order of the component L .And then we immediately obtain the consequence 3.1(iii)—which we will name similarly:(iii) L A p ( L ) = ∅ , and hence A p ( H ) = ∅ ;as before this will again be fundamental. urthermore, later in the paper we will wish to work under hypothesis (H1): and then we willget by Remark 2.5 that either (H-QC) holds for G ; or else O p ′ ( G ) = 1 = O p ( G )—in which caseby Lemma 2.2, the components of G have order divisible by p . That is, we will then get thecondition (ii) L —giving us that hypothesis, when we want to set up to apply Theorem 4.1.Now we introduce some further terminology; including the definition of the A i needed in thestatement of Theorem 4.1 above. We will use these concepts throughout this and the remainingsections; the image-posets A G,L and A i will play a fundamental role in our constructions andarguments in the proof. Definition 4.2. For any subgroup T of G , we view the usual projection map just on A p -posets: π T : A p (cid:0) N G ( T ) (cid:1) → A p (cid:0) Aut G ( T ) (cid:1) ∪ { } . Define the image poset A G,T as the restriction of π T to A p (cid:0) N G ( T ) (cid:1) − A p (cid:0) C G ( T ) (cid:1) . Equivalently: A G,T := Im[ π T : A p (cid:0) N G ( T ) (cid:1) → A p (cid:0) Aut G ( T ) (cid:1) ∪ { } ] − { } . In Lemma 6.2, we will give some alternative descriptions of the image-poset A G,T , in the casewhere T is any subgroup of G with p not dividing the order of Z ( T ). In addition, if T is quasisimple,we will show that ˜ H ∗ ( A G,T ) = 0 (see Theorem 6.5). For later applications when T is a component L of G , this will be relevant to establishing the hypothesis of Theorem 4.1.Next we define the posets A i , based on corresponding centralizer-subgroups C i ( H ). We notethat the definition of these objects depends on the choice of the ordering of our G -orbit of com-ponents L , . . . , L t of G ; and the same holds for the definition of the subgroups C i ( H ) andmaps ψ C i ( H ) shortly thereafter. But this will not be an issue—cf. our comment after Defini-tion 3.7: namely we will only need the fact that the ψ C i ( H ) are poset maps—which holds for anychoice of that ordering. Definition 4.3. Let L , . . . , L t be a G -orbit of components of G . We define: • H := T i N G ( L i ) and N := L . . . L t ≤ H . • For 0 ≤ i ≤ t , let C i ( H ) := C H ( L i +1 . . . L t ), with C t ( H ) := C H (1) = H .Note that L i ≤ C i ( H ). • We will use the abbreviation A := A p (cid:0) C H ( N ) (cid:1) ,and write π for the identity map A p (cid:0) C H ( N ) (cid:1) → A p (cid:0) C H ( N ) (cid:1) . • For i ≥ 1, let π i be the map: π i : A p (cid:0) C i ( H ) (cid:1) → A p (cid:0) Aut C i ( H ) ( L i ) (cid:1) ∪ { } , which is induced via taking the quotient by C C i ( H ) ( L i ) = C i − ( H ). • For i ≥ 1, the image-poset A i is the poset: A i := A C i ( H ) ,L i . Using (i) L for faithfulness in Lemma 6.2, we may write A p ( L i ) ⊆ A i .Note also that an element E of A i is the quotient of an element of A p ( C i ( H )) by C C i ( H ) ( L i );so we have A i ⊆ A p (Aut C i ( H ) ( L i )) ⊆ A p (Aut H ( L i )). For the next few paragraphs, we explore the general properties of the objects defined above. So wefix a component L of G , and an ordering of its G -orbit L , . . . , L t . We may take L t := L . Finallylet H , N , C i ( H ) and A i be as in the definitions above. Remark 4.4. Note that: C ( H ) = C H ( L . . . L t ) = C H ( N ) = C G ( N ) , and by convention: C t ( H ) = C H (1) = H. Further the restriction π i : A p (cid:0) C i ( H ) (cid:1) − A p (cid:0) C C i ( H ) ( L i ) (cid:1) → A i gives a quotient map in the topological sense. oreover, we have a normal series of H given by: C H ( N ) = C ( H ) E C ( H ) E . . . E C t ( H ) = H. We also have that H , N and C H ( N ) are normal subgroups of G .Further L i ≤ C i ( H ) and C C i ( H ) ( L i ) = C i − ( H ); so Aut C i ( H ) ( L i ) = C i ( H ) /C i − ( H ), and A i isthe image of π i restricted to A p (cid:0) C i ( H ) (cid:1) − A p (cid:0) C i − ( H ) (cid:1) , since π − i (1) = A p (cid:0) C i − ( H ) (cid:1) . ♦ With the above properties established, we will now able to construct the map ψ H —still neededfor the statement of Theorem 4.1. This will be a variant of Segev’s map ψ : A p ( H ) → X , whichwe had described in Definition 3.7.For the purposes just of proving Theorem 4.1, it would be sufficient to now simply define ψ H essentially via the condition in Definition 3.7: namely for E ∈ A p ( H ), as the projection of E on Aut H ( L i E ), where i E is the largest index j for which E (cid:2) C H ( L j ). However, for the purposesof verifying the A i -related hypotheses of that theorem in later applications, it will be convenientto see below that we can “inductively” build up ψ H from successive approximations ψ i , using thefiltration of H given by the C i ( H ); the definition via i E then emerges naturally from this process: Definition 4.5 (The maps ψ i and ψ H ) . For 0 ≤ i ≤ t , set W i := A ∗ A ∗ . . . ∗ A i ; note X := W t .Define ψ C i ( H ) : A p (cid:0) C i ( H )) → W i on E ∈ A p (cid:0) C i ( H ) (cid:1) by: ψ C i ( H ) ( E ) := π k ( E ) , k = max { j ≤ i : E (cid:2) C H ( L j ) } = π k ( E ) , k = min { j ≤ i : E ≤ C j ( H ) } . Notice since E ≤ C i ( H ) that E centralizes L j for all j > i ; hence in the top line above, wecould remove the restriction “ j ≤ i ”; that is, the indicated “ k ” is indeed the index that wecalled i E in Definition 3.7: namely the largest index j for which E (cid:2) C H ( L j ). In particular,we see that ψ C i ( H ) ( E ) = π i E ( E ) lies in the image-poset A i E (and in no other A j ), and hence(since i E ≤ i ) in W i .We had seen that A p ( L i ) ⊆ A i ; this says for E ∈ A p ( L i ) that i E = i . Thus for the restrictionof ψ C i ( H ) to A p ( L i ), the usual projection becomes just the natural “inclusion”. This is the analogueof the property “ ψ | A p ( E ( G )) = π ” that we saw before Remark 3.6. Notation. We usually write ψ H , or just ψ , to abbreviate ψ C t ( H ) ; and ψ i for ψ C i ( H ) . ♦ The following is an easy consequence of the definition above and Remark 4.4: Lemma 4.6. The maps ψ C i ( H ) are order-preserving. Moreover, for all ≤ i ≤ k ≤ t , we have acommutative diagram: A p (cid:0) C i ( H ) (cid:1) (cid:127) _ (cid:15) (cid:15) ψ Ci ( H ) / / W i (cid:127) _ (cid:15) (cid:15) A p (cid:0) C k ( H ) (cid:1) ψ Ck ( H ) / / W k At this point, much as in our discussion after Definition 3.5, we see that ψ H maps A p ( H ) tothe formal join X —of posets A i , related to the individual A p (cid:0) Aut H ( L i ) (cid:1) , for L i now just in theorbit of L ; and the “trivial-action part” A containing the kernel C H ( N ) of the action of H oneach member of the orbit of L . We have now completed our preparations for: The main argument for Theorem 4.1 Thus we can finally prove our main technical result Theorem 4.1. In view of our preliminarydiscussion of its hypothesis, the proof will parallel the part of the proof of Segev’s Theorem 2 inSection 3, after we obtained the A p ( H )-form of the hypothesis in Remark 3.6: roof of Theorem 4.1. As usual in proving (H-QC), we assume that O p ( G ) = 1. We show that,under the hypotheses of this theorem, ˜ H ∗ (cid:0) A p ( G ) (cid:1) = 0; recall we had earlier called this “(Goal)”.To that end, we will follow the outline of Segev’s argument in our implications-analysis inRemark 3.9. In brief summary: We will prove that conditions (C), (D) and (E) hold. We will seethat (C) and (E) follow from our definition of the appropriate new X (and of the analogous K ).And our hypothesis, of nonzero-ness in homology of the natural poset map ψ H : A p ( H ) → X , wasdesigned to give exactly condition (D).We now begin the detailed proof:We recall that as “ H ”, we are here using the local kernel—namely the kernel of G on the orbitof L . Also we have a new “ X ”, defined as the join of the A i (see Definition 4.3). With thesechanges, our basic setup will be as in [Seg96].Just as in our earlier analysis of [Seg96], we need to first reduce to the subcase A p ( H ) ( A p ( G ).This closely follows our earlier discussion of the proof of Segev’s Theorem 2: Note that in anycase for A p ( H ), our hypothesis of ( ψ H ) ∗ = 0 forces ˜ H ∗ (cid:0) A p ( H ) (cid:1) = 0 for its domain-space. Sowhen A p ( H ) = A p ( G ), we in fact have the nonzero homology of A p ( G ) required for (Goal). Thusfor this case, the hypothesis already includes the conclusion (H-QC) of the result—as required.With this reduction to A p ( H ) ( A p ( G ) (so that H < G ) in hand, we again obtain consequencesas in earlier Remark 3.2—though now we use (iii) L above to get A p ( H ) = ∅ , in place of the usethere of 3.1(iii). So in analogy with 3.2(ii) we get:We have t ≥ p ≥ L i in our G -orbit for L .Furthermore from 3.2(i) we get nontriviality of the fundamental decomposition for A p ( G ): Let’s infact review that basic setup of [Seg96] from the discussion around Remark 3.2. Set Y := N G ( H ),with Z := A p ( G ) − A p ( H ), and Y := Y ∩ Z = N G ( H ) − A p ( H ). We recall also the earlierretraction r : Y → A p ( H ), and set V := r ( Y ). Then 3.2(i) gives us:The decomposition A p ( G ) = Y ∪ Z is nontrivial; with Y, Z, Y , V = ∅ .Then from Remark 3.3 we have the corresponding Mayer-Vietoris exact sequence: . . . → ˜ H k +1 (cid:0) A p ( G ) (cid:1) → ˜ H k ( Y ) → ˜ H k ( Y ) ⊕ ˜ H k ( Z ) → ˜ H k (cid:0) A p ( G ) (cid:1) → . . . . As before, we will get the desired conclusion ˜ H ∗ ( A p ( G )) = 0, if the map a ∗ induced in homologyby the inclusion a : Y ⊆ Y is not surjective. (Cf. condition (A) in Remark 3.4.)We saw by Remark 2.7 that the retraction r : Y → A p ( H ), given by r ( E ) = E ∩ H , is ahomotopy equivalence. The analogue of the earlier commutative diagram in Remark 3.10 is givenhere by diagram (4.1) below; and we now see similarly that we get our conclusion ˜ H ∗ (cid:0) A p ( G ) (cid:1) = 0,if in fact the r -translated map b ∗ induced by the inclusion b : V ⊆ A p ( H ) is not surjective.