2 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER G is a p-group, and where the extension is central. In both of these cases, BH^p is the hom

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1 EXTENSIONS OF p-local FINITE GROUPS C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, AND B. OLIVER Abstract. A p-local finite group consists of a finite p-group S, together with a pair of categories which encode conjugacy" relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we study and classify extensions of p-local finite groups, and also compute the fundamental group of the classifying space of a p-local finite group. A p-local finite group consists of a finite p-group S, together with a pair of categories (F; L), of which F is modeled on the conjugacy (or fusion) in a Sylow subgroup of a finite group. The category L is essentially an extension of F and contains just enough extra information so that its p-completed nerve has many of the same properties as p-completed classifying spaces of finite groups. We recall the precise definitions of these objects in Section 1, and refer to [BLO2] and [5A1] for motivation for their study. In this paper, we study extensions of saturated fusion systems and of p-local finite groups. This is in continuation of our more general program of trying to understand to what extent properties of finite groups can be extended to properties of p-local finite groups, and to shed light on the question of how many (exotic) p-local finite groups there are. While we do not get a completely general theory of extensions of one p-local finite group by another, we do show how certain types of extensions can be described in manner very similar to the situation for finite groups. From the point of view of group theory, developing an extension theory for p-local finite groups is related to the question of to what extent the extension problem for groups is a local problem, i.e., a problem purely described in terms of a Sylow p- subgroup and conjugacy relations inside it. In complete generality this is not the case. For example, strongly closed subgroups of a Sylow p-subgroup S of G need not correspond to normal subgroups of G. However, special cases where this does happen include the case of existence of p-group quotients (the focal subgroup theorems, see [Go, xx ]) and central subgroups (described via the Z -theorem of Glauberman [Gl]). From the point of view of homotopy theory, one of the problems which comes up when looking for a general theory of extensions of p-local finite groups is that while an extension of groups 1! K!! G! 1 always induces a (homotopy) fibration sequence of classifying spaces, it does not in general induce a fibration sequence of p-completed classifying spaces. Two cases where this does happen are those where 1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D20. Key words and phrases. Classifying space, p-completion, finite groups, fusion. C. Broto is partially supported by MCYT grant BFM N. Castellana is partially supported by MCYT grant BFM J. Grodal is partially supported by NSF grants DMS and DMS R. Levi is partially supported by EPSRC grant GR/M7831. B. Oliver is partially supported by UMR 7539 of the CNRS. 1

2 2 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER G is a p-group, and where the extension is central. In both of these cases, BH^p is the homotopy fiber of the map B ^p! BG^p. Thus, also from the point of view of homotopy theory, it is natural to study extensions of p-local finite groups with p- group quotient, and to study central extensions of p-local finite groups. The third case we study is that of extensions with quotient of order prime to p; and the case of p and p 0 -group quotients to some extent unify to give a theory of extensions with p-solvable quotient. Recall in this connection that by a previous result of ours [5A1, Proposition C], solvable p-local finite groups all come from p-solvable groups. In all three of these situations, we develop a theory of extensions which parallels the situation for finite groups. We now describe the contents of this paper in more detail, stating simplified versions of our main results on extensions. Stronger and more precise versions of some of these theorems will be stated and proven later. In Section 3, we construct a very general theory of fusion subsystems (Proposition 3.8) and linking subsystems (Theorem 3.9) with quotient a p-group or a group of order prime to p. As a result we get the following theorem (Corollary 3.10), which for a p-local finite group (S; F; L), describes a correspondence between covering spaces of the geometric realization jlj and certain p-local finite subgroups of (S; F; L). Theorem A. Suppose that (S; F; L) is a p-local finite group. Then there is a normal subgroup H C ß 1 (jlj) which is minimal among all those whose quotient is finite and p- solvable. Any covering space of the geometric realization jlj whose fundamental group contains H is homotopy equivalent to jl 0 j for some p-local finite group (S 0 ; F 0 ; L 0 ), where S 0» S and F 0 F. Moreover, the p-local finite group (S 0 ; F 0 ; L 0 ) of Theorem A can be explicitly described in terms of L, as we will explain in Section 3. In order to use this theorem, it is useful to have ways of finding the finite p-solvable quotients of ß 1 (jlj). This can be done by iteration, using the next two theorems. In them, the maximal p-group quotient of ß 1 (jlj), and the maximal quotient of order prime to p, are described solely in terms of the fusion system F. When G is an infinite group, we define O p (G) and O p0 (G) to be the intersection of all normal subgroups of G of p-power index, or index prime to p, respectively. These clearly generalize the usual definitions for finite G (but are not the only possible generalizations). Theorem B (Hyperfocal subgroup theorem for p-local finite groups). For a p-local finite group (S; F; L), the natural homomorphism is surjective, with kernel equal to S! ß 1 (jlj)=o p (ß 1 (jlj)) ο = ß1 (jlj^p ) Ω OF p def (S) = g 1 ff(g) 2 fi S fi ff g 2 P» S; ff 2 O p (Aut F (P )) : For any saturated fusion system F over a p-group S, we let O p0 (F) F be the smallest fusion subsystem of F (in the sense of Definition 1.1) which contains all automorphism groups O p0 (Aut F (P )) for P» S. Equivalently, O p0 (F) is the smallest subcategory of F with the same objects, and which contains all restrictions of all automorphisms in F of p-power order. This subcategory is needed in the statement of the next theorem.

3 EXTENSIONS OF P -LOCAL FINITE GROUPS 3 Theorem C. For a p-local finite group (S; F; L), the natural map is surjective, with kernel equal to Out F (S)! ß 1 (jlj)=o p0 (ß 1 (jlj)) Out 0 F(S) def = Ω ff 2 Out F (S) fi fi ffjp 2 Mor O p 0 (F) (P; S); some F-centric P» Sff : Theorem Bisproved as Theorem 2.5, and Theorem C is proved as Theorem 5.5. In fact, we give a purely algebraic description of these subsystems of p-power index" or of index prime to p" (Definition 3.1), and then show in Sections 4.1 and 5.1 that they in fact all arise as finite covering spaces of jlj (see Theorems 4.4 and 5.5). Similar results were also discovered independently by Puig [Pu4]. Then, in Sections 4.2 and 5.2, we establish converses to these concerning the extensions of a p-local finite group, which include the following theorem. Theorem D. Let (S; F; L) be a p-local finite group. Suppose we are given a fibration sequence jlj^p! E! BG, where G is a finite p-group or has order prime to p. Then there exists a p-local finite group (S 0 ; F 0 ; L 0 ) such that jl 0 j^p ' E^p. This is shown as Theorems 4.7 and 5.8. Moreover, when G is a p-group, we give in Theorem 4.7 an explicit algebraic construction of the p-local finite group (S 0 ; F 0 ; L 0 ). Finally, in Section 6, we develop the theory of central extensions of p-local finite groups. Our main results there (Theorems 6.8 and 6.13) give a more elaborate version of the following theorem. Here, the center of a p-local finite group (S; F; L) is defined to be the subgroup of elements x 2 Z(S) such that ff(x) =x for all ff 2 Mor(F c ). Theorem E. Suppose that A is a central subgroup of a p-local finite group (S; F; L). Then there exists a canonical quotient p-local finite group (S=A; F=A; L=A), and the canonical projection of jlj onto jl=aj is a principal fibration with fiber BA. Conversely, for any principal fibration E! jlj with fiber BA, where A is a finite abelian p-group, there exists a p-local finite group (e S; e F; e L) such that j e Lj ' E. Furthermore, this correspondence sets up a 1 1 correspondence between central extensions of L by A and elements in H 2 (jlj; A). One motivation for this study was the question of whether extensions of p-local finite groups coming from finite groups can produce exotic p-local finite groups. In the case of central extensions, we are able to show that (S; F; L) comes from a finite group if and only if (S=A; F=A; L=A) comes from a group (Corollary 6.14). For the other types of extensions studied in this paper, this is still an open question. We have so far failed to produce exotic examples in this way, and yet we have also been unable to show that exotic examples cannot occur. This question seems to be related to some rather subtle and interesting group theoretic issues relating local to global structure; see Corollary 4.8 for one partial result in this direction. This paper builds on the earlier paper [5A1] by the same authors, and many of the results in that paper were originally motivated by this work on extensions. The authors would like to thank the University of Aberdeen, Universitat Aut onoma de Barcelona, Université Paris 13, and Aarhus Universitet for their hospitality. In particular, the origin of this project goes back to a three week period in the spring of 2001, when four of the authors met in Aberdeen.

