___!. But such a morphism clearly cannot send * *Q to itself. Thus Q is strongly closed in F. If P S does not contain Q, then NPQ (* *P )=P is a nontrivial p-subgroup of OutF (P ), which is in fact normal there. To see * *normality notice that if ff 2 AutF (P ) then ff extends to ~ff2 AutF (P Q) since Q normal* * in F, so for all x 2 NPQ (P ) we have ffcxff-1 = (~ffcx~ff-1)|P = c~ff(x)2 AutPQ (P ). H* *ence such a subgroup P cannot be F-radical. Thus, all F-radical subgroups of S contain Q. T* *his shows (a) ) (b). Condition (b) clearly implies (c), and so it remains to show (c) ) (a). Assum* *e that Q is weakly closed in F, and that all F-radical subgroups contain Q. Then by Al* *perin's fusion theorem, each morphism in F is a composite of morphisms, each of which i* *s the restriction of a morphism between subgroups containing Q, and which necessarily* * sends Q to itself (since_Q is weakly closed). In other words, each ' 2 Hom F(P, P 0) * *extends to a morphism ' 2 Hom F(P Q, P 0Q) which sends Q to itself, and hence Q is norm* *al in F. 2. Centric and radical subgroups determine saturation Given a fusion system which is not known to come from a group (or a block), it turns out to be difficult in general to show that it is saturated when using th* *e definition directly. This is one of the obstacles one encounters when trying to construct * *p-local finite groups which do not come from groups. The main result of this section, Theorem 2.2, says that it suffices to check * *the axioms of saturation on the centric subgroups, in the sense that any fusion system whi* *ch satisfies these axioms for its centric subgroups generates a saturated fusion s* *ystem in a way made precise below. In fact, our result is stronger than that. We prove* * that it suffices to check the axioms of saturation on those subgroups which are cent* *ric and radical, and a much weaker condition on the centric subgroups which are not rad* *ical. Before stating the main results, we make some definitions. Definition 2.1. Let F be any fusion system over a finite p-group S, and let H b* *e a set of subgroups of S closed under conjugation. (a)F is H-generated if every morphism in F is a composite of restrictions of mo* *r- phisms in F between subgroups in H. (b)F is H-saturated if conditions (I) and (II) hold in F for all subgroups P 2 * *H. In terms of these definitions, Alperin's fusion theorem for abstract fusion s* *ystems (in the form shown in [BLO2 , Theorem A.10]) can be reformulated by saying that if* * F is a SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 7 saturated fusion system over S, and H is the family of F-centric, F-radical sub* *groups of S, then F is H-generated. Our main result in this section can be thought of as a converse to this form * *of the fusion theorem. In practice, it often simplifies the task of deciding whether * *a fusion system is saturated or not. As one example, the proof of [BLO2 , Proposition 9* *.1] _ the proof that the fusion systems constructed there are saturated _ becomes far sim* *pler when we can use Theorem 2.2, applied with H the set of F-centric subgroups of S. Theorem 2.2. Let F be a fusion system over a finite p-group S and let H be a set of subgroups of S closed under conjugation in F that contains all F-centric, F-* *radical subgroups of S. Assume that F is H-generated and H-saturated, and that (*)each F-conjugacy class of subgroups of S which are F-centric but not in H co* *ntains at least one subgroup P such that OutS (P ) \ Op(Out F(P )) 6= 1. Then F is saturated. Note that the condition that H contain all F-centric, F-radical subgroups of * *S is in fact implied by (*), but we keep it in the statement for the sake of emphasis. We first discuss the relation between conditions (I) and (II) in Definition 1* *.3, and certain other, similar conditions on fusion systems. We recall the definition o* *f N' for any given ' 2 Mor F(P, Q), N' = {x 2 NS(P ) | 'cx'-1 2 AutS('(P ))}. Lemma 2.3. Let F be a fusion system over a p-group S, and let H be a set of sub* *groups of S closed under F-conjugacy. Consider the following conditions on F: (I)H :For each fully normalized subgroup P 2 H, P is fully centralized, and Aut* *S(P ) 2 Sylp(Aut F(P )). (I0)H :Each P 2 H is F-conjugate to a fully centralized subgroup P 02 H such th* *at Aut S(P 0) 2 Sylp(Aut F(P 0)). (II)H :For each P 2 H, and each '_2 Hom F(P, S) such that '(P ) is fully centra* *lized in F, ' extends to a morphism ' 2 Hom F(N', S). (IIA)H :Each F-conjugacy class P H contains a fully normalized subgroup bP2 P with the following property: for all P 2 P, there exists ' 2 Hom F(NS(P ), N* *S(Pb)) such that '(P ) = bP. (IIB)H :For each fully normalized subgroup bP2 H and each ' 2 AutF (Pb), there * *is a _ morphism ' 2 Hom F(N', NS(Pb)) which extends '. Then (a) (I)H () (I0)H ; and (b) (I)H + (II)H =) (IIA)H + (IIB)H =) (II)H . Proof.(a) Condition (I)H clearly implies (I0)H , since every P S is conjugate* * to a fully normalized subgroup. To see the converse, assume P 2 H is fully normalize* *d. By (I0)H we can choose P 02 H which is F-conjugate to P , fully centralized, and s* *atisfies 8 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER AutS(P 0) 2 Sylp(Aut F(P 0)). Then | AutS(P 0)|.|CS(P 0)| = |NS(P 0)| |NS(P )| = | AutS(P )|.|CS(P )| | AutS(P 0)|.|CS(P 0* *)| : the first inequality holds since P is fully normalized, and the second by the a* *ssumptions on P 0. Thus all of these inequalities are equalities, and so P is fully centra* *lized and AutS(P ) 2 Sylp(Aut F(P )). (b) Assume (I)H and (II)H hold; we next prove that this implies (IIA)H and (IIB* *)H . We first check condition (IIB)H . Let b'= 'PbO' where 'Pbis the inclusion of bP* *in S. Since bPis fully normalized, condition_(I)H implies that bPis also fully centra* *lized. By condition (II)H , b'extends to ' 2 Hom F(Nb', S), where Nb'= {g 2 NS(Pb) | b'cgb'-12 AutS('b(Pb))} = {g 2 NS(Pb) | 'cg'-1 2 AutS(Pb)} =* * N' . _ Furthermore, Im(' ) NS(Pb), since bPC Nb'. Next we check that (IIA)H holds. Fix an F-conjugacy class P H, and choose a fully normalized subgroup bP 2 P. Since (I)H holds, Pb is also fully centraliz* *ed, and AutS(Pb) 2 Sylp(Aut F(Pb)). Thus for any P 2 P and any ' 2 IsoF(P, bP), there e* *xists Ø 2 AutF (P ) such that 'Ø AutS(P )Ø-1'-1 AutS(Pb). Then N'ffldef={g 2 NS(P ) | 'ØcgØ-1'-1 2 AutS(Pb)} = NS(P ) , _ _ and hence the morphism 'Ø extends to ' 2 Hom F(NS(P ), S) by (II)H . Then '(P )* * = _ 'Ø(P ) = bP, and hence Im(' ) NS(Pb). It remains to prove the last implication. Assume (IIA)H and (IIB)H ; we must * *prove (II)H . Fix P 2 H and ' 2 Hom F (P, S) such that P 0def='(P ) is fully central* *ized in F. Using (IIA)H , choose a fully normalized subgroup bPwhich is F-conjugate to * *P, P 0, and morphisms _ 2 Hom F(NS(P ), NS(Pb)) and _02 Hom F(NS(P 0), NS(Pb)) such that _(P ) = _0(P 0) = bP. Set e'= (_0|P0) O' O(_|P)-1 2 AutF (Pb). For each x 2 N', there exists y 2 NS(P 0) such that 'cx'-1 = cy as elements of Aut(P 0). Then as automorphisms of bP, e'c_(x)e'-1= c_0(y). This