(Cf. condition (B) in Remark 3.4.)We now make appropriate adjustments to the earlier constructions of Segev. Recall we havealready defined some new variants, of the older constructions given at Definitions 3.5 and 3.7: X = A ∗ A ∗ . . . ∗ A t , with poset map ψ H : A p ( H ) → X .In the case where A = ∅ , we simply take X = A ∗ . . . ∗ A t .In particular we saw, in defining ψ H as ψ C t ( H ) at Definition 4.5, that for any E ∈ A p ( H ), taking i E to be the largest index j with E (cid:2) C H ( L j ), we have ψ H ( E ) = π i E ( E ) ∈ A i E . Furthermore forour new choice of X above, we define the subcomplex K in parallel with earlier Definition 3.8:Namely we write A ˆ i for the maximal sub-join given by the A j for j = i ; and then take: K := S ti =0 K ( A ˆ i ) (but exclude i = 0 when A = ∅ ).This definition means that every chain σ from K has no contribution from some A i σ , i.e. forat least one index i σ . Recall that our initial reduction to the nontrivial-action case H < G guarantees t ≥ p ≥ K = ∅ .These adjusted-constructions provide the remaining terms needed for our desired overall com-mutative diagram, in analogy with that in earlier Remark 3.10: H n ( Y ) a ∗ (cid:15) (cid:15) ( r | Y ) ∗ / / ˜ H n ( V ) b ∗ (cid:15) (cid:15) ( ψ H | V ) ∗ / / ˜ H n ( K ) =0 c ∗ (cid:15) (cid:15) ˜ H n ( Y ) r ∗ ∼ = / / ˜ H n ( A p ( H )) ( ψ H ) ∗ =0 / / ˜ H n ( X )That is: to verify the properties in the diagram, and hence complete our proof, it suffices (in viewof the discussion of implications in Remark 3.9) to establish conditions (C), (D) and (E).We show first that ψ H maps K ( V ) into K : Claim: (C) holds; that is, ψ H (cid:0) K ( V ) (cid:1) ⊆ K .Here we have a partial analogue of the proof of Segev’s inclusion j : V ⊆ D p ( H ) (for his Theorem 1,as we described in Section 3): We saw that our earlier reduction to H < G gives V = ∅ . So weconsider any σ ∈ K ( V ); and we need to get ψ H ( σ ) ∈ K . We can take σ = ( A < . . . < A l ), wherethere is E ∈ A p ( G ) with E (cid:2) H and A l = E ∩ H . So each A k ≤ A l ≤ E . Recall that H is thelocal-kernel: so the elements e ∈ E − H induce nontrivial permutations on the G -orbit { L , . . . , L t } .Let i < i < · · · < i p (with p ≥ 2) be the indices for some such nontrivial e -orbit. We will showthat each ψ H ( A k ) lies in A b i : Recall we have ψ H ( A k ) = π i Ak ( A k )—where i A k is the largest index j with A k (cid:2) C H ( L j ). We claim that i A k = i : If A k ≤ C H ( L i ), then i A k = i because we justsaw that A k (cid:2) C H ( L i Ak ). Otherwise A k (cid:2) C H ( L i ); but then as A k centralizes e , A k is alsonontrivial on the remaining members of the e -orbit, including L i p with i p > i ; so i is not the largest index j with A k (cid:2) C H ( L j ), as required above. Thus we have shown that i A k = i , andhence ψ H ( A k ) 6∈ A i —in fact for all k ; so that ψ H ( σ ) ∈ K ( A b i ) ⊆ K , as needed.Condition (D), i.e. ( ψ H ) ∗ = 0, holds since it is already part of the hypothesis; indeed this wasthe main motivation for the design of the hypothesis.Hence it remains to establish condition (E). Claim: (E) holds, that is, c : K ⊆ K ( X ) induces the zero map in homology.We had already included essentially this observation in Definition 3.8; for convenience we reviewthe argument here: Recall we followed [Seg96, p958] in describing the construction of a standardcontractible simplicial complex ˆ K , such that K ⊆ ˆ K ⊆ K ( X ); namely we gave the definition:ˆ K := S i St K ( X ) ( v i ),where the v i ∈ A i are fixed vertices, and St K ( X ) ( v i ) is the star of v i in K ( X ). This complex iscontractible (cf. [AS92, (5.1)]). Notice that at this point, we are using that A i = ∅ for all i , so thatthese vertices v i exist. So to see that A i = ∅ : For i ≥ 1, we recall that p divides the order of each L i by (ii) L , while Z ( L i ) is a p ′ -group by (i) L ; so we can use Lemma 6.2 to see that A i ⊇ A p ( L i ) = ∅ .For i = 0, if A = ∅ , then we had excluded it from the definition of X and hence of K . (If wehad A i = ∅ for some i , then K = K ( X ); and the claim would not hold unless ˜ H ∗ ( X ) = 0—inwhich case we would not be using this X .)We now complete the proof of Theorem 4.1: From (4.1), 0 = c ∗ ◦ ( ψ H | V ) ∗ = ( ψ H ) ∗ ◦ b ∗ ; soas ( ψ H ) ∗ = 0, b ∗ cannot be surjective. From our earlier analogue of (B), we get ˜ H ∗ (cid:0) A p ( G ) (cid:1) = 0—that is, (Goal).For the “it suffices” part of the statement of the theorem, note that ψ H | B = ψ H ◦ f , for theinclusion f : B ⊆ A p ( H ); so that ( ψ H | B ) ∗ 6 = 0 forces ( ψ H ) ∗ = 0. (cid:3) In the remainder of the section, we present: Further technical results via the generalized Segev method e now give some definitions and technical properties that can be useful for establishing, inapplications, the hypotheses of Theorem 4.1. These are inspired by the filtration of A p ( H ) deter-mined by the subposets A p (cid:0) C i ( H ) (cid:1) of sub groups of H (from Definition 4.3): which we use now forapproaching the posets A i (of A p -posets of quotient groups of the C i ( H )) appearing as the factorsin the join X —where of course X is the codomain-space for our crucial mapping ψ H . Indeed wecould roughly view the process of such successive subgroup-by-quotient replacements as graduallytransforming the filtration of A p ( H ) via the A p (cid:0) C i ( H ) (cid:1) , into our filtration of X determined bythe sub-joins W i (from the A j for j ≤ i ).So below we consider maps taking the i -th subgroup-term A p (cid:0) C i ( H ) (cid:1) to a version of itself—inwhich we roughly “shrink the top part” to a join-factor, given by the poset of quotients of itsmembers appearing in A i . We can then view our earlier map ψ i as being defined downward (inthe ordering on components) from the i -th stage of these shrinkings; and then our final map ψ H arises as we proceed upward through the successive “approximations” ψ i . Thus our filtration-transforming process inductively builds toward our goal of homology propagation from A p ( H )to A p ( G ): and along the way we will be propagating good behavior in homology (in the sense ofRemark 3.11) for the intermediate maps ψ i leading to our final ψ t = ψ H .Hence for 1 ≤ i ≤ j ≤ t , we define the following maps: ϕ i : A p (cid:0) C i ( H )) → A p (cid:0) C i − ( H ) (cid:1) ∗ A i (4.2) ϕ i ( E ) := ( E ∈ A p (cid:0) C i − ( H ) (cid:1) if E ≤ C i − ( H ) ,π i ( E ) ∈ A i if E (cid:2) C i − ( H ) . (4.3) Φ i,j : A p (cid:0) C i ( H ) (cid:1) ∗ A i +1 ∗ . . . ∗ A j → A p (cid:0) C i − ( H ) (cid:1) ∗ A i ∗ A i +1 ∗ . . . ∗ A j (4.4) Φ i,j ( E ) := ( ϕ i ( E ) ∈ A p (cid:0) C i − ( H ) (cid:1) ∗ A i if E ∈ A p (cid:0) C i ( H ) (cid:1) ,E ∈ A k if E ∈ A k , i + 1 ≤ k ≤ j. (4.5)The following lemma summarizes the relations between the maps Φ, ϕ and ψ : Lemma 4.7 (Relations Φ − ψ ) . The following hold:(1) Φ i,j = ϕ i ∗ Id A i +1 ∗ . . . ∗ Id A j .(2) Φ i,i = ϕ i .(3) ψ i = Φ ,i ◦ . . . ◦ Φ i − ,i ◦ Φ i,i .(4) If ϕ i induces an epimorphism in all the homology groups of degree ≤ n ,then so does Φ i,j in all the homology groups of degree ≤ n + ( j − i ) , for all j ≥ i .