4 4 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER 1. A quick review of p-local finite groups We first recall the definitions of a fusion system, and a saturated fusion system, in the form given in [BLO2]. For any group G, and any pair of subgroups H; K» G, we set N G (H; K) =fx 2 G j xhx 1» Kg; let c x denote conjugation by x (c x (g) =xgx 1 ), and set Hom G (H; K) = Φ c x 2 Hom(H; K) fi fi x 2 NG (H; K) Ψ ο = NG (H; K)=C G (H): By analogy, we also write Aut G (H) = Hom G (H; H) = Φ c x 2 Aut(H) fi fi x 2 NG (H) Ψ ο = NG (H)=C G (H): Definition 1.1 ([Pu2] and [BLO2, Definition 1.1]). A fusion system over a finite p- group S is a category F, where Ob(F) is the set of all subgroups of S, and which satisfies the following two properties for all P; Q» S: ffl Hom S (P; Q) Hom F (P; Q) Inj(P; Q); and ffl each ' 2 Hom F (P; Q) is the composite of an isomorphism in F followed by an inclusion. The following additional definitions and conditions are needed in order for these systems to be very useful. If F is a fusion system over a finite p-subgroup S, then two subgroups P; Q» S are said to be F-conjugate if they are isomorphic as objects of the category F. Definition 1.2 ([Pu2], see [BLO2, Def. 1.2]). Let F be a fusion system over a p-group S. ffl A subgroup P» S is fully centralized in F if jc S (P )j jc S (P 0 )j for all P 0» S which is F-conjugate to P. ffl A subgroup P» S is fully normalized in F if jn S (P )j jn S (P 0 )j for all P 0» S which is F-conjugate to P. ffl F is a saturated fusion system if the following two conditions hold: (I) For all P» S which is fully normalized in F, P is fully centralized in F and Aut S (P ) 2 Syl p (Aut F (P )). (II) If P» S and ' 2 Hom F (P; S) are such that 'P is fully centralized, and if we set N ' = fg 2 N S (P ) j 'c g ' 1 2 Aut S ('P )g; then there is ' 2 Hom F (N ' ;S) such that 'j P = '. If G is a finite group and S 2 Syl p (G), then by [BLO2, Proposition 1.3], the category F S (G) defined by letting Ob(F S (G)) be the set of all subgroups of S and setting Mor FS (G)(P; Q) = Hom G (P; Q) is a saturated fusion system. An alternative pair of axioms for a fusion system being saturated have been given by Radu Stancu [St]. He showed that axioms (I) and (II) above are equivalent to the two axioms: (I 0 ) Inn(S) 2 Syl p (Aut F (S)).

5 EXTENSIONS OF P -LOCAL FINITE GROUPS 5 (II 0 ) If P» S and ' 2 Hom F (P; S) are such that 'P is fully normalized, and if we set N ' = fg 2 N S (P ) j 'c g ' 1 2 Aut S ('P )g; then there is ' 2 Hom F (N ' ;S) such that 'j P = '. The following consequence of conditions (I) and (II) above will be needed several times throughout the paper. Lemma 1.3. Let F be a saturated fusion system over a p-group S. Let P; P 0» S be a pair of F-conjugate subgroups such that P 0 is fully normalized inf. Then there is a homomorphism ff 2 Hom F (N S (P );N S (P 0 )) such that ff(p )=P 0. Proof. This is shown in [BLO2, Proposition A.2(b)]. In this paper, it will sometimes be necessary to work with fusion systems which are not saturated. This is why we have emphasized the difference between fusion systems, and saturated fusion systems, in the above definitions. We next specify certain collections of subgroups relative to a given fusion system. Definition 1.4. Let F be a fusion system over a finite p-subgroup S. ffl A subgroup P» S is F-centric if C S (P 0 ) = Z(P 0 ) for all P 0» S which is F- conjugate to P. ffl A subgroup P» S is F-radical if Out F (P ) is p-reduced; i.e., if O p (Out F (P ))=1. ffl For any P» S which is fully centralized in F, the centralizer fusion system C F (P ) is the fusion system over C S (P ) defined by setting Hom CF (P )(Q; Q 0 )= Φ ffjq fi fi ff 2 HomF (QP; Q 0 P ); ffj P =Id P ; ff(q)» Q 0Ψ : A subgroup P» S is F-quasicentric if for all P 0» S which is F-conjugate to P and fully centralized in F, C F (P 0 ) is the fusion system of the p-group C S (P 0 ). ffl F c F q F denote the full subcategories of F whose objects are the F-centric subgroups, and F-quasicentric subgroups, respectively, of S. If F = F S (G) for some finite group G, then P» S is F-centric if and only if P is p-centric in G (i.e., Z(P ) 2 Syl p (C G (P ))), and P is F-radical if and only if N G (P )=(P C G (P )) is p-reduced. Also, P is F-quasicentric if and only if C G (P ) contains a normal subgroup of order prime to p and of p-power index. In fact, when working with p-local finite groups, it suffices to have a fusion system F c defined on the centric subgroups of S, and which satisfies axioms (I) and (II) above for those centric subgroups. In other words, fusion systems defined only on the centric subgroups are equivalent to fusion systems defined on all subgroups, as described in the following theorem. Theorem 1.5. Fix a p-group S and a fusion system F over S. (a) Assume F is saturated. Then each morphism in F is a composite of restrictions of morphisms between subgroups of S which are F-centric, F-radical, and fully normalizedinf. Moreprecisely, for each P; P 0» S and each ' 2 Iso F (P; P 0 ), there are subgroups P = P 0 ;P 1 ;:::;P k = P 0, subgroups Q i hp i 1;P i i (i = 1;:::;k) which are F-centric, F-radical, and fully normalized in F, and automorphisms ' i 2 Aut F (Q i ), such that ' i (P i 1) =P i for all i and ' = ' k ffi ffi ' 1 j P.