shows that _(N* *') Ne'. By (IIB)H , e'extends to a morphism b'2 Hom F(Ne', NS(Pb)). Now fix x 2 N', and let y 2 NS(P 0) be such that 'cx'-1 = cy as elements of A* *ut(P 0). The elements b'_(x), _0(y) 2 NS(Pb) induce the same conjugation action on bP, a* *nd thus differ by an element in CS(Pb). Also, since P 0is fully centralized, _0(CS(P 0)* *) = CS(Pb), and hence b'_(x) 2 _0(y).CS(Pb) = _0(y.CS(P 0)) _0(NS(P 0)). _ * * 0 Thus b'_(N') _0(NS(P 0)), and so b'O_ factors through some ' 2 Hom F(N', NS(P* * )) which extends '. This finishes the proof of condition (II)H . As an immediate consequence of Lemma 2.3, we obtain the following alternative characterization of the conditions of saturation: a fusion system F over S is s* *aturated if and only if it satisfies the conditions (I0)H , (IIA)H and (IIB)H where H is* * the set of all subgroups P S. SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 9 Notation. Following the notation introduced in Lemma 2.3 for the conditions sta* *ted there, we also write (-)Q or (-)>Q for (-)H when H = {Q} or H = {P | Q P S}, respectively. Given a fusion system F over S, let S be the set of all subgroups* * of S. For P S, let S P S>P be the sets of subgroups of S which contain, or strict* *ly contain, P . We will now prove two lemmas which allow us to prove Theorem 2.2 by induction on the number of F-conjugacy classes of subgroups of S not in H. Lemma 2.4. Let F be a fusion system over a finite p-group S, and let H be a set* * of subgroups of S closed under conjugacy. Let P be a conjugacy class of subgroups * *of S which is maximal among those not in H. Assume F is H-generated and H-saturated. Then the following hold for any P 2 P which is fully normalized in F: (a)NF (P ) is S>P -saturated. (b)Each ' 2 AutF (P ) is a composite of restrictions of morphisms in NF (P ) be* *tween subgroups strictly containing P . (c)F is (H [ P)-saturated if NF (P ) is S P -saturated. Proof.By a proper P-pair will be meant a pair (Q, P ), where P Q NS(P ) and P 2 P. Two proper P-pairs (Q, P ) and (Q0, P 0) will be called F-conjugate if t* *here is an isomorphism ' 2 IsoF(Q, Q0) such that '(P ) = P 0. A proper P-pair (Q, P ) * *will be called fully normalized if |NNS(P)(Q)| |NNS(P0)(Q0)| for all (Q0, P 0) in * *the same F-conjugacy class. The proof of the lemma is based on the following statements, whose proof will* * be carried out in Steps 1 to 4. (1)If (Q, P ) is a fully normalized proper P-pair, then Q is fully centralized * *in F and AutNS(P)(Q) 2 Sylp(Aut NF(P)(Q)). (2)For each proper P-pair (Q, P ), and each fully normalized proper P-pair (Q0,* * P 0) which is F-conjugate to (Q, P ), there is some _ 2 Hom F(NNS(P)(Q), NNS(P0)(* *Q0)) such that _(P ) = P 0and _(Q) = Q0. (3)There is a subgroup Pb 2 P which is fully centralized in F, and which has the property that for all P 2 P, there is a morphism ' 2 Hom F(NS(P ), NS(Pb)) s* *uch that '(P ) = bP. (4)Let (Q, P ) be a proper P-pair such that P is fully normalized in F. If Q is* * fully normalized in NF (P ), then (Q, P ) is fully normalized. If Q is fully centr* *alized in NF (P ), then Q is fully centralized in F. Note that point (3) implies that bPis fully normalized in F, and that any other* * P 02 P which is fully normalized in F has the same properties. Assuming points (1)-(4) have been shown, the lemma is proven as follows: (a) We show that conditions (I) and (II) hold in NF (P ) for all Q 2 S>P . If * *Q P is fully normalized in NF (P ), then the proper P-pair (Q, P ) is fully normali* *zed