(5) If ϕ i induces a monomorphism in all the homology groups of degree ≤ n ,then so does Φ i,j in all the homology groups of degree ≤ n + ( j − i ) , for all j ≥ i .Proof. Parts (1), (2) and (3) are straightforward. For (4) and (5), note that the tensor productis exact over the field Q , and that Φ i,j = ϕ i ∗ Id A i +1 ∗ . . . ∗ Id A j induces in homology the tensorproduct map ( ϕ i ) ∗ ⊗ (Id A i +1 ) ∗ ⊗ . . . ⊗ (Id A j ) ∗ (see equation 2.1). (cid:3) We consider now the two properties below, which will encode sufficient conditions to propagatehomology from the posets A p ( C i ( H )) to A p ( H ). Then in Proposition 4.9, we will indicate one wayof guaranteeing the conditions of Theorem 4.1 from these properties. Later we will indicate stillother ways of using these properties.We begin by describing the common setup for both properties: Definition 4.8 (The map ϕ for a normal L ) . Assume L E G is a normal component of G ; anddenote by π : A p ( G ) → A p (cid:0) Aut G ( L ) (cid:1) ∪ { } the map induced by the quotient. Set A := A G,L ; and efine ϕ : A p ( G ) → A p (cid:0) C G ( L ) (cid:1) ∗ A by: ϕ ( E ) = ( E ∈ A p (cid:0) C G ( L ) (cid:1) if E ≤ C G ( L ) ,π ( E ) ∈ A if E (cid:2) C G ( L ) . We will apply this to successive components L i via our filtration. ♦ In this context, we define first the epimorphism-property: Property E(n). We have an n ≥ 0, for which in the setup of Definition 4.8, we get: ϕ ∗ : ˜ H m (cid:0) A p ( G ) (cid:1) → ˜ H m ( A p (cid:0) C G ( L ) (cid:1) ∗ A ) is an epimorphism for all m ≤ n .We then define analogously the monomorphism-property: Property M(n). We have an n ≥ 0, for which in the setup of Definition 4.8, we get: ϕ ∗ : ˜ H m (cid:0) A p ( G ) (cid:1) → ˜ H m ( A p (cid:0) C G ( L ) (cid:1) ∗ A ) is a monomorphism for all m ≤ n .The hypotheses of Proposition 4.9 below give one possible approach to conditions which are suffi-cient, in terms of the above Properties E( n ) and M( n ), to establish the hypothesis of Theorem 4.1.For 4.9, we will not assume the inductive hypotheses (H1) or (H L ( p )) mentioned earlier (though we will later assume them in some other results which involve the properties). Instead we will assumea special inductive assumption on the homology of certain relevant p -subgroup posets. Note alsothat n + t − i ≥ i ≤ n + t —since i ≤ t and n ≥ Proposition 4.9. Let L be a component of G of order divisible by p , and let { L , . . . , L t } denoteits G -orbit under conjugation. Let H := T i N G ( L i ) and N := L · · · L t . Suppose that thereexists n ≥ such that one of the following holds:(1) ˜ H n (cid:0) A p ( H ) (cid:1) = 0 , and for each ≤ i ≤ t , (cid:0) L i , C i ( H ) (cid:1) has Property M( n − t + i ).(2) ˜ H n ( A p (cid:0) C H ( N ) (cid:1) ∗ A ∗ . . . ∗ A t ) = 0 ,and for each ≤ i ≤ t , (cid:0) L i , C i ( H ) (cid:1) has Property E( n − t + i ).Then the hypotheses of Theorem 4.1 hold, so that G satisfies (H-QC).Proof. We will show that (1) (resp. (2)) fulfills the hypotheses of Theorem 4.1. By Lemma 4.7(3)with t in the role of “ i ” there, we have ψ H = ψ t = Φ ,t ◦ . . . ◦ Φ t,t . We now fix some index i ≥ i,t in the above composition. Note first that C C i ( H ) ( L i ) = C i − ( H ): sowe can use L i , C i ( H ) in the roles of “ L , G ” in the setup of Definition 4.8, to see that the roleof “ ϕ ” there is played here by the map ϕ i defined just before Lemma 4.7. Now by hypothesis wehave Property M( n + t − i ) (resp. Property E( n + t − i )) for the pair (cid:0) L i , C i ( H ) (cid:1) . This meansthat ϕ i : A p (cid:0) C i ( H ) (cid:1) → A p (cid:0) C i − ( H ) (cid:1) ∗ A i induces a monomorphism (resp. an epimorphism) in allhomology groups of degree ≤ n − t + i . So by (5) (resp. (4)) of Lemma 4.7, with t, n + t − i in theroles of “ j, n ”, we see that Φ i,t induces monomorphisms (resp. epimorphisms) in all the homologygroups of degree ≤ ( n − t + i ) + ( t − i ) = n . So the composition ψ H of all the Φ i,t induces amonomorphism (resp. an epimorphism) in the n -th homology. Since the n -th homology group ofthe domain (resp. codomain) of ψ H is nonzero by hypothesis, ψ H does not induce the zero map inthe n -th homology group of A p ( H )—so we can apply Theorem 4.1, and complete the proof. (cid:3) To finish this section, we present a local version of Theorem 1 of [Seg96]. For that purpose,we define a local-diagonal poset D p ( H ), analogous to that of [Seg96]—which we described afterRemark 3.4. So let L , . . . , L t be an orbit of components of G , and let H := T i N G ( L i ) be thelocal kernel on these components. Let D p ( H ) be the subposet of elements A ∈ A p ( H ) such thatthere exists J ⊆ { , . . . , t } with | J | ≥ C A ( { L j : j ∈ J } ) = C A ( L i ) for all i ∈ J .We now state our version of Theorem 1 of Segev. In contrast with the original theorem of Segev,we do not require O p ′ ( G ) = 1, nor any extra inductive hypothesis on the components: heorem 4.10. Let L be a component of G of order divisible by p , and L , . . . , L t its G -orbit.Let H be the local kernel T i N G ( L i ) . If D p ( H ) → A p ( H ) is not surjective in homology, then G satisfies (H-QC).Proof. As usual for (H-QC) we assume that O p ( G ) = 1; and we must show that ˜ H ∗ (cid:0) A p ( G ) (cid:1) = 0.We first follow the initial arguments in the proof of Segev’s Theorem 1 in our discussion in Section 3.Indeed those arguments also provide the basic setup for the proof of our Theorem 4.1 in this section.So we begin by essentially quoting a number of features from our proof of Theorem 4.1:Since we no longer have O p ′ ( G ) = 1 and its consequences in Remark 3.1, we will use insteadthe partial replacements that we called (i) L , (ii) L , and (iii) L .We also obtain the usual reduction to A p ( H ) ( A p ( G ), and hence nontrivial action H < G :Namely in any case for H , the hypothesis of non-surjectivity of a map to ˜ H ∗ (cid:0) A p ( H ) (cid:1) guaranteesthat that this homology is at least nonzero; so that in the trivial-action case where A p ( H ) = A p ( G ),this nonzero-ness already includes the conclusion for (H-QC).So using that reduction along with (iii) L , as in Remark 3.2 we obtain t ≥ p ≥ A p ( G ) = Y ∪ Z , with Y, Z, Y , V = ∅ . This lastgives the usual Mayer-Vietoris sequence in Remark 3.3. Hence from our implications-analysis inRemark 3.4, it will suffice to prove that (B) holds.We now follow Segev’s argument as in our discussion of the proof of his Theorem 1:Namely we will show that V ⊆ D p ( H ). That is, we need to show that b : V → A p ( H ) is notsurjective in homology, where we recall that V = { E ∩ H : E ∩ H > , E ∈ A p ( G ) − A p ( H ) } . Forif we can establish that non-surjectivity, we can exploit the commutative diagram:˜ H ∗ ( V ) b (cid:15) (cid:15) i / / ˜ H ∗ ( D p ( H )) d w w ♣♣♣♣♣♣♣♣♣♣♣ ˜ H ∗ (cid:0) A p ( H ) (cid:1) where the maps are induced by the inclusions. Since d is not surjective by hypothesis, b = d ◦ i isnot surjective—and hence (B) holds. Therefore ˜ H ∗ ( A p ( G )) = 0, so that we would then completethe proof.Thus it remains to prove that V ⊆ D p ( H ). The proof follows Segev’s original; and indeedessentially the same idea is used for Claim (C) in our proof of Theorem 4.1: We saw abovethat V = ∅ ; so we now consider any A ∈ V : Then A = E ∩ H , where E ∈ A p ( G ) − A p ( H )satisfies E ∩ H > 1. Note that any nontrivial e ∈ E − H must induce a nontrivial action onthe G -orbit { L , . . . , L t } . Let O be a nontrivial orbit of this e -action—so that |O| = p ≥ 2. Notethat nontrivial elements of A ≤ C G ( e ) must exhibit isomorphic actions on all members L i of O ;hence we get C A ( O ) = C A ( L i ) for each L i ∈ O , where |O| ≥ A ∈ D p ( H ). So we have now completed the proof. (cid:3) Some more concrete consequences of the generalized method In this section, we establish several corollaries of our primary technical Theorem 4.1; includingour main result Theorem 1.6.First for Segev’s Theorem 2 in [Seg96] (which we stated as Theorem 1.4 here), we will deducethe local version from our Theorem 4.1—after assuming one of the inductive hypotheses (H1)or (H L ( p )). Recall we saw that Theorem 4.1 is an analogue of Segev’s Theorem 2 which assumesthe A p ( H )-form of the hypothesis in Remark 3.6; below we instead assume hypotheses on theindividual components L i , in analogy with the original hypotheses of Theorem 2: orollary 5.1. Let L be a component of G , and L , . . . , L t its G -orbit. We write H := ∩ i N G ( L i ) for the local-kernel on the orbit; and we recall the posets A i from Definition 4.3. Assume that G satisfies (H1) or (H L ( p ) ).If A p ( L i ) → A i is nonzero in homology for each i ≥ , then G satisfies (H-QC).In particular, this holds if A p ( L ) → A is nonzero in homology, where A is one of the following: A H,L , A G,L , A p (cid:0) Aut H ( L ) (cid:1) , A p (cid:0) Aut G ( L ) (cid:1) , A p (cid:0) Aut( L ) (cid:1) . Proof. As usual for (H-QC), we assume that O p ( G ) = 1. By Remark 2.5, we can assumethat (H L ( p )) holds. In particular, p divides the order of L ; and as in the proof of Theorem 4.1,we have the partial replacements (i) L , (ii) L , and (iii) L for Remark 3.1. And then as in Defini-tion 4.3 (which used (i) L for faithfulness in Lemma 6.2), we see that the maps A p ( L i ) → A i in thehypothesis may be regarded as inclusions A p ( L i ) ⊆ A i .First, we prove the “In particular” reduction of the Theorem:Fix i , and choose g ∈ G with L i = L g . Let A be one of the posets in the statement: A H,L , A G,L , A p (cid:0) Aut H ( L ) (cid:1) , A p (cid:0) Aut G ( L ) (cid:1) , A p (cid:0) Aut( L ) (cid:1) . Since we saw in Definition 4.3 that A i ⊆ Aut H ( L i ), we can extend our earlier inclusion-statementto A p ( L i ) ⊆ A i ⊆ A g for any of these A —where A g is the analogous A -poset for L g obtained viathe conjugation action. Now we are assuming for the “In particular” statement that A p ( L ) → A isnot the zero map in homology; then by the conjugation action, neither is A p ( L i ) = A p ( L g ) → A g .Hence A p ( L i ) → A i is not the zero map in homology—and we have reduced to the main hypothesisof Corollary 5.1, as claimed.Now we turn to the main proof of Corollary 5.1:In view of our discussion of the hypothesis, we will see that the proof parallels the deductionof