6 6 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER (b) Assume conditions (I) and (II) in Definition 1.2 are satisfied for all F-centric subgroups P» S. Assume also that each morphism in F is a composite of restrictions of morphisms between F-centric subgroups of S. Then F is saturated. Proof. Part (a) is Alperin's fusion theorem for saturated fusion systems, in the form shown in [BLO2, Theorem A.10]. Part (b) is a special case of [5A1, Theorem 2.2]: the case where H is the set of all F-centric subgroups of S. Theorem 1.5(a) will be used repeatedly throughout this paper. The following lemma is a first easy application of the theorem, and provides a very useful criterion for a subgroup to be quasicentric or not. Lemma 1.6. Let F be a saturated fusion system over a p-group S. Then the following hold for any P» S. (a) Assume that P» Q» P C S (P ) and Id 6= ff 2 Aut F (Q) are such that ffj P and ff has order prime to p. Then P is not F-quasicentric. =Id P (b) Assume that P is fully centralized in F, and is not F-quasicentric. Then there are P» Q» P C S (P ) and Id 6= ff 2 Aut F (Q) such that Q is F-centric, ffj P = Id P, and ff has order prime to p. Proof. (a) Fix any P 0 which isf-conjugate to P and fully centralized in F. By axiom (II), there is ' 2 Hom F (Q; S) such that '(P )=P 0 ; set Q 0 = '(Q). Thus 'ff' 1 j CQ 0 (P 0 ) is an automorphism in C F (P 0 ) whose order is not a power of p, so C F (P 0 ) is not the fusion system of C S (P 0 ), and P is not F-quasicentric. (b) Assume that P is fully centralized in F and not F-quasicentric. Then C F (P ) strictly contains the fusion system of C S (P ) (since C F (P 0 ) is isomorphic as a category to C F (P ) for all P 0 which isf-conjugate to P and fully centralized in F). Since C F (P ) is saturated by [BLO2, Proposition A.6], Theorem 1.5(a) implies there is a subgroup Q» C S (P ) which is C F (P )-centric and fully normalized in C F (P ), and such that Aut CF (P )(Q) Ψ Aut CS (P )(Q). Since Q is fully normalized, Aut CS (P )(Q) is a Sylow p- subgroup of Aut CF (P )(Q), and hence this last group is not a p-group. Also, by [BLO2, Proposition 2.5(a)], PQ is F-centric since Q is C F (P )-centric. Since orbit categories both of fusion systems and of groups will play a role in certain proofs in the last three sections, we define them here. Definition 1.7. (a) If F is a fusion system over a p-group S, then O c (F) (the centric orbit category of F) is the category whose objects are the F-centric subgroups of S, and whose morphism sets are given by Mor O c (F)(P; Q) = Rep F (P; Q) def = Qn Hom F (P; Q): Let Z F : O c (F)! Z (p) -mod be the functor which sends P to Z(P ) and ['] (the ' class of ' 2 Hom F (P; Q)) to Z(Q)! 1 Z(P ). (b) If G is a finite group and S 2 Syl p (G), then O c S (G) (the centric orbit category of G) is the category whose objects are the subgroups of S which are p-centric in G, and where Mor O c S (G)(P; Q) =QnN G (P; Q) ο = MapG (G=P; G=Q): Let Z G : OS c (G)! Z (p)-mod be the functor which sends P to Z(P ) and [g] (the class of g 2 N G (P; Q)) to conjugation by g 1.

7 EXTENSIONS OF P -LOCAL FINITE GROUPS 7 We now turn to linking systems associated to abstract fusion systems. Definition 1.8 ([BLO2, Def. 1.7]). Let F be a fusion system over the p-group S. A centric linking system associated to F is a category L whose objects are the F- centric subgroups of S, together with a functor ß : L!F c, and distinguished" monomorphisms P ffi P! Aut L (P ) for each F-centric subgroup P» S, which satisfy the following conditions. (A) ß is the identity on objects. For each pair of objects P; Q 2 L, Z(P ) acts freely on Mor L (P; Q) by composition (upon identifying Z(P ) with ffi P (Z(P ))» Aut L (P )), and ß induces a bijection Mor L (P; Q)=Z(P ) ο=! Hom F (P; Q): (B) For each F-centric subgroup P» S and each x 2 P, ß(ffi P (x)) = c x 2 Aut F (P ). (C) For each f 2 Mor L (P; Q) and each x 2 P, the following square commutes in L: P f! Q # P ffi P (x) f #! Q: ffi Q (ß(f)(x)) A p-local finite group is defined to be a triple (S; F; L), where S is a finite p-group, F is a saturated fusion system over S, and L is a centric linking system associated to F. The classifying space of the triple (S; F; L) is the p-completed nerve jlj^p. For any finite group G with Sylow p-subgroup S, a category L c S(G) was defined in [BLO1], whose objects are the p-centric subgroups of G, and whose morphism sets are defined by Mor L c S (G)(P; Q) =N G (P; Q)=O p (C G (P )): Since C G (P ) = Z(P ) O p (C G (P )) when P is p-centric in G, L c S(G) is easily seen to satisfy conditions (A), (B), and (C) above, and hence is a centric linking system associated to F S (G). Thus (S; F S (G); L c S(G)) is a p-local finite group, with classifying space jl c S (G)j^p ' BG^p (see [BLO1, Proposition 1.1]). It will be of crucial importance in this paper that any centric linking system associated to F can be extended to a quasicentric linking system; a linking system with similar properties, whose objects are the F-quasicentric subgroups of S. We first make more precise what this means. Definition 1.9. Let F be any saturated fusion system over a p-group S. A quasicentric linking system associated to F consists of a category L q whose objects are the F- quasicentric subgroups of S, together with a functor ß : L q!f q, and distinguished monomorphisms P C S (P ) which satisfy the following conditions. (A) q ffi! P Aut L q(p ); ß is the identity on objects and surjective on morphisms. For each pair of objects P; Q 2 L q such that P is fully centralized, C S (P ) acts freely on Mor L q(p; Q) by composition (upon identifying C S (P ) with ffi P (C S (P ))» Aut L q(p )), and ß induces a bijection Mor L q(p; Q)=C S (P ) ο=! Hom F (P; Q):

8 8 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER (B) q For each F-quasicentric subgroup P» S and each g 2 P, ß sends ffi P (g) 2 Aut L q(p ) to c g 2 Aut F (P ). (C) q Mor L q(p; Q). For each f 2 Mor L q(p; Q) and each x 2 P, f ffi ffi P (x) = ffi Q (ß(f)(x)) ffi f in (D) q For each F-quasicentric subgroup P» S, there is a morphism P 2 Mor L q(p; S) such that ß( P ) = incl S P 2 Hom(P; S), and such that for each g 2 P C S (P ), ffi S (g) ffi P = P ffi ffi P (g) in Mor L q(p; S). If P and P 0 are F-conjugate and F-quasicentric, then for any Q» S, Mor L q(p; Q) ο = Mor L q(p 0 ;Q) and Hom F (P; Q) ο = HomF (P 0 ;Q), while the centralizers C S (P ) and C S (P 0 ) need not have the same order. This is why condition (A) q makes sense only if we assume that P is fully centralized; i.e., that C S (P ) is as large as possible. When P is F-centric, then this condition is irrelevant, since every subgroup P 0 which is F-conjugate to P is fully centralized (C S (P 0 )=Z(P 0 ) ο = Z(P )). Note that (D) q is a special case of (C) q when P is F-centric; this is why the axiom is not needed for centric linking systems. We also note the following relation between these axioms: Lemma In the situation of Definition 1.9, axiom (C) q implies axiom (B) q. Proof. Fix an F-quasicentric subgroup P» S, and an element g 2 P. We apply (C) q with f = ffi P (g). For each x 2 P, if we set y = ß(ffi P (g))(x), then ffi P (g) ffi ffi P (x) = ffi P (y) ffi ffi P (g). Since ffi P is an injective homomorphism, this implies that gx = yg, and thus that y = c g (x). So ß(ffi P (g)) = c g. When (S; F; L) is a p-local finite group, and L q is a quasicentric linking system associated to F, then we say that L q extends L if the full subcategory of L q with objects the F-centric subgroups of S is isomorphic to L via a functor which commutes with the projection functors to F and with the distinguished monomorphisms. In [5A1, Propositions 3.4 & 3.12], we constructed an explicit quasicentric linking system L q associated to F and extending L, and showed that it is unique up to an isomorphism of categories which preserves all of these structures. So from now on, we will simply refer to L q as the quasicentric linking system associated to (S; F; L). Condition (D) q above helps to motivate the following definition of inclusion morphisms in a quasicentric linking system. Definition Fix a p-local finite group (S; F; L), with associated quasicentric linking system L q. (a) A morphism P 2 Mor L q(p; S) is an inclusion morphim if it satisfies the hypotheses of axiom (D) q : if ß( P ) = incl S P, and if ffi S (g) ffi P = P ffi ffi P (g) in Mor L q(p; S) for all g 2 P C S (P ). If P = S, then we also require that S =Id S. (b) A compatible set of inclusions for L q is a choice of morphisms f Q P g for all pairs of F-quasicentric subgroups P» Q, such that Q P 2 Mor L q(p; Q), such that R P = R ffi Q Q P for all P» Q» R, and such that SP is an inclusion morphism for each P. The following properties of quasicentric linking systems were also proven in [5A1]. Proposition The following hold for any p-local finite group (S; F; L), with associated quasicentric linking system L q.

9 EXTENSIONS OF P -LOCAL FINITE GROUPS 9 (a) The inclusion L L q induces a homotopy equivalence jlj ' jl q j between geometric realizations. More generally, for any full subcategory L 0 L q which contains as objects all subgroups of S which are F-centric and F-radical, the inclusion L 0 L q induces a homotopy equivalence jl 0 j'jl q j. (b) Let ' 2 Mor L q(p; R) and ψ 2 Mor L q(q; R) be any pair of morphisms in L q with the same target group such that Im(ß('))» Im(ß(ψ)). Then there is a unique morphism χ 2 Mor L q(p; Q) such that ' = ψ ffi χ. Proof. The homotopy equivalences jl q j'jlj'jl 0 j are shown in [5A1, Theorem 3.5]. Point (b) is shown in [5A1, Lemma 3.6]. Point (b) above will be frequently used throughout the paper. In particular, it makes it possible to embed the linking system of S (or an appropriate full subcategory) in L q, depending on the choice of an inclusion morphisms P as defined above, for each object P. Such inclusion morphisms always exist by axiom (D) q. In the following proposition, L S (S)j Ob(L q ) denotes the full subcategory of L S (S) whose objects are the F-quasicentric subgroups of S. In general, for a functor F : C!C 0, and objects c; d 2 Ob(C), we let F c;d denote the map from Mor C (c; d) tomor C 0(F (c);f(d)) induced by F. Proposition Fix a p-local finite group (S; F; L). let L q be its associated quasicentric linking system, and let ß : L q!f q be the projection. Then any choice of inclusion morphisms P = SP 2 Mor L q(p; S), for all F-quasicentric subgroups P» S, extends to a unique inclusion of categories such that ffi P;S (1) = P ffi : L S (S)j Ob(L q )!L q for all P ; and such that (a) ffi P;P (g) =ffi P (g) for all g 2 P C S (P ), and (b) ß(ffi P;Q (g)) = c g 2 Hom(P; Q) for all g 2 N S (P; Q). In addition, the following hold. (c) If we set Q P for L q. = ffi P;Q(1) for all P» Q, then f Q P g is a compatible set of inclusions (d) For any P C Q» S, where P and Q are both F-quasicentric and P is fully centralized in F, and any morphism ψ 2 Aut L q(p ) which normalizes ffi P;P (Q), there is a unique ψ b 2 AutL q(q) such that ψ b ffi Q = Q P P ffi ψ. Furthermore, for any g 2 Q, ψffi P;P (g)ψ 1 = ffi P;P (ß(b ψ)(g)). (e) Every morphism ' 2 Mor L q(p; Q) in L q is a composite ' = Q P 0 ffi ' 0 for a unique morphism ' 0 2 Iso L q(p; P 0 ), where P 0 =Im(ß(')). Proof. For each P and Q, and each g 2 N S (P; Q), there is by Proposition 1.12(b) a unique morphism ffi P;Q (g) such that ffi S (g) ffi P = Q ffi ffi P;Q (g): This defines ffi on morphism sets, and also allows us to define Q = ffi P P;Q(1). Then by the axioms in Definition 1.9, f Q P g is a compatible set of inclusions for L q, and ffi is a functor which satisfies (a), (b), and (c). Point (e) is a special case of Proposition 1.12(b) (where P 0 =Im(ß('))).