the A p ( H )-form of the hypothesis of Segev’s Theorem 2, in our earlier discussion leading up toRemark 3.6.As a brief initial overview: The earlier direct product group E ( G ) = L × · · · × L n will now bereplaced by the “inner automorphism” part of H , namely H := C H ( N ) L · · · L t —which this timeis instead a central product, over a p ′ -center. We still get that A p ( H ) is the join of correspondingfactors A p (cid:0) C H ( N ) (cid:1) and A p ( L i ); with homology given by the tensor product of the homology ofthe individual terms. Now our hypothesis for Corollary 5.1 is that each of the latter terms gives anonzero image in ˜ H ∗ ( A i ) when i ≥ 1; while we get ˜ H ∗ ( A p (cid:0) C H ( N ) (cid:1) ) = 0 using (H L ( p )), and thismaps via the identity to ˜ H ∗ ( A ). Hence the tensor product of these nonzero images is nonzero inthe homology ˜ H ∗ ( X ) of the join X of the A i . Thus for “ B ” given by A p ( H ), we get that ψ H | B is nonzero in homology—namely the hypothesis of Theorem 4.1, which we quote to complete theproof of (H-QC).We now expand the above rough outline to a more formal argument: Let ψ H : A p ( H ) → X denote the map in Theorem 4.1; we want to verify the hypothesis there that the map ψ H | B shouldbe nonzero in homology. Now ψ H | B arises from B = A p ( H ) ⊆ A p ( H ) followed by ψ H . Andwe still have the analogue of the property in the discussion before Remark 3.6, as indicated inDefinition 4.5, that the restriction of ψ H to each L i is the “inclusion” A p ( L i ) ⊆ A i above (andthe identity on A p (cid:0) C H ( N ) (cid:1) = A ). This means that the restriction ψ H := ψ H | A p ( H ) just givesthe standard homotopy equivalence of A p ( H ) for the central-product group H , with the join ofthe A p -posets for its factors C H ( N ) and L i (cf. [Qui78, Prop 2.6] or Proposition 2.1). That is, wehave the maps: A p (cid:0) H ) ψ H → A p (cid:0) C H ( N ) (cid:1) ∗ A p ( L ) . . . ∗ A p ( L t ) j → A ∗ A ∗ . . . ∗ A t = X, where ψ H is a homotopy equivalence, and j is an inclusion. In particular, we have ψ H | B = j ◦ ψ H .Furthermore, letting j i : A p ( L i ) → A i denote the inclusion, with j : A p (cid:0) C H ( N ) (cid:1) → A p (cid:0) C H ( N ) (cid:1) he identity map, we have j = j ∗ j ∗ . . . ∗ j t as a fuller expression for the inclusion in the joinof the posets. Hence the map induced by j in homology is j ∗ = ( j ) ∗ ⊗ ( j ) ∗ ⊗ . . . ⊗ ( j t ) ∗ . Themap j ∗ is nonzero since each map ( j i ) ∗ for i ≥ ψ H | B ) ∗ = ( j ◦ ψ H ) ∗ = j ∗ ◦ ( ψ H ) ∗ (where ψ H isa homotopy equivalence) is a nonzero map. This give the hypothesis for Theorem 4.1, which wequote to complete the proof of (H-QC). (cid:3) Note that Theorem 1.6 is just the particular choice A = Aut H ( L ) in Corollary 5.1. The twofinal statements follow, since they involve the further-specialized case of equality—that is, wehave L = Aut H ( L ), so that we are considering the image just of the identity map: and thehomology of A p ( L ) is nonzero by Theorem 6.5, since L is quasisimple with p ′ -center by (i) L .We give next an alternative corollary of Theorem 4.1: It provides an elimination result forthe case where (roughly) outer p -automorphisms do occur, but they are “separated”—rather thanbeing embedded diagonally across several components: the model such case would be just a directproduct of automorphism groups of individual components. In particular, the value t = 1 coversthe case of a normal component: Corollary 5.2. Let L be a component of G and L , . . . , L t its G -orbit. Assume further that G satisfies (H1) or (H L ( p ) ), and also that:(i) For each ≤ i ≤ t , there exists a subgroup F i such that: L i ≤ F i ≤ N G ( L i ) , [ F i , C G ( L i )] = 1 , and F i ∩ C G ( L i ) is a p ′ -group.(ii) A p (cid:0) N G ( L i ) (cid:1) = A p (cid:0) F i C G ( L i ) (cid:1) for all i .(iii) If i = j then [ F i , F j ] = 1 .Then G satisfies (H-QC).Proof. In view of Remark 2.5, since we are trying to prove (H-QC), we can work directly un-der (H L ( p )). As usual for (H-QC) we assume that O p ( G ) = 1.We will prove that the above special conditions on the F i yield ( ψ H ) ∗ = 0—giving the hypothesesof Theorem 4.1, and hence ˜ H ∗ (cid:0) A p ( G ) (cid:1) = 0.By hypothesis (i), F i ≤ C G (cid:0) C G ( L i ) (cid:1) ; so C G ( L i ) ≤ C G ( F i ), and hence C G ( L i ) = C G ( F i ). Itfollows that Z ( F i ) = F i ∩ C G ( F i ) is a p ′ -group, again using (i).When t ≥ i = j , we get [ F i , F j ] = 1 by hypothesis (iii), so F i ≤ C G ( F j ); and then we seethat F i ∩ F j ≤ Z ( F i ) ∩ Z ( F j ) is a p ′ -group. Together with hypothesis (ii) and Lemma 2.3, theseobservations allow us to conclude that:(5.1) A p ( H ) = A p (cid:0) F . . . F t C H ( F . . . F t ) (cid:1) , and C H ( F . . . F t ) = C G ( F . . . F t ) = C G ( L . . . L t ). Moreover, F . . . F t C H ( F . . . F t ) is a centralproduct by p ′ -centers. Let N := L . . . L t , so that C G ( N ) = C H ( N ) = C H ( F . . . F t ). (Noticethat (5.1) and these conditions show that here, H itself has some of the formal properties of thegroup H in the proof of Corollary 5.1.)Now we verify the hypotheses of Theorem 4.1: Namely we will show that the map given thereby ψ H : A p ( H ) → A p (cid:0) C H ( N ) (cid:1) ∗ A ∗ . . . ∗ A t =: X is not zero in homology—because it is ahomotopy equivalence, onto the join X —which has nonzero homology as we will see below. Recallfrom Definitions 4.2 and 4.3 that we have A i = Im( A p (cid:0) C i ( H ) (cid:1) → [ A p (cid:0) Aut C i ( H ) ( L i ) (cid:1) ∪{ } ] ) −{ } ,where we recall that C i ( H ) = C H ( L i +1 . . . L t ).First, we claim that A i = A p (cid:0) F i /Z ( F i ) (cid:1) . When t ≥ i = j , by hypothesis (iii) alongwith C G ( F j ) = C G ( L j ) above, we have: F i ≤ \ j = i C G ( F j ) = \ j = i C G ( L j ) ≤ C i ( H ) . oreover using (i), [ F i , C i − ( H )] ≤ [ F i , C G ( L i )] = 1, and F i ∩ C i − ( H ) ≤ F i ∩ C G ( L i ) is a p ′ -group. Therefore, if A, B ∈ A p ( F i ) with AC i − ( H ) = BC i − ( H ), then it is not hard to seethat A = B . Now for A ∈ A p (cid:0) N G ( L ) (cid:1) , by (ii) we have A ≤ A × A , where A is theprojection of A onto F i , and A is the projection of A onto C G ( L i ). If we take the quotientby C i − ( H ), we see that AC i − ( H ) /C i − ( H ) = A . Hence the image of the map that goesfrom A p (cid:0) C i ( H ) (cid:1) into A p (cid:0) Aut C i ( H ) ( L i ) (cid:1) ∪ { } is exactly A p (cid:0) F i /Z ( F i ) (cid:1) ∪ { } ; so we do indeedhave A i = A p (cid:0) F i /Z ( F i ) (cid:1) .Second, we prove that the map ϕ i : A p (cid:0) C i ( H ) (cid:1) → A p (cid:0) C i − ( H ) (cid:1) ∗ A i is a homotopy equivalencefor each i : In the case A ∈ A p (cid:0) F i /Z ( F i ) (cid:1) , A can be viewed in A p ( F i ) since Z ( F i ) is a p ′ -group, so: ϕ − (cid:0) ( A p (cid:0) C i − ( H ) (cid:1) ∗ A i ) ≤ A (cid:1) = A p (cid:0) AC i − ( H ) (cid:1) , which is contractible since O p (cid:0) AC i − ( H ) (cid:1) ≥ A > 1. In the other case A ∈ A p (cid:0) C i − ( H ) (cid:1) , we see: ϕ − (cid:0) ( A p (cid:0) C i − ( H ) (cid:1) ∗ A i ) ≤ A (cid:1) = A p ( A ) , which is also contractible. Therefore, by Quillen’s fiber lemma (cf. Proposition 2.1), ϕ i is ahomotopy equivalence. And then using Lemma 4.7: each Φ i,t is a homotopy equivalence via therelation Φ i,t = ϕ i ∗ Id A i +1 ∗ . . . ∗ Id A t ; and so we get that ψ H is a homotopy equivalence, since wein fact get ψ H = Φ ,t ◦ . . . ◦ Φ t,t as the composition of homotopy equivalencesIt remains to show that ψ H is not the zero map in homology. By (H L ( p )), C H ( N ) sat-isfies (H-QC); and O p (cid:0) C H ( N ) (cid:1) = 1, since C H ( N ) is normal in G with O p ( G ) = 1. There-fore ˜ H ∗ ( A p (cid:0) C H ( N ) (cid:1) ) = 0. Now, for each i , the quotient F i /Z ( F i ) is an almost simple group, soby [AK90] (cf. Theorems 6.4 and 6.5 below) we have that ˜ H ∗ ( A p (cid:0) F i /Z ( F i ) (cid:1) ) = 0. Finally bythe homology decomposition of the join of spaces (see equation 2.1), we get nonzero homology forthe join X = A p (cid:0) C H ( N ) (cid:1) ∗ A p (cid:0) F /Z ( F )) ∗ . . . ∗ A p (cid:0) F t /Z ( F t ) (cid:1) ; and so since ψ H is a homotopyequivalence of A p ( H ) with X , it is not the zero map in homology. And this is what remained tobe established. (cid:3) Some properties of the image poset in the generalized method This section begins with some properties of the image poset A G,L . We then show ˜ H ∗ ( A G,L ) = 0when L is quasisimple—extending the almost-simple case in Aschbacher-Kleidman [AK90] to thealmost-quasisimple case, via posets of this “Aut G ( L )”-form. Later, we also give concrete sufficientconditions, on the centralizers of whatever outer p -automorphisms of a component L do occur in G ,which will allow us to establish (H-QC). Definition 6.1. Let L be a subgroup of G . If A ∈ A p (cid:0) N G ( L ) (cid:1) is an elementary abelian p -subgroupsuch that A ∩ (cid:0) LC G ( L ) (cid:1) = 1, then we say that A is a p -outer of L in G . (We emphasize thatthe word outer is here being used in the “purely outer” sense—that A contains no nontrivial innerautomorphisms.) We define the