10 10 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER If ffi P;Q (g) = ffi P;Q (g 0 ) for g; g 0 2 N G (P; Q), then ffi S (g) ffi P = ffi S (g 0 ) ffi P, and hence g = g 0 by [5A1, Lemma 3.9]. Thus each ffi P;Q is injective. It remains to prove (d). Set ' = ß(ψ) 2 Aut F (P ) for short. Since ψ normalizes ffi P;P (Q), for all g 2 Q there is h 2 Q such that ψffi P;P (g)ψ 1 = ffi P;P (h), and this implies the relation 'c g ' 1 = c h in Aut F (P ). Thus Q is contained in N '. So by axiom (II) (and since P is fully centralized), ' extends to ' 2 Iso F (Q; Q 0 ) for some P C Q 0» S. Let b ψ0 2 Iso L q(q; Q 0 ) be any lifting of ' to L q. By axiom (A) q (and since ψ is an isomorphism), there is x 2 C S (P ) such that Q0 P ffi ffi P (x) ffi ψ = b ψ0 ffi Q P. By (C) q, b ψ0 ffi Q (Q)b ψ 1 0 = ffi Q 0(Q 0 ), and hence after restriction, ffi P (x) ffi ψ conjugates ffi P;P (Q) to ffi P;P (Q 0 ). Since ψ normalizes ffi P;P (Q), this shows that ffi P (x) conjugates ffi P;P (Q) toffi P;P (Q 0 ), and hence (since ffi P;P is injective) that xqx 1 = Q 0. We thus have the following commutative diagram: Q P P Q ψ! P ffi P (x)! P # b ψ0! Q 0 Q0 P # ffi P (x) 1! P Q P # ffi Q;Q 0 (x) 1! Q. So if we set b ψ = ffiq;q 0(x) 1 ffi b ψ0, then b ψ 2 AutL q(q) and Q P ffi ψ = b ψ ffi Q P. The uniqueness of ψ b follows from [5A1, Lemma 3.9]. Finally, for any g 2 Q, bψffi Q (g)b ψ 1 = ffi Q (ß(b ψ)(g)) by (C)q, and hence ψffi P;P (g)ψ 1 = ffi P;P (ß(b ψ)(g)) since morphisms have unique restrictions (Proposition 1.12(b) again). Once we have fixed a compatible set of inclusions f Q P g in a linking system L q, then for any ' 2 Mor L q(p; Q), and any P 0» P and Q 0» Q such that ß(')(P 0 )» Q 0, there is a unique morphism ' 2 Mor L q(p 0 ;Q 0 ) such that Q Q 0 ffi ' = ' ffi PP the restriction of '. 0. We think of ' as Note, however, that all of this depends on the choice of morphisms P 2 Mor L q(p; S) which satisfy the hypotheses of axiom (D) q, and that not just any lifting of the inclusion incl 2 Hom F (P; S) can be chosen. To see why, assume for simplicity that P is also fully centralized. From the axioms in Definition 1.9 and Proposition 1.12(b), we see that if P ; 0P 2 Mor L q(p; S) are two liftings of incl 2 Hom F (P; S), then 0P = P ffi ffi P (g) for some unique g 2 C S (P ). But if P satisfies the conditions of (D) q, then 0P also satisfies those conditions only if g 2 Z(C S (P )). One situation where the choice of inclusion morphisms is useful is when describing the fundamental group of jlj or of its p-completion. For any group, we let B( ) denote the category with one object, and with morphism monoid the group. Recall that jlj ' jl q j (Proposition 1.12(a)), so we can work with either of these categories; we will mostly state the results for jl q j. Let the vertex S be the basepoint ofjl q j. For each morphism ' 2 Mor L q(p; Q), let J(') 2 ß 1 (jl q j) denote the homotopy class of the loop Q ' P 1 in jl q j (where paths are composed from right to left). This defines a functor J : L q!b(ß 1 (jl q j)); where all objects are sent to the unique object of B(ß 1 (jl q j)), and where all inclusion morphisms are sent to the identity. Let j : S! ß 1 (jl q j) denote the composite of J with the distinguished monomorphism ffi S : S! Aut L (S). The next proposition describes how J is universal among functors of this type, and also includes some other technical results for later use about the structure of ß 1 (jlj).

11 EXTENSIONS OF P -LOCAL FINITE GROUPS 11 Proposition Let (S; F; L) be a p-local finite group, and let L q be the associated quasicentric linking system. Assume a compatible set of inclusions f Q P g has been chosen for L q. Then the following hold. (a) For any group, and any functor : L q!b( ) which sends inclusions to the identity, there is a unique homomorphism μ : ß 1 (jl q j)! such that = B( μ ) ffi J. (b) For g 2 P» S with P F-quasicentric, J(ffi P (g)) = J(ffi S (g)). In particular, J(ffi P (g)) = 1 in ß 1 (jl q j) if and only if ffi P (g) is nulhomotopic as a loop based at the vertex P of jl q j. (c) If ff 2 Mor L q(p; Q), and ß(ff)(x) =y, then j(y) =J(ff)j(x)J(ff) 1 in ß 1 (jl q j). (d) If x and y are F-conjugate elements of S, then j(x) and j(y) are conjugate in ß 1 (jl q j). Proof. Clearly, any functor : L q!b( ) induces a homomorphism μ = ß 1 (j j) between the fundamental groups of their geometric realizations. If sends inclusion morphisms to the identity, then = B( ) μ ffi J by definition of J. The other points follow easily, using condition (C) q for quasicentric linking systems. Point (d) is shown by first reducing to a map between centric subgroups of S which sends x to y. We finish this introductory section with two unrelated results which will be needed later in the paper. The first is a standard, group theoretic lemma. Lemma Let Q C P be p-groups. If ff is a p 0 -automorphism of P which acts as the identity on Q and on P=Q, then ff =Id P. Equivalently, the group of all automorphisms of P which restrict to the identity on Q and on P=Q is a p-group. Proof. See [Go, Corollary 5.3.3]. The following proposition will only be used in Section 4, but we include it here because it seems to be of wider interest. Note, for any fusion system F over S, any subgroup P» S fully normalized in F, and any P 0 which iss-conjugate to P, that P 0 is also fully normalized in F since N S (P 0 )iss-conjugate to N S (P ). Proposition Let F be a saturated fusion system over a p-group S. Then for any subgroup P» S, the set of S-conjugacy classes of subgroups F-conjugate to P and fully normalized in F has order prime to p. Proof. By [BLO2, Proposition 5.5], there is an (S; S)-biset Ω which, when regarded as a set with (S S)-action, satisfies the following three conditions. (a) The isotropy subgroup of each point inωis of the form P ' def = f(x; '(x)) j x 2 Pg for some P» S and some ' 2 Hom F (P; S). (b) For each P» S and each ' 2 Hom F (P; S), the two structures of (S P )-set on Ω obtained by restriction and by Id ' are isomorphic. (c) jωj=jsj 1 (mod p).

12 12 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER Note that by (a), the actions of S 1 and 1 S on Ω are both free. Now fix a subgroup P» S. Set S 2 =1 S for short, and let Ω 0 Ω be the subset such that Ω 0 =S 2 = (Ω=S 2 ) P. In other words, Ω 0 is the set of all x 2 Ω such that for each g 2 P, there is some h 2 S satisfying (g; h) x = x. Since the action of S 2 on Ω is free, this element h 2 S is uniquely determined for each x 2 Ω 0 and g 2 P. Let (x): P! S denote the function such that for each g 2 P, (g; (x)(g)) x = x. The isotropy subgroup at x of the (P S)-action is thus the subgroup P (x) = f(g; (x)(g)) j g 2 Pg; and by (a), (x) 2 Hom F (P; S). This defines a map : Ω 0! Hom F (P; S): By definition, for each ' 2 Hom F (P; S), 1 (') is the set of elements of Ω fixed by P '. By condition (b) above, the action of P P on Ω induced by the homomorphism 1 ' 2 Hom(P P; S S) is isomorphic to the action defined by restriction, and thus j 1 (')j = j 1 (incl)j. This shows that the point inverses of all have the same fixed order k. Now let Rep F (P; S) = Hom F (P; S)=Inn(S): the set of S-conjugacy classes of morphisms from P to S. Let P be the set of S-conjugacy classes of subgroups F-conjugate to P, and let P fn P be the subset of classes of subgroups fully normalized in F. If x 2 Ω 0 and (x) =', then for all s 2 S and g 2 P, (1;s) x =(g; s'(g)) x =(g; c s ffi '(g)) (1;s) x; and this shows that ((1;s) x) =c s ffi (x). Thus induces a map μ : (Ω=S 2 ) P =S 2 Im( ) =Ω 0 =S 2! Rep F (P; S)!P; where Rep F (P; S) = Hom F (P; S)=Inn(S). Furthermore, jω=s 2 j 1 (mod p) by (c), and thus j(ω=s 2 ) P j 1 (mod p). For each P 0 which isf-conjugate to P, there are jsj=jn S (P 0 )j distinct subgroups in the S-conjugacy class [P 0 ]. Hence there are j Aut F (P )j jsj=jn S (P 0 )j elements of Hom F (P; S) whose image lies in [P 0 ]. Since each of these is the image of k elements in Ω 0, this shows that j μ 1 ([P 0 ])j = k j Aut F (P )j=jn S (P 0 )j: Thus jn S (P 0 )j fi fi k j AutF (P )j for all P 0 F-conjugate to P, and so μ 1 ([P 0 ]) has order a multiple of p if P 0 is not fully normalized in F (if [P 0 ] 2PrP fn ). Hence j μ 1 (P fn )j j(ω=s 2 ) P j 1 (mod p) : So if we set m = jn S (P 0 )j for [P 0 ] 2 P fn (i.e., the maximal value of jn S (P 0 )j for P 0 F-conjugate to P ), then j μ 1 Aut (P fn )j = jp fn j k j F (P )j ; m and thus jp fn j is prime to p. Using a similar argument, one can also show that the set Rep fc F(P; S) of elements of Rep F (P; S) whose image is fully centralized also has order prime to p.

13 EXTENSIONS OF P -LOCAL FINITE GROUPS The fundamental group of jlj^p The purpose of this section is to give a simple description of the fundamental group of jlj^p, for any p-local finite group (S; F; L), purely in terms of the fusion system F. The result is analogous to the (hyper-) focal subgroup theorem for finite groups, as we explain below. In Section 1, we defined a functor J : L q!b(ß 1 (jlj)), for any p-local finite group (S; F; L), and a homomorphism j = J ffi ffi S from S to ß 1 (jlj). Let fi : S! ß 1 (jlj^p ) be the composite of j with the natural homomorphism from ß 1 (jlj) toß 1 (jlj^p ). In [BLO2, Proposition 1.12], we proved that fi : S! ß 1 (jlj^p ) is a surjection. In this section, we will show that Ker(fi) isthehyperfocal subgroup of F, defined by Puig [Pu4] (see also [Pu3]). Definition 2.1. For any saturated fusion system F over a p-group S, the hyperfocal subgroup of F is the normal subgroup of S defined by O p F (S) =Ω g 1 ff(g) fi fi g 2 P» S; ff 2 O p (Aut F (P )) ff : We will prove, for any p-local finite group (S; F; L), that ß 1 (jlj^p ) ο = S=O p F (S). This is motivated by Puig's hyperfocal theorem, and we will also need that theorem in order to prove it. Before stating Puig's theorem, we first recall the standard focal subgroup theorem. If G is a finite group and S 2 Syl p (G), then this theorem says that S [G; G] (the focal subgroup) is the subgroup generated by all elements of the form x 1 y for x; y 2 S which are G-conjugate (cf. [Go, Theorem 7.3.4] or [Suz2, 5.2.8]). The quotient group S=(S [G; G]) is isomorphic to the p-power torsion subgroup of G=[G; G], and can thus be identified as a quotient group of the maximal p-group quotient G=O p (G). Since G=O p (G) isap-group, G = S O p (G), and hence G=O p (G) ο = S=(S O p (G)). Hence S O p (G)» S [G; G]. This subgroup S O p (G) is what Puig calls the hyperfocal subgroup, and is described by the hyperfocal subgroup theorem in terms of S and fusion. For S 2 Syl p (G) asabove, let O p G (S) bethe normal subgroup of S defined by O p (S) G =Op F (S) =Ω fi S (G) g 1 ff(g) fi ff g 2 P» S; ff 2 O p (Aut G (P )) = Ω [g; x] fi fi g 2 P» S; x 2 NG (P ) of order prime to p ff : Lemma 2.2 ([Pu3]). Fix a prime p, a finite group G, and a Sylow subgroup S 2 Syl p (G). Then O p G (S) =S Op (G). Proof. This is stated in [Pu3, x1.1], but the proof is only sketched there, and so we elaborate on it here. Following standard notation, for any P» S and any A» Aut(P ), we write [P; A] =hx 1 ff(x) j x 2 P; ff 2 Ai. Thus O p G (S) is generated by the subgroups [P; O p (Aut G (P ))] for all P» S. It is clear that O p (S)» S G Op (G); the problem is to prove the opposite inclusion. Set G = O p (G) and S = S G for short. Then [G ;G ] has index prime to p in G, so it contains S. By the focal subgroup theorem (cf. [Go, Theorem 7.3.4]), applied to S 2 Syl p (G ), S is generated by all elements of the form x 1 y for x; y 2 S which are G -conjugate. Combined with Alperin's fusion theorem (in Alperin's original version [Al] or in the version of Theorem 1.5(a)), this implies that S is generated by all subgroups [P; N G (P )] for P» S such that N S (P ) 2 Syl p (N G (P )). (This last condition is equivalent to P being fully normalized in F S (G ).) Also, N G (P ) is