corresponding posets of p -outers by:I G ( L ) := { A ∈ A p (cid:0) N G ( L ) (cid:1) : A is p -outer on L } ˆI G ( L ) := I G ( L ) ∪ { } . If P ⊆ A p ( G ), let I P ( L ) := P ∩ I G ( L ) and ˆI P ( L ) := I P ( L ) ∪ { } .Sometimes we work in subposets A ⊆ A G,L of the poset in Definition 4.2. Since the members A of A are then quotients of subgroups of N G ( L ) modulo C G ( L ), the p -outer condition becomes ineffect just A ∩ (cid:0) L/Z ( L ) (cid:1) = 1. So we get:I A ( L ) = A − N A ( L ) ,in the language of the inflated poset N ( − ).We say that L admits only cyclic p -outers in G if I G ( L ) is non-empty, and its members arecyclic (rather than of rank ≥ p -outers have order p —and there are noposet inclusions among them. For example, if Out( L ) has cyclic Sylow p -subgroups, then eitherthe cyclic-only case or the empty case holds. ater we will use the following properties: Assume that A ∈ I G ( L ), for an L which admits onlycyclic p -outers; set A := A G,L and N := N A ( L ). Then when E ∈ A >A , by the order- p propertyof cyclic-only, we have E = A × B for nontrivial B ∈ A p ( L ), so that E ∈ N A ( L ); thus we seethat A >A = N >A . ♦ We now use the above notion of p -outers to further study the properties of the image poset A G,L in Definition 4.2. So recall that if L ≤ G is a subgroup and π : A p (cid:0) N G ( L ) (cid:1) → A p (cid:0) Aut G ( L ) (cid:1) ∪ { } is the map induced by the quotient N G ( L ) → Aut G ( L ), then: A G,L = π ( A p (cid:0) N G ( L ) (cid:1) ) − { } = π ( A p (cid:0) N G ( L ) (cid:1) − A p (cid:0) C G ( L ) (cid:1) ) . In the following lemma, we will characterize the poset A G,L in terms of the p -outers I G ( L ) of L .And then after that, we will prove that ˜ H ∗ ( A G,L ) = 0 if L is a quasisimple subgroup of G ,by using [AK90]. This quasisimple result can be seen as a small improvement of the almost-simple case of Quillen’s conjecture in [AK90], since we show that the conjecture also holds for theposet A G,L consisting of quotients, and not only for posets consisting of subgroups, such as A p ( T )with F ∗ ( T ) = L .Thus as just indicated, we first provide some different descriptions of the image poset. (We usethe bar notation ¯ E to denote the image of an element E under the map π —notably when E isa p -outer.) Lemma 6.2 (Description of A G,L ) . Let L ≤ G be a subgroup with p ′ -center, and let: π : A p (cid:0) N G ( L ) (cid:1) → A p (cid:0) Aut G ( L ) (cid:1) ∪ { } be the map induced by the quotient N G ( L ) → Aut G ( L ) . Then since Z ( L ) is a p ′ -group, π em-beds A p ( L ) ∼ = A p ( L ) into A p (cid:0) Aut G ( L ) (cid:1) , and hence into A G,L . So we will identify A p ( L ) with its(barred) image via π ; and it is also convenient if slightly abusive to just write unbarred L in placeof L itself. Then we have that: A G,L = [ E ∈ ˆI G ( L ) A p ( LE ) . Equivalently, A G,L is isomorphic to the poset: { EC G ( L ) : E ∈ A p (cid:0) N G ( L ) (cid:1) − N G (cid:0) C G ( L ) (cid:1) } = { EC G ( L ) : E ∈ A p (cid:0) N G ( L ) (cid:1) , C E ( L ) = 1 } . In particular, we have that: I A G,L ( L ) = { E : E ∈ I G ( L ) } , and if I G ( L ) = ∅ (i.e. there are no p -outers), then A G,L = A p ( L ) .Proof. Since Z ( L ) is a p ′ -group by hypothesis, we see that:Members of A p ( L ) act faithfully on L .In particular, we have a natural poset isomorphism A p ( L ) → A p (cid:0) L/Z ( L ) (cid:1) induced by the quotientmap L → L/Z ( L ). Therefore, without loss of generality: In the remainder of the proof, we assume that Z ( L ) = 1.Let A, B ∈ A p ( L ) such that ¯ A ≤ ¯ B . Then AC G ( L ) ≤ BC G ( L ). Since A and B commutewith C G ( L ), and A ∩ C G ( L ) = 1 = B ∩ C G ( L ) by the condition Z ( L ) = 1, we get A ≤ B . Thisshows that A p ( L ) embeds into A p (cid:0) Aut G ( L ) (cid:1) naturally via π .Now we show that A G,L = S E ∈ ˆI G ( L ) A p ( LE ): Consider any E ∈ A p (cid:0) N G ( L ) (cid:1) . Then we canwrite E = (cid:0) E ∩ LC G ( L ) (cid:1) × E , for some complement E to E ∩ (cid:0) LC G ( L ) (cid:1) . So for that purely-outerpart we have E ∈ ˆI G ( L ). Write E for the projection of E ∩ (cid:0) LC G ( L ) (cid:1) into L . Note that E and E commute, and ( E E ) ∩ C G ( L ) = 1. Then: E = EC G ( L ) / C G ( L ) = E E C G ( L ) / C G ( L ) ∼ = E E . Finally, since E ≤ L , we have that E ∈ A p ( LE ). y the isomorphism theorems, the poset A G,L is isomorphic to the poset of subgroups: { EC G ( L ) : E ∈ A p (cid:0) N G ( L ) (cid:1) − A p (cid:0) C G ( L ) (cid:1) } . If E ∈ A p (cid:0) N G ( L ) (cid:1) − A p (cid:0) C G ( L ) (cid:1) , write E = E C E ( L ) for some complement E to C E ( L ) in E .In this case note that E = 1. Then EC G ( L ) = E C G ( L ), and E / ∈ N G (cid:0) C G ( L ) (cid:1) . Therefore: A G,L ∼ = { EC G ( L ) : E ∈ A p (cid:0) N G ( L ) (cid:1) − A p (cid:0) C G ( L ) (cid:1) } = { EC G ( L ) : E ∈ A p (cid:0) N G ( L ) (cid:1) − N G (cid:0) C G ( L ) (cid:1) } . The “In particular” part is clear from these descriptions of A G,L . (cid:3) In particular if L is simple, then it acts faithfully on itself. And in that case, we can regardits mapping into the quotient Aut G ( L ) just as an inclusion; and so using Lemma 6.2 we canwrite A G,L = S A ∈ ˆI G ( L ) A p ( L ¯ A ) ⊆ A p (cid:0) Aut G ( L ) (cid:1) . This subposet may not be a poset of subgroups;i.e. may not be of the form A p ( T ) for some almost-simple group T ≤ Aut G ( L )—but it is somewhatanalogous. Our aim now is to show that A G,L behaves like the Quillen poset of an almost-simplegroup, so that ˜ H ∗ ( A G,L ) = 0. We use the generalized version of Robinson’s Lemma due toAschbacher-Smith [AS93, Sec 5], together with the proofs in [AK90] of the almost-simple case ofthe conjecture. Lemma 6.3 ([AS93, Lemma 0.14]) . Suppose that a q -hyperelementary group H acts on a poset Y with ˜ H ∗ ( Y ) = 0 . Then ˜ χ ( Y H ) ≡ q . In particular, Y H is non-empty. The almost simple case of (H-QC) is a consequence of the above lemma and the main theoremsof [AK90]. We summarize the results that we will need from [AK90] in the theorem below. Theorem 6.4 (Aschbacher-Kleidman) . Let T be almost-simple. The following are equivalent:(1) For all hyperelementary nilpotent p ′ -subgroups H of F ∗ ( T ) , the fixed-point set S p ( T ) H isnot empty.(2) p = 2 , F ∗ ( T ) = L (4) and | | T : F ∗ ( T ) | .Moreover, (H-QC) holds for almost-simple groups. Theorem 3 of [AK90] establishes (H-QC) for almost-simple groups by using the equivalence ofthe above theorem. However, the proof of Theorem 3 has a small gap—which can be easily fixed,as we show below: Theorem 6.5 (Homology for A G,L ) . Suppose that L ≤ G is a quasisimple subgroup with p ′ -center,and let A G,L be the image poset. Then ˜ H ∗ ( A G,L ) = 0 .Proof. Without loss of generality, we can assume that Z ( L ) = 1. Also write A := A G,L . Recallfrom Lemma 6.2 that A p ( L ) ⊆ A .Assume first that part (2) of Theorem 6.4 does not hold for T := hAi ≤ Aut G ( L ). Then thereexists a q -hyperelementary p ′ -subgroup H ≤ L such that S p ( T ) H is the empty set. Note that H acts on A , and in particular, A H ⊆ S p ( T ) H is empty. By Lemma 6.3, ˜ H ∗ ( A ) = 0.Thus we may assume instead that part (2) of Theorem 6.4 does hold; that is, L = L (4)and 4 | | T : L | . In this case, following the original proof of the almost-simple case on p211of [AK90], we take P ∈ Syl ( L ), which is cyclic of order 5. In the original proof of Aschbacher-Kleidman, it is stated that A p (cid:0) Aut( L ) (cid:1) P has exactly three points: a subgroup of order 2, and twosubgroups of order 16. But this is not correct, since we have in fact five points in A p (Aut( L )) P .Namely, A p ( L ) P is discrete with two points (two subgroups of order 16), and A p (cid:0) Aut( L ) (cid:1) P isdiscrete with five points (two subgroups of order 16, and three subgroups of order 2). In any case,if a subposet Y ⊆ A p (cid:0) Aut( L ) (cid:1) containing A p ( L ) is stable under the action of P , then Y P is adiscrete poset of cardinality between 2 and 5. Therefore, ˜ χ ( Y P ) is between 1 and 4 mod 5, and in Recall that a q -hyperelementary group is a group H such that O q ( H ) is cyclic. articular is nonzero. By Lemma 6.3, ˜ H ∗ ( Y ) = 0. In particular, this holds if Y = A , since P ≤ L and L acts on A ; and also holds if Y = A p ( T ) for some almost-simple group T with F ∗ ( T ) = L .This fixes the small gap in [AK90], for showing nonzero homology of A p ( T ); and also finishes ourproof for A . (cid:3) Remark 6.6. The above theorem can be strengthened (details will appear elsewhere) to showthat A G,L in fact satisfies the Lefschetz-character version of Quillen’s conjecture. That is, we havethe stronger conclusion that ˜ χ ( A G,Lg ) = 0 for some g ∈ L . ♦ Remark 6.7. Lemma 6.2 and Theorem 6.5 provide some properties for the image poset A G,L ,for L ≤ G quasisimple with p ′ -center, which suggest that it behaves much like an A p -poset.Moreover, in view of Lemma 6.2, the image poset A G,L can be easily described in terms ofthe p -outer poset I G ( L ) = A p (cid:0) N G ( L ) (cid:1) − N G (cid:0) LC G ( L ) (cid:1) . On the other hand, the potentially-largerposet A