14 14 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER generated by O p (N G (P )) and the Sylow subgroup N S (P ), so [P; N G (P )] is generated by [P; O p (N G (P ))]» O p G (S) and [P; N S (P )]» [S ;S ]. Thus S = ho p G (S); [S ;S ]i. Since O p (S) is normal in S (hence also normal in S G ), this shows that S =O p G (S) is equal to its commutator subgroup, which for a p-group is possible only if S =O p G (S) is trivial, and hence S = O p (S). G By Proposition 1.14, the key to getting information about ß 1 (jlj) is to construct functors from L or L q to B( ), for a group, which send inclusions to the identity. The next lemma is our main inductive tool for doing this. Whenever ff 2 Mor L q(p; P 0 ) and fi 2 Mor L q(q; Q 0 ) are such that P» Q, P 0» Q 0, we write ff = fij P to mean that ff is the restriction of fi in the sense defined in Section 1; i.e., Q0 P 0 ffi ff = fi ffi Q. P Lemma 2.3. Fix a p-local finite group (S; F; L), and let L q be its associated quasicentric linking system. Assume a compatible set of inclusions f Q Pg has been chosen for L q. Let H 0 be a set of F-quasicentric subgroups of S which is closed under F- conjugacy and overgroups. Let P be an F-conjugacy class of F-quasicentric subgroups maximal among those not in H 0, set H = H 0 [P, and let L H 0 subcategories with these objects. Assume, for some group, that 0 : L H 0!B( ) L H L q be the full is a functor which sends inclusions to the identity. Fix some P 2 P which is fully normalized in F, and fix a homomorphism P : Aut L q(p )!. Assume that () for all P Φ Q» N S (P ) such that Q is fully normalized in N F (P ), and for all ff 2 Aut L q(p ) and fi 2 Aut L q(q) such that ff = fij P, P (ff) = 0 (fi). Then there is a unique extension of 0 to a functor : L H!B( ) which sends inclusions to the identity, and such that (ff) = P (ff) for all ff 2 Aut F (P ). Proof. The uniqueness of the extension is an immediate consequence of Theorem 1.5(a) (Alperin's fusion theorem). To prove the existence of the extension, we first show that () implies the following (a prori stronger) statement: () for all Q; Q 0» S which strictly contain P, and for all fi 2 Mor L q(q; Q 0 ) and ff 2 Aut L q(p ) such that ff = fij P, P (ff) = 0 (fi). To see this, note first that it suffices to consider the case where P is normal in Q and Q 0. By assumption, ß(fi)(P )=ß(ff)(P )=P, hence ß(fi)(N Q (P ))» N Q 0(P ), and therefore fi restricts to a morphism fi 2 Mor L q(n Q (P );N Q 0(P )) by Proposition 1.12(b) (applied with ' = fi ffi Q N Q (P ) and ψ = Q0 N Q 0 (P ) ). Since N Q(P );N Q 0(P ) Ψ P, by the induction hypothesis, 0 (fi) = 0 (fi), so we are reduced to proving that P (ff) = 0 (fi). Thus fi 2 Mor NL q (P )(Q; Q 0 ). We now apply Alperin's fusion theorem (Theorem 1.5(a)) to the morphism ß(fi) in the fusion system N F (P ) (which is saturated by [BLO2, Proposition A.6]). Thus ß(fi) =' k ffi ffi' 1, where each ' i 2 Hom NF (P )(Q i 1;Q i )isthe restriction to Q i 1 of an automorphism b' i 2 Aut NF (P )(R i ), where R i Q i 1;Q i is an N F (P )-centric subgroup of N S (P ) which is fully normalized in N F (P ), and where Q = Q 0 and Q 0 = Q k. Each R i contains P, and hence is F-centric by [BLO2, Lemma 6.2]. For each b' i, also regarded as an automorphism in F, wechoose a lifting b fii 2 Aut L (R i ), and let fi i 2 Mor L q(q i 1;Q i ) be its restriction. By (A) q, fi = fi k ffi ffi fi 1 ffi ffi Q (g) for

15 some g 2 C S (Q), and hence EXTENSIONS OF P -LOCAL FINITE GROUPS 15 0 (fi) = 0 (fi k ) 0 (fi 1 ) 0 (ffi Q (g)) = 0 (b fik ) 0 (b fi1 ) 0 (ffi NS (P )(g)) = P (b fik j P ) P (b fi1 j P ) P (Id P )= P (fij P )= P (ff) ; where the third equality follows from (). This finishes the proof of (). We can now extend to be defined on all morphisms in L H not in L H 0. Fix such a morphism ' 2 Mor L q(p 1 ;Q). Set P 2 = ß(')(P 1 )» Q; then P 1 ;P 2 2 P, and ' = Q P 2 ffi ' 0 for some unique ' 0 2 Iso L q(p 1 ;P 2 ). By Lemma 1.3 (and then lifting to the linking category), there are isomorphisms ' i 2 Iso L q(n S (P i );N i ), for some N i» N S (P ) containing P, which restrict to isomorphisms ' i 2 Iso L q(p i ;P). Set ψ = ' 2 ffi' 0 ffi' Aut L q(p ). We have thus decomposed ' 0 as the composite ' 1 2 ffiψ ffi' 1, and can now define (') = (' 0 )= 0 (' 2 ) 1 P (ψ) 0 (' 1 ): Now let ' 0 =(' 0 2) 1 ffiψ 0 ffi' 0 1 be another such decomposition, where ' 0 i is the restriction of ' 0 i 2 Iso L q(n S (P i );N 0 i). We thus have a commutative diagram P ψ ' 1 P 1 ' 0 1! P # P ψ ψ ' 2 # P 2 ' 0 # ψ 0 ' 0 2! P, where for each i, ' i and ' 0 i are restrictions of isomorphisms ' i and ' 0 i defined on N S (P i ). To see that the two decompositions give the same value of ('), it remains to show that P (ψ 0 ) 0 (' 0 1 ffi (' 1) 1 )= 0 (' 0 2 ffi (' 2) 1 ) P (ψ): And this holds since 0 (' 0 i ffi (' i) 1 )= P (' 0 i ffi (' i) 1 ) by (). We have now defined on all morphisms in L H, and it sends inclusion morphisms to the identity by construction. By construction, sends composites to products, and thus the proof is complete. Lemma 2.3 provides the induction step when proving the following proposition, which is the main result needed to compute ß 1 (jlj^p ). Proposition 2.4. Fix a p-local finite group (S; F; L), and let L q be its associated quasicentric linking system. Assume a compatible set of inclusions f Q P g has been chosen for L q. Then there is a unique functor : L q!b(s=o p F (S)) which sends inclusions to the identity, and such that (ffi S (g)) = g for all g 2 S. Proof. The functor will be constructed inductively, using Lemma 2.3. Let H 0 Ob(L q ) be a subset (possibly empty) which is closed under F-conjugacy and overgroups. Let P be an F-conjugacy class of F-quasicentric subgroups maximal among those not in H 0, set H = H 0 [P, and let L H 0 with these objects. Assume that 0 : L H 0!B(S=O p F (S)) L H L q be the full subcategories has already been constructed, such that 0 (ffi S (g)) = g for all g 2 S (if S 2 H 0 ), and such that 0 sends inclusions to the identity.