p (cid:0) Aut G ( L ) (cid:1) cannot necessarily be described in terms of the elements of A p (cid:0) N G ( L ) (cid:1) : sincein general the map π : A p (cid:0) N G ( L ) (cid:1) → A p (cid:0) Aut G ( L ) (cid:1) ∪ { } is not surjective. That is, in some situa-tions, taking quotient by C G ( L ) might be generating new p -outers in L , which were not containedin N G ( L ) before—as the example below shows.The above remarks provide motivation for using the image poset A G,L : indeed we have givenabove a description of its elements—in terms of the p -outers that are already contained in G .Moreover, Theorem 4.1 shows that in fact we can reduce our (H-QC) analysis to studying thisposet, and the behavior of A p ( L ) → A G,L in homology. Even more, in some situations it can beshown that there is a topological section, and hence an inclusion of A G,L into A p (cid:0) N G ( L ) (cid:1) —thoughwe won’t require such a section in our arguments. ♦ Example 6.8. Let L be a simple group of Lie type over the field of q p elements. Let A ≤ Aut( L )be a cyclic group of field automorphisms of order p of the field q p , and let a generate A . Weuse a to define different actions on two distinct components isomorphic to L ; take two copies L and L of L , and let B := h b i be defined via the morphisms: φ : b a p ∈ Aut( L ) ,φ : b a ∈ Aut( L ) . Consider the semidirect product G := ( L × L ) ⋊ φ × φ B . We have that C G ( L ) = L h b p i . Thuscosets which have order p in the quotient B/L have elements inducing only inner automorphismsof L ; so that elements of B which induce a nontrivial outer automorphism on L must lie in cosetswhich have order p in the quotient, and so those elements have order p (or more) rather than p ,as elements of G . It follows then that I G ( L ) = ∅ ; and hence A G,L = A p ( L ). On the otherhand, we have Aut G ( L ) = G/C G ( L ) ∼ = L ( B/ h b p i ) ∼ = L h a p i ; so we see that A G,L = A p ( L ) ( A p (cid:0) Aut G ( L ) (cid:1) .Thus the p -outers of L in G do not suffice to exhibit A p (cid:0) Aut G ( L ) (cid:1) . Moreover, A G,L = A p ( L )and A p (Aut G ( L )) = A p ( L h a p i ) are not homotopy equivalent in general. ♦ We close this section with a useful proposition that gives a sufficient condition—in terms ofthe centralizers of whatever p -outers of L do arise in G , in its hypothesis (2)—to show that themap A p ( L ) → A G,L is nonzero in homology. Thus it establishes (H-QC) for G via Corollary 5.1,with A G,L in the role of “ A ”: Proposition 6.9. Suppose that G satisfies (H1). Let L be a component of G satisfying:(1) I G ( L ) is either empty, or it consists only of cyclic p -outers.Furthermore, there exists k ≥ such that:(2) the induced map ˜ H k ( A p (cid:0) C L ( E ) (cid:1) ) → ˜ H k (cid:0) A p ( L ) (cid:1) is the zero map for all E ∈ I G ( L ) ; and(3) ˜ H k (cid:0) A p ( L ) (cid:1) = 0 .Then G satisfies (H-QC). roof. As usual for (H-QC) we may assume that O p ( G ) = 1. By Theorem 2.4, we can also assumethat O p ′ ( G ) = 1. Set A := A G,L and N := N A ( L ); We will apply Corollary 5.1, by establishingits hypothesis that A p ( L ) → A is not the zero map in homology.As in Lemma 6.2 we have: A = [ E ∈ ˆI G ( L ) A p ( LE ) ⊆ A p (cid:0) Aut G ( L ) (cid:1) . Recall also that I A ( L ) = { E : E ∈ I G ( L ) } . Applying hypothesis (1) to such E , we get the analoguefor the resulting E : namely we see that I A ( L ) is either empty, or it consists only of cyclic p -outers.Now since the remainder of our argument takes place entirely in A , below we simplify notation by writing just E (rather than E ) for members of A .Recall from Definition 6.1 that since A ⊆ Aut( L ), we have A − N = I A ( L ).Assume first that I A ( L ) is empty. Then A = N = A p ( L ), so that A p ( L ) → A is the identitymap; and the homology of A p ( L ) is nonzero using hypothesis (3); so by Corollary 5.1 we get ourconclusion of (H-QC). (Recall this special case A p ( L ) = A gives the final statement in Theorem 1.6.)So we now assume instead that I A ( L ) is nonempty; and we consider some member E . Recall(as in Remark 2.7) that we have the homotopy equivalence N >E ≃ A p (cid:0) C L ( E ) (cid:1) , via the usualretraction r : A A ∩ L , with homotopy inverse A AE ; with a similar equivalence moregenerally for N ≃ A p ( L ). We will informally use i to denote these homotopy inverses.Let s denote the number of choices for E ; that is, I A ( L ) = { E , . . . , E s } . Since these givethe members of A − N , we can proceed by building up from N to A —via “adding one E i at atime”. Actually it will be more convenient, for use in our Mayer-Vietoris sequence below, to in factadjoin each time the larger set A ≥ E i : though in fact only the bottom member E i is really being newly added at that point—since in view of our cyclic-only hypothesis (1), from Definition 6.1 wehave A >E i ⊆ N . Namely we set X := N , and define the general term inductively via: X i +1 := X i ∪ A ≥ E i +1 ;in particular, we get X s = A . Furthermore using Definition 6.1 as just indicated, we get: A ≥ E i +1 ∩ X i = ( { E i +1 } ∪ A >E i +1 ) ∩ X i = N >E i +1 ∩ X i = N >E i +1 .Now consider the Mayer-Vietoris exact sequence corresponding to the above decomposition of X i +1 .Since the added-in posets A ≥ E i are contractible, they give zero-terms in reduced homology; hencethe sequence in degree k takes the form: . . . / / ˜ H k +1 ( X i +1 ) / / ˜ H k ( N >E i +1 ) / / ˜ H k ( X i ) / / ˜ H k ( X i +1 ) / / . . . ˜ H k ( A p ( C L ( E i +1 )) ) ∼ = i ∗ O O =0 / / ˜ H k (cid:0) A p ( L ) (cid:1) O O where the map in the lower line is zero in view of hypothesis (2). That determines by compo-sition a zero map into ˜ H k ( X i )—which by commutativity gives zero for the other composite mapinto ˜ H k ( X i ). In view of the isomorphism i ∗ induced by the homotopy inverse i mentioned earlier,we conclude that ˜ H k ( N >E i +1 ) → ˜ H k ( X i ) is also zero. Exactness in the sequence then gives amonomorphism: ˜ H k ( X i ) ֒ → ˜ H k ( X i +1 ) . Composing these maps over all i then gives a monomorphism:˜ H k (cid:0) A p ( L ) (cid:1) ∼ = → ˜ H k (cid:0) N A ( L ) (cid:1) = ˜ H k ( N ) = ˜ H k ( X ) ֒ → ˜ H k ( X s ) = ˜ H k ( A ) , where the isomorphism is induced by the more general homotopy inverse i mentioned above. Notethat this map is induced by the inclusion A p ( L ) → A , and it is nonzero if ˜ H k (cid:0) A p ( L ) (cid:1) = 0—which olds here by hypothesis (3). By Corollary 5.1, we conclude that ˜ H ∗ (cid:0) A p ( G ) (cid:1) = 0—as requiredfor (H-QC). (cid:3) Remark 6.10. Hypothesis (1) of Proposition 6.9 may hold reasonably often: for example, whenthe p -Sylows of Out( L ) are cyclic, as we observed in Definition 6.1. And notice that hypothesis (3) always holds—for at least some value of k , in view of Theorem 6.5. So it would then suffice toverify hypothesis (2) just for any of the values of k that already do satisfy hypothesis (3). ♦ Elimination of components of sporadic type HS (for the prime 2) In this section, we prove part (2) of Corollary 1.5: That is, we show that under (H1), if G has a component isomorphic to HS (the Higman-Sims sporadic group), then G satisfies (H-QC).Recall that Out(HS) is of order 2: thus for odd p , we have A p (HS) = A p (cid:0) Aut(HS) (cid:1) , so thatwe can immediately apply Corollary 5.1 to get (H-QC); hence in this section, we assume insteadthat p = 2.Here we will still proceed via Corollary 5.1: we will prove that A (HS) → A (cid:0) Aut(HS) (cid:1) is not thezero map in homology. To that end, we will inspect the second homology groups of these posets—via examining their Euler characteristics, and using the structure of the centralizers of the 2-outers.We performed some of the relevant computations in GAP [GAP] with the package [FPSC].Thus the main result of this section is the following theorem: Theorem 7.1. Take p := 2 . Let L := HS denote the Higman-Sims sporadic group, and alsotake A := Aut(HS) . Then we have ˜ χ (cid:0) A ( L ) (cid:1) = 1767424 and ˜ χ (cid:0) A ( A ) (cid:1) = 1204224 ; and thefollowing hold:(1) m ( L ) = 4 and m ( A ) = 5 .(2) A ( L ) → A ( A ) is a -equivalence.(3) ˜ H n (cid:0) A ( A ) (cid:1) = 0 for n ≥ .(4) ˜ H (cid:0) A ( L ) (cid:1) ⊆ ˜ H (cid:0) A ( A ) (cid:1) .In particular, if G satisfies (H1) and L is a component of G , then G satisfies (H-QC) for p = 2 . For the “In particular” part of the above theorem: Note that Out( HS ) is cyclic of order 2, so wehave the cyclic-only situation—which would give us hypothesis (1) of Proposition 6.9; however, wedo not get the zero-maps for centralizers which would be needed for hypothesis (2) there. Instead,we will show that we can still exploit a Mayer-Vietoris sequence very similar to the one used inproving that Proposition—but now based just on the subset of un-cooperative centralizers; in orderto then show that A (HS) → A (cid:0) Aut(HS) (cid:1) is nonzero in homology, and so again complete theproof of (H-QC) as there, via Corollary 5.1.Recall from Definition 2.6 that if H ≤ G , then N G ( H ) = { B ∈ A p ( G ) : B ∩ H = 1 } . ByRemark 2.7, this poset is homotopy equivalent to A p ( H ) via the retraction r : N G ( H ) → A p ( H )defined by B B ∩ H , with inverse given by the inclusion A p ( H ) ֒ → N G ( H ). Moreover, we sawthere that if we have E ∈ A p ( G ) − N G ( H ), then N G ( H ) >E is homotopy equivalent to A p (cid:0) C H ( E ) (cid:1) via the same retraction r ( B ) = B ∩ H , but now with the inverse i : A p (cid:0) C H ( E ) (cid:1) → N G ( H ) givenby i ( B ) = BE .Also recall the formula:˜ χ (cid:0) A p ( G ) (cid:1) = X E ∈A p ( G ) ∪{ } ( − m p ( E ) − p m p ( E )( m p ( E ) − / ;see for example [JM12]. Proof of Theorem 7.1. The values of the Euler characteristic in the statement follow from theabove formula via direct computation—e.g. in GAP. n the following argument for conclusions (1) and (2), we refer to Table 5.3m of [GLS98] for theassertions on the structure of the subgroups of Aut(HS). Since L has index 2 in A , I A ( L ) consistsonly of cyclic 2-outers. Indeed, if E ∈ I A ( L ), then E is of type 2 C or 2 D , in view of Table 5.3m.We recall below the structure of the centralizers of these involutions:The centralizer C L (2 C ) is a non-split extension: given by a normal elementary abelian 2-groupof 2-rank 4, under action by O − (2). In particular, O (cid:0) C L (2 C ) (cid:1) > A (cid:0) C L (2 C ) (cid:1) is con-tractible. So the inclusion of this centralizer in L is certainly zero in homology. Conclusion (1)follows from Table 5.6.1 of [GLS98] and the fact that m (cid:0) C L (2 C )2 C (cid:1) = 5. (Indeed using thecentralizer in the following paragraph, we also get m (cid:0) C L (2 D )2 D (cid:1) = 5.)On the other hand, C L (2 D ) ∼ = S , and A ( S ) is homotopy equivalent to a wedge of 512spheres of dimension 2: This holds since A ( S ) is simply connected, while the Bouc poset B ( S )of non-trivial radical 2-subgroups has dimension 2, and is homotopy equivalent to A ( S )—where ˜ χ (cid:0) A ( S ) (cid:1) = 512. These assertions can be directly computed, or else checked with GAPand the package [FPSC]. In particular, it is not clear that the inclusion of this centralizer in L should induce the zero map in homology—notably in dimension 2. So the 2 D -centralizers do notallow us to complete hypothesis (2) for Proposition 6.9.Write A := A ( A ), and N := N A ( L ). Since we are working in A = Aut( L ), using Definition 6.1we have A − N = I A ( L ); indeed we mentioned earlier that we have the cyclic-only there, sothat I A ( L ) consists of groups of order p , with no inclusion relations among them.Moreover, if E ∈ I A ( L ) and E is of type 2 C , then N >E r ≃ A (cid:0) C L (2 C ) (cid:1) (using the earlierretraction r ) is contractible. Correspondingly we set J A ( L ) := { E ∈ A : E is of type 2 D } ;and N J := N ∪{A ≥ E : E ∈ J A ( L ) } . Then we see that N J ֒ → A is a homotopy equivalence, in viewof Proposition 2.1—since beyond N J , we are adding only A ≥ E for E of type 2 C —and this posetis E ∗ A >E , where A >E is contractible. Further, the inclusion N ֒ → N J is a 2-equivalence—alsovia Proposition 2.1: for if E ∈ J A ( L ), then again using Definition 6.1, and the earlier retraction r : A >E = N >E r ≃ A (cid:0) C L ( E ) (cid:1) ∼ = A (cid:0) C L (2 D ) (cid:1) , which we saw is a wedge of 2-spheres, and so is 1-connected. Since A ( L ) ֒ → N is a homotopyequivalence (cf. Remark 2.7), we conclude that the composition A ( L ) ⊆ N ⊆ N J ⊆ A = A ( A )of our three inclusions, giving A ( L ) ֒ → A ( A ), is a 2-equivalence—and hence we have establishedconclusion (2).For conclusions (3) and (4): We saw that N J ≃ A = A ( A ); and we obtained N J from N by aprocess of adding the A ≥ E for E of type 2 D , glued through the 2-spheres in the link Lk A ( E ).Therefore, the process does not change the homology groups of degree ≥ 4; so we concludethat ˜ H n (cid:0) A ( A ) (cid:1) = ˜ H n (cid:0) A ( L ) (cid:1) = 0 for all n ≥ 4. We now examine that process in more detail—paralleling the proof of Proposition 6.9: Assume we have s members of type 2 D ; order themas J A ( L ) = { E , . . . , E s } . Again set X := N , and:(7.1) X i +1 := X i ∪ A ≥ E i +1 ;in particular, we now get X s = N J . Again we have: A ≥ E i +1 ∩ X i = N >E i +1 r ≃ A (cid:0) C L ( E i +1 ) (cid:1) ∼ = A (cid:0) C L (2 D ) (cid:1) , which we saw is a wedge of 2-spheres. We apply the Mayer-Vietoris sequence to the decompositionof X i +1 given in the right side of equation (7.1). Below we describe the relevant terms of thislong exact sequence. Note since each N >E is a wedge of 2-spheres that we get values of 0 in thecorresponding places for its homology in dimensions = 2:0 → ˜ H n ( X i ) → ˜ H n ( X i +1 ) → , n ≥ / / ˜ H ( X i ) / / ˜ H ( X i +1 ) / / ˜ H ( N >E i +1 ) / / ˜ H ( X i ) / / ˜ H ( X i +1 ) / / H (cid:0) C L ( E i +1 ) (cid:1) ∼ = i ∗ O O / / ˜ H ( A ( L )) O O → ˜ H ( X i ) → ˜ H ( X i +1 ) → n ≥ 4: Exactness in the top line gives an isomorphism in homology for eachpair ( i, i + 1); and we compose these over i to get a homology isomorphism between X = N and X s = N J in those degrees. We compose these with our earlier equivalences, namely A ( L ) → N along with N J → A = A ( A ), so that we extend to a homology isomorphism between A ( L )and A ( A )—establishing conclusion (3). Next for degree 3: Exactness at the left of the middlesequence gives monomorphisms ˜ H ( X i ) → ˜ H ( X i +1 ), which we compose over i to get a monomor-phism ˜ H ( N ) → ˜ H ( N J ); and then composing with the same two earlier homotopy equivalencesgives our desired monomorphism ˜ H (cid:0) A ( L ) (cid:1) ⊆ ˜ H (cid:0) A ( A ) (cid:1) , establishing conclusion (4).Finally, we show that with the above computation, we can prove the “In particular” part of thetheorem; we will apply Corollary 5.1 with A ( A ) in the role of “ A ”: Namely we will show that theinclusion A ( L ) ⊆ A ( A ) is nonzero in homology. Note that for the even degrees 2 m other than 2itself, we get ˜ H m (cid:0) A ( L ) (cid:1) = 0 = ˜ H m ( A ): namely for 2 m = 0 since both posets are connected,and for 2 m ≥ χ ( A ( L )) and ˜ χ ( A ) are positive, we conclude thatboth the homology groups ˜ H (cid:0) A ( L ) (cid:1) and ˜ H ( A ) are nonzero. So the epimorphism in degree 2provided by the 2-equivalence in conclusion (2) cannot be the zero map in homology. Then theproof finishes with Corollary 5.1 as promised. (cid:3) Elimination of certain alternating components (for the prime 2) In this section, we prove part (3) of Corollary 1.5. That is, we will apply the results of earliersections—to establish (H-QC) when G has an alternating component of type A or A , for p = 2.After that, we will discuss how to generalize these results to arbitrary alternating groups A n arising as components of the group G . We focus on the case p = 2, since for odd p , Out( A n ) isa p ′ -group and hence we can establish (H-QC) directly via Theorem 1.6. Finally, we will discussthe case of (H-QC) for a group G having exactly two components, both isomorphic to A andinterchanged in G : We show that (H-QC) holds for this group, first by using our methods; andthen also investigating it via the viewpoint of Theorems 1–3 in [Seg96]. This comparison showsthat the hypotheses of our theorems may be more widely applicable, and easier to check.We begin below with a direct application of Proposition 6.9 to some alternating components.We refer to [GLS98] for the assertions on the centralizers in alternating groups. Corollary 8.1. Assume p = 2 . Suppose that O ′ ( G ) = 1 , or that G satisfies (H1). If G has acomponent L isomorphic to A , then G satisfies (H-QC).Proof. In view of Theorem 2.4, we can assume that O ( G ) = 1 = O ′ ( G ). We check the hypothesesof Proposition 6.9 for the component L of G isomorphic to A .Although the quotient group Out( A ) is elementary of rank 2, the automorphisms in one of thethree nontrivial cosets have preimages in Aut( A ) which have order 4 (the class denoted 4 C ). Soif E ∈ A (cid:0) Aut( A ) (cid:1) satisfies E ∩ A = 1, then | E | = 2; and hence I Aut( A ) ( A ) contains only cyclic 2-outers—giving hypothesis (1) of Proposition 6.9: Moreover, for the other outer automorphismgroups E that do have elements of order 2 (the classes denoted 2 D and 2 B, C ), we see that C A ( E )is isomorphic to either D or S . In the former case, note A ( D ) is discrete with 5 points; and n the latter case O ( S ) > 1, so A ( S ) is contractible. To check hypotheses (2) and (3) of theProposition 6.9, it remains to find a value k ≥ 0, such that ˜ H k (cid:0) A ( A ) (cid:1) = 0, while A (cid:0) C A ( E ) (cid:1) ֒ →A ( A ) induces the zero map in the k -th homology group, for all E ∈ I Aut( A ) ( A ). We take k := 1:For A ( A ) is homotopy equivalent with the Tits building for Sp (2)—a wedge of 16 1-spheres, sothat ˜ H (cid:0) A ( A ) (cid:1) = 0; while the A -posets of the centralizers are of dimension 0 or contractible,and hence have zero homology in dimension 1—so that the induced maps in ˜ H are indeed zero,as desired.Thus we may apply Proposition 6.9, to conclude that ˜ H ∗ (cid:0) A ( G ) (cid:1) = 0. (cid:3) Corollary 8.2. Assume p = 2 . Suppose that O ′ ( G ) = 1 , or that G satisfies (H1). If G has acomponent L isomorphic to A , then G satisfies (H-QC).Proof. We proceed in a