16 16 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER Fix P 2 P which is fully normalized in F, and let ffi P;P : N S (P )! Aut L q(p ) be the homomorphism of Proposition Then Im(ffi P;P ) is a Sylow p-subgroup of Aut L q(p ), since Aut S (P ) 2 Syl p (Aut F (P )) by axiom (I). We identify N S (P ) as a subgroup of Aut L q(p ) to simplify notation. Then Aut L q(p )=O p (Aut L q(p )) ο = NS (P ) ffi N S (P ) O p (Aut L q(p )) = N S (P )=N 0 ; where by Lemma 2.2, N 0 is the subgroup generated by all commutators [g; x] for g 2 Q» N S (P ), and x 2 N AutL q (P )(Q) of order prime to p. In this situation, conjugation by x lies in Aut F (Q) by Proposition 1.13(d), and thus [g; x] =g c x (g) 1 2 OF p (S). We conclude that N 0» OF p (S), and hence that the inclusion of N S(P )into S extends to a homomorphism P : Aut L q(p )! S=O p F (S): We claim that condition () in Lemma 2.3 holds for 0 and P. To see this, fix P Φ Q» S such that P C Q and Q is fully normalized in N F (P ), and fix ff 2 Aut L q(p ) and fi 2 Aut L q(q) such that ff = fij P. We must show that P (ff) = 0 (fi). Upon replacing ff by ff k and fi by fi k for some appropriate k 1 (mod p), we can assume that both automorphisms have order a power of p. Since Q is fully normalized, Aut NS (P )(Q) is a Sylow subgroup of Aut NF (P )(Q). Hence (since any two Sylow p-subgroups of a finite group G are conjugate by an element of O p (G)), there is an automorphism fl 2 O p (Aut NL q (P )(Q)) such that flfifl 1 = ffi Q (g) for some g 2 N S (Q) N S (P ). In particular, 0 (fl) = 1 since it is a composite of automorphisms of order prime to p. Set fl = flj P ; then fl 2 O p (Aut L q(p )) and hence P (fl) =1. Using axiom (C) q and Lemma 1.12, we see that flfffl 1 = ffi P (g); and thus (since P (fl) =1and 0 (fl) = 1) that 0 (fi) = 0 (ffi Q (g)) = g = P (ffi P (g)) = P (ff): Thus, by Lemma 2.3, we can extend 0 to a functor defined on L H. Upon continuing this procedure, we obtain a functor defined on all of L q. For any finite group G, ß 1 (BG^p ) ο = G=Op (G). (This is implicit in [BK, xvii.3], and shown explicitly in [BLO1, Proposition A.2].) Hence our main result in this section can be thought of as the hyperfocal theorem for p-local finite groups. Theorem 2.5 (Hyperfocal subgroup theorem for p-local finite groups). Let (S; F; L) be a p-local finite group. Then ß 1 (jlj^p ) ο = S=O p F (S): More precisely, the natural map fi : S! ß 1 (jlj^p ) is surjective, and Ker(fi) =O p F (S). Proof. Let : L!B(S=OF p (S)) be the functor of Proposition 2.4, and let j j be the induced map between geometric realizations. Since jb(s=of p (S))j is the classifying space of a finite p-group and hence p-complete, j j factors through the p-completion jlj^p. Consider the following commutative diagram: S j H HH! ß 1 (jlj) H HH ( )^p fi H H # j ß 1 (jlj^p ) H HH ß 1 (j j) H HH μ = ß 1 (j j^p ) H H j! S=O p F (S).

17 EXTENSIONS OF P -LOCAL FINITE GROUPS 17 Here, fi is surjective by [BLO2, Proposition 1.12]. Also, by construction, the composite μ ffi fi = ß 1 (j j) ffi j is the natural projection. Thus Ker(fi)» OF p (S), and it remains to show the opposite inclusion. Fix g 2 P» S and ff 2 Aut F (P ) such that ff has order prime to p; we want to show that g 1 ff(g) 2 Ker(fi). Since Ker(fi) is closed under F-conjugacy (Proposition 1.14(d)), for any ' 2 Iso F (P; P 0 ), g 1 ff(g) is in Ker(fi) if '(g 1 ff(g)) is in Ker(fi). In particular, since '(g 1 ff(g)) = '(g) 1 ('ff' 1 )('(g)), we can assume that P is fully centralized in F. Then, upon extending ff to an automorphism of P C S (P ), which can be assumed also to have order prime to p (replace it by an appropriate power if necessary), we can assume that P is F-centric. In this case, by Proposition 1.14(c), j(g) and j(ff(g)) are conjugate in ß 1 (jlj) by an element of order prime to p, and hence are equal in ß 1 (jlj^p ) since this is a p-group. This shows that g 1 ff(g) 2 Ker(fi), and finishes the proof of the theorem. The following result, which will be useful in Section 5, is of a similar nature, but much more elementary. Proposition 2.6. Fix a p-local finite group (S; F; L), and let L q be its associated quasicentric linking system. Then the induced maps ß 1 (jlj)! ß 1 (jf c j) and ß 1 (jl q j)! ß 1 (jf q j) are surjective, and their kernels are generated by elements of p-power order. Proof. We prove this for jl q j and jf q j; a similar argument applies to jlj and jf c j. Recall from Section 1 that we can regard ß 1 (jl q j) as the group generated by the loops J(ff), for ff 2 Mor(L q ), with relations given by composition of morphisms and making inclusion morphisms equal to 1. In a similar way, we regard ß 1 (jf q j) as being generated by loops J(') for ' 2 Mor(F q ). Since every morphism ff 2 Hom F q(p; Q) has a lifting to a morphism of L q, the map ß # : ß 1 (jl q j)! ß 1 (jf q j) induced by the projection functor ß : L q!f q is an epimorphism. Write JC for the normal subgroup of ß 1 (jl q j) generated by the loops j(g) =J(ffi P (g)), In particular, JC is for all F-quasicentric subgroups P» S and all g 2 C S (P ). generated by elements of p-power order. Since ß # (j(g)) = 1, JC is contained in the kernel of ß #, and we have a factorization bß # : ß 1 (jl q j)=jc! ß 1 (jf q j). Define an inverse s: ß 1 (jf q j)! ß 1 (jl q j)=jc as follows. Given a morphism ff 2 Hom F q(p; Q), choose a lifting μff in L q, and set s(ff) =[J(μff)]. If μff; μff 0 2 Hom L q(p; Q) are two liftings of ff, then there is an isomorphism χ: P 0! P in L q where P 0 is fully centralized, and an element g 2 C S (P 0 ), such that μff 0 = μff ffi (χ ffi ffi P 0(g) ffi χ 1 ). Since χffiffi P 0(g)ffiχ 1 represents a loop that belongs to JC,[J(μff)] = [J(μff 0 )]. Thus the definition of s does not depend on the choice of μff. It remains to show that s preserves the relations among the generators. But we clearly have that s([j(incl Q P )]) = [J( Q P )]=1. Also, if ff and fi are composable morphisms of F q, and μff and μ fi are liftings to L q, then μff ffi μ fi is a lifting of ff ffi fi, and hence s(ff)s(fi) =[J(μff)][J( μ fi)] = [J(μff ffi μ fi)] = s(ff ffi fi). This shows that ß # induces an isomorphim ß 1 (jl q j)=jc ο = ß1 (jf q j). 3. Subsystems with p-solvable quotient In this section, we prove some general results about subsystems of saturated fusion systems and p-local finite subgroups: subsystems with p-group quotient or quotient of