similar way to the previous proof—namely we check the hypotheses ofProposition 6.9: Again we may assume that O ( G ) = 1 = O ′ ( G ). This time Out( A ) hasorder 2, and so in particular has cyclic 2-Sylow subgroups; so we are in the cyclic-only situation,giving hypothesis (1). For E ∈ Aut( A ) giving a 2-outer of A , we have C A ( E ) ∼ = S ∼ = Sp (2).Now A (cid:0) Sp (2) (cid:1) is in fact homotopy equivalent with A ( A ), namely a wedge of 16 1-spheres,with homology concentrated in degree 1. To verify hypotheses (2) and (3), we will choose thevalue k := 2: For A ( A ) is homotopy equivalent to the building for L (2), a wedge of 64 2-spheres,so that ˜ H (cid:0) A ( A ) (cid:1) = 0. On the other hand, we saw that A (cid:0) C A ( E ) (cid:1) has homology concentratedin dimension 1, and hence has zero homology in dimension 2, so that A (cid:0) C A ( E ) (cid:1) ֒ → A ( A )induces the zero map in homology, as desired.Thus we may apply Proposition 6.9, to conclude that ˜ H ∗ (cid:0) A ( G ) (cid:1) = 0. (cid:3) We now discuss possible ways of generalizing the above results:The proofs given above fail for A n with odd n . For example, take n = 5: Then A ( A ) has 5 con-nected components, each contractible; so that reduced homology is concentrated in ˜ H (cid:0) A ( A ) (cid:1) ,a space of dimension 4. By contrast, A ( S ) is a connected wedge of 1-spheres—so that reducedhomology is concentrated in degree 1. Therefore the inclusion A ( A ) ֒ → A ( S ) induces the zeromap in homology. One of the reasons for this behavior in homology is that the centralizers ofouter 2-subgroups E ≤ S have centralizer in A equal to S , which is discrete (so of dimension 0)with 3 points. That is, the centralizers have the same maximum nonzero homological dimensionas A . A similar (but more complex) situation arises with A and S . The difficulty for odd n isthat we have a “leap” of the largest dimension of a nonzero homology group from A n to S n (atleast for small n ). It would be interesting to study these behaviors for n ≥ r ≥ r ] thelargest integer n with n ≤ r (the floor function). Problem. Let m a := [ n/ − n ≥ 5, and m s := [( n + 1) / − n = 2 , 4. Showthat m a (resp. m s ) is the largest integer such that A ( A n ) (resp. A ( S n )) has nonzero homologyin degree m a (resp. m s ).Suppose that the statements in the above Problem have been established. Since C A n ( E ) ∼ = S n − is the centralizer of a particular outer involution of A n , we see that A ( C A n ( E )) and A ( A n ) sharethe same largest dimension for a nonzero homology group if and only if m s ( n − 2) = m a ( n ):[(( n − 2) + 1) / − n/ − , that is, if and only if [( n − / 2] = [ n/ . This equality holds if and only if n is odd. Therefore, modulo the above Problem, for even n wecan proceed as in Corollary 8.2, and deduce that (H-QC) holds for G , if it satisfies (H1) and hasa component of type A n . e conclude this section with an application of Theorem 4.1, to establish (H-QC) for certaingroups with A -components. We will establish Property E(2), and use that to directly verify thehypothesis of Theorem 4.1—rather than using the inductive procedure of Proposition 4.9. It wouldbe interesting to investigate if this kind of proof can show (H-QC) in groups with A n componentsfor larger odd n . Example 8.3. Take p := 2. Let G := ( A × A ) ⋊ ( E × R ), where E ∼ = C acts diagonally byouter involutions on both copies of A , and R ∼ = C interchanges the components.Thus our G -orbit has length t = 2. Note that the kernel on the components is H = ( A × A ) ⋊ E .We will apply Theorem 4.1 to show (H-QC) for G for p = 2; that is, we will show that ψ H is not thezero map in homology. We will accomplish this by first establishing Property E(2) for the ϕ -mapscorresponding to the components.Let L and L be the copies of A which are components of G , and set N := L L = F ∗ ( G ).Note that G interchanges L and L , and that O ( G ) = 1 = O ′ ( G ). With the notation ofDefinition 4.3, we have: C ( H ) = C H ( L L ) = 1 , C ( H ) = C H ( L ) = L , C ( H ) = H. As promised above, we will first establish Property E(2) for the pairs ( L i , C i ( H )); in fact we willshow that ϕ i induces an epimorphism in all homology groups, for the relevant values 1 ≤ i ≤ t = 2.For i = 1: Using C ( H ) = 1 and C ( H ) = L above, we see that ϕ maps A ( L ) to: A (1) ∗ A = ∅ ∗ A L ,L = A ( L ).Thus ϕ is the identity map, and in particular induces an epimorphism in all homology groups.For i = 2, the proof will be much lengthier:This time using C ( H ) = L and C ( H ) = H , we see that ϕ maps A ( H ) to A ( L ) ∗ A ,where A = A H,L ∼ = A ( S ). In view of Lemma 6.2, since we have C H ( L ) = L , we can regard A as the set { L A : A ∈ A ( H ) , C A ( L ) = 1 } ; where L A is a coset of L in the subset A of thequotient H/C H ( L ) = H/L . We want ϕ to induce an epimorphism in homology.In fact, we will prove that ϕ is even a 2-equivalence: namely an epimorphism in degree 2, buta homology isomorphism in smaller degrees. For this, we will apply the variant of Quillen’s fiberlemma which we gave as Proposition 2.1. Set: Y := A ( L ) ∗ A ,and note that the join Y is 2-dimensional. In view of this proposition, it is enough to show, forall y ∈ Y , that W y := ϕ − ( Y ≤ y ) ∗ Y >y is 1-connected.Assume first the case where y ∈ A ( L ). Here we see ϕ − ( Y ≤ y ) = A ( L ) ≤ y is contractible; sothat W y is also contractible, and in particular is 1-connected.So we may assume instead the other case, where y ∈ A . We saw above that we may regard y as L F , for some F ∈ A ( H ) such that C F ( L ) = F ∩ L = 1 (since C H ( L ) = L ). Therefore: ϕ − ( Y ≤ y ) = A ( L ) ∪ { F ∈ A ( H ) − A ( L ) : L F ≤ L F } = A ( L F ) . Assume first the case where F := F ∩ L C H ( L ) = F ∩ L L > 1. Then L F contains theprojections F i of F on the L i —with F > F ∩ L = 1. We see that F ≤ Z ( L F ), sothat 1 < F ≤ O ( L F ); thus A ( L F ) is contractible, and hence so is W y .Hence we may assume the remaining case where F ∩ L C H ( L ) = 1—so that F acts faithfullyon L by outer automorphisms. Then F has order 2, and A ( L F ) = A ( S ) is 0-connected(i.e. connected). On the other hand, Y >y = ( A ) >y . Now since H/C H ( L ) = H/L ∼ = L E ∼ = S ,we can regard A = A H,L in the alternative form of A ( L E ) = A ( S ); and in this representation, Notice this is our first use of the specific A -structure in our hypothesis. ur element y ∈ A is an elementary abelian ˜ F of order 2, inducing outer automorphisms on L .Hence: A ( S ) >y = N S ( L ) > ˜ F ≃ A (cid:0) C L ( ˜ F ) (cid:1) = A ( S ) , which is ( − W y ≃ A ( S ) ∗ A ( S ) is infact 1-connected, by the classical join property mentioned after (2.1)—with 0 , − n, m ”. (Moreover, W y is homotopy equivalent to a bouquet of:dim H (cid:0) A ( S ) (cid:1) · dim ˜ H (cid:0) A ( S ) (cid:1) = 16 · y ∈ Y = A ( L ) ∗ A , that W y is 1-connected (and indeed oftencontractible). By Proposition 2.1, we see that ϕ is a 2-equivalence; and in particular, it is anepimorphism in all homology groups of degree at most 2. Since Y has dimension 2, its homologygroups vanish in degree > 2; so in fact ϕ is an epimorphism in all the homology groups. This factfor ϕ in particular completes the proof of Property E(2) for all (i.e. both) ϕ i .Now to complete the hypothesis of Theorem 4.1, we want ˜ H ( A (cid:0) C H ( N ) (cid:1) ∗ A ∗ A ) = 0. Inthis case, C H ( N ) = 1 since there are no other components outside our orbit; while A = A ( L )and A = A ( S ). Hence using (2.1):˜ H ( A (cid:0) C H ( N ) (cid:1) ∗ A ∗ A ) = ˜ H (cid:0) A ( A ) ∗ A ( S ) (cid:1) = ˜ H (cid:0) A ( A ) (cid:1) ⊗ ˜ H (cid:0) A ( S ) (cid:1) = 0 ;for = 0 follows by the almost-simple case in Theorem 6.4, while the more detailed expressionfollows by direct computation—since A ( A ) is a wedge of 4 0-spheres, and A ( S ) is a wedgeof 16 1-spheres. Hence applying the Theorem, (H-QC) also holds for G . (Along the way, thecomputation has shown that (H-QC) holds for H as well.)Finally, we approach the proof using the viewpoint of Segev’s theorems:First note that Segev’s Theorems 2 and 3 (which we stated as Theorem 1.4) cannot be appliedhere: We have that H is the kernel on components, and H > F ∗ ( H ) = A × A —so that we can’t useTheorem 3. Moreover, if L is a component of H , then Aut H ( L ) = S ; so A ( L ) → A (cid:0) Aut H ( L ) (cid:1) is the zero map in homology: For A ( L ) has 5 connected components, each contractible, and hencehas reduced homology given by 4-dimensional ˜ H ; while A (cid:0) Aut H ( L ) (cid:1) = A ( S ) is connected, andso has zero in ˜ H . Hence we don’t get the nonzero map required for the hypothesis of Theorem 2.Nevertheless, it is possible to get (H-QC) here, by applying Theorem 1 of [Seg96]: Notefirst that D ( H ) = A ( H ) − (cid:0) A ( L ) ∪ A ( L ) (cid:1) . And we can conclude, using computationsin GAP, that D ( H ) → A ( H ) is not surjective in homology: since H (cid:0) A ( H ) (cid:1) has dimension 384,while H (cid:0) D ( H ) (cid:1) has dimension only 36. Hence ˜ H ∗ (cid:0) A ( G ) (cid:1) = 0 by Theorem 1 of [Seg96]. ♦ References [Asc00] M. Aschbacher. 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