by (4), and hence condition (I) holds in NF (P ) by (1). It remains to show condi* *tion (II). Also, by (4) again, if P Q NS(P ) and Q is fully centralized in NF (* *P ), then it is fully centralized in F. Hence (II) holds automatically for morphism* *s ' 2 Hom NF(P)(Q, NS(P )), since it holds in F. 10 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER (b) Fix ' 2 AutF (P ). Since F is H-generated, there are subgroups P = P0, P1, . .,.Pk = P in P, and Pi Qi S, and morphisms 'i 2 Hom F(Qi, S) (0 i k - 1), such that 'i(Pi) = Pi+1 and ' = 'k-1|Pk-1O. .O.'0|P0. Upon replacing each Qi by NQi(Pi) Pi, we can assume that Qi NS(Pi). By (3), there are morphisms Øi2 Hom F(NS(Pi), NS(P )) for each* * i such that Øi(Pi) = P , where we take Ø0 = Øk to be the identity. Upon replacing* * each 'iby ØiO'iOØ-1i2 Hom F(Øi(Qi), S), we can arrange that Pi= P for all i. Thus ' * *is a composite of restrictions of morphisms in NF (P ) between subgroups strictly co* *ntaining P . (c) Assume that NF (P ) is S P -saturated. By Lemma 2.3, it is enough to check* * that Conditions (I0)P, (IIA)P, and (IIB)P are satisfied in F. Condition (IIA)P follo* *ws from point (3). Since Aut F(P ) = Aut NF(P)(P ), it is clear that Condition (IIB)P * *holds in F. Finally, since AutS(P ) = AutNS(P)(P ), and since the properties of bPas des* *cribed in point (3) hold for every fully normalized subgroup, (I)P also holds, and thi* *s proves that F is (H [ P)-saturated. In order to finish the proof, it remains to prove points (1)-(4). Step 1: For any proper P-pair (Q, P ), let KP Aut(Q) be defined by KP = {' 2 Aut(Q) | '(P ) = P } . If the pair (Q, P ) is fully normalized, then Q is fully KP-normalized in F in * *the sense of [BLO2 , Definition A.1]. Hence by [BLO2 , Proposition A.2(a)], Q is fully * *centralized and Aut NS(P)(Q) = AutS(Q) \ KP 2 SylpAut F(Q) \ KP = Sylp(Aut NF(P)(Q)). More precisely, this follows from the proof of [BLO2 , Proposition A.2], where* * we need only know that F satisfies the axioms of saturation on subgroups containing Q a* *nd its conjugates. Step 2: Let (Q0, P 0) be any fully normalized proper P-pair of subgroups of S * *which is F-conjugate to (Q, P ). Let ' 2 IsoF(Q, Q0) such that '(P ) = P 0. Since (* *Q0, P 0) is fully normalized, Q0is fully centralized and AutNS(P0)(Q0) 2 Sylp(Aut NF(P0)* *(Q0)) by (1). Since ' AutNS(P)(Q)'-1 is a p-subgroup of AutNF(P0)(Q0), there is some morphi* *sm ff 2 AutNF(P0)(Q0) such that ff' AutNS(P)(Q)'-1ff-1 AutNS(P0)(Q0). Since F is H-saturated, ff' extends to a morphism fff'2 Hom F (Nff', S) by (II)* *Q, where Nff'= {x 2 NS(Q) | ff'cx'-1ff-1 2 AutS(Q0)} NNS(P)(Q) . Set _ = fff'|NNS(P)(Q)2 Hom F(NNS(P)(Q), S). By construction, Im(_) NNS(P0)(Q* *0). Moreover _|Q = ff'|Q, _|P = ff'|P, and hence _(P ) = P 0and _(Q) = Q0. Step 3: We first show, for any P, P 02 P, that there are P 002 P, and morphis* *ms _ 2 Hom F(NS(P ), NS(P 00)) and _02 Hom F(NS(P 0), NS(P 00)), such that _(P ) =* * P 00= _0(P 0). Let T be the set of all sequences 0 , = P = P0, Q0, '0; P1, Q1, '1; . .;.Pk-1, Qk-1, 'k-1; Pk = P SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 11 such that Pi Qi NS(Pi), 'i 2 Hom F(Qi, NS(Pi+1)), and 'i(Pi) = Pi+1. Let Tr T be the subset of those , for which there is no 1 i k - 1 such that Qi= NS(Pi) = 'i-1(Qi-1). Let T --R! Tr be the "reduction" map, which removes any Pisuch that Qi= NS(Pi) = 'i-1(Qi-1) (and replaces 'i-1and 'iby their composite). Define I(,) = {0 i k - 1 | Qi NS(Pi) and 'i(Qi) NS(Pi+1)}. If , 2 T and I(,) 6= ;, define ~(,) = min [Qi: Pi] p. i2I(,) The main observation needed to prove point (3) is that there exists an element * *, 2 Tr such that I(,) = ;. Note first that T 6= ;, since F is H-generated (and since Q* * P implies NQ(P ) P ). Hence (by the existence of the retraction functor R) Tr 6* *= ;. Fix an element , 2 Tr such that I(,) 6= ;. We will construct b,2 Tr such that* * either I(b,) = ;, or ~(b,) > ~(,). For each i 2 I(,), choose a fully normalized proper* * P-pair (Q00i, Pi00) which is F-conjugate to (Qi, Pi), and apply (2) to choose homomorp* *hisms 0 _i2 Hom F NNS(Pi)(Qi), S and _i2 Hom F NNS(Pi+1)('i(Qi)), S such that _i(Pi) = _0i(Pi+1) = Pi00and _i(Qi) = _0i(Qi+1) = Q00i. Set eQi= NNS(Pi)(Qi) Qi and Qe0i= _0iNNS(Pi+1)('i(Qi)) _0i(Qi). Note that if (Q, P ) is a proper P-pair with P Q NS(P ), then NNS(P)(Q) Q. Thus upon replacing the sequence (Pi, Qi, 'i; Pi+1) in , by (Pi, eQi, _i; Pi00, eQ0i, (_0i)-1; Pi+1) and similarly for the other components of I(,), we obtain a new element ,0 2 T , such that either I(,0) = ; or ~(,0) > ~(,) (by construction [Qei: Pi] > [Qi : P* *i] and [Qe0i: P 0] > [Qi : Pi]). Then b,= R(,0) 2 Tr is also such that either I(b,) =* * ; or ~(b,) > ~(,). Since the function ~ is bounded above, it follows by induction that there is * *, 2 Tr such that I(,) = ;. Write 0 , = P0, Q0, '0; . .;.Pk-1, Qk-1, 'k-1; Pk 2 Tr (P0 = P, Pk = P ). The assumption I(,) = ; implies that for each i, either Qi= NS(Pi) (hence |NS(P* *i)| |NS(Pi+1)|), or 'i(Qi) = NS(Pi+1) (hence |NS(Pi)| |NS(Pi+1)|). Thus when , 2 Tr, there is no 1 i k - 1 such that |NS(Pi)| < |NS(Pi-1)| a* *nd also |NS(Pi)| < |NS(Pi+1)|. So if we choose 0 j k such that |NS(Pj)| is max* *imal, then |NS(P )| |NS(P1)| |NS(P2)| . . .|NS(Pj)|, and |NS(Pj)| |NS(Pj-1)| . . .|NS(Pk-1)| |NS(P 0)|. Since I(,) = ;, this implies that Qi= NS(Pi) for all i < j, and that 'i(Qi) = N* *S(Pi+1) for all j i k - 1. So upon setting P 00= Pj, we obtain homomorphisms _ = 'j-1O . .O.'0 2 Hom F(NS(P ), NS(P 00)) and _0= ('k-1O . .O.'j)-1 2 Hom F(NS(P 0), NS(P 00)) such that _(P ) = P 00= _0(P 0). 12 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER This was shown for an arbitrary pair of subgroups P, P 02 P. By successively applying the above construction to the subgroups in the conjugacy class P, it n* *ow follows easily that there is some bP2 P such that for all P 2 P, there is a mor* *phism ' 2 Hom F(NS(P ), NS(Pb)) such that '(P ) = bP. Note that bPis fully normalized* * since NS(Pb) contains an injective image of any other NS(P ) for P 2 P. For the same * *reason, bPis fully centralized in F: its centralizer contains an injective image of the* * centralizer of any other subgroup in the conjugacy class P. Step 4: Fix a proper P-pair (Q, P ) such that P is fully normalized in F. By (* *3), the pair (NS(P ), P ) is F-conjugate to (NS(Pb), bP). Hence for every P 02 P, there* * is _ 2 Hom F(NS(P 0), NS(P )) such that _(P 0) = P . Assume Q is fully normalized in NF (P ). Let (Q0, P 0) be any proper P-pair * *F- conjugate to (Q, P ), and choose _ as above. Set Q00= _(Q0). Then _ sends NNS(P* *0)(Q0) injectively into NNS(P)(Q00). So |NNS(P0)(Q0)| |NNS(P)(Q00)| |NNS(P)(Q)|; where the last inequality holds since Q is fully normalized in NF (P ). This sh* *ows that the pair (P, Q) is fully normalized. Finally, assume Q is fully centralized in NF (P ), and let Q0 be any other su* *bgroup in the F-conjugacy class of Q. Fix ' 2 IsoF(Q, Q0), and set P 0= '(P ). Again, choose _ as above, and set Q00= _(Q0). Then |CS(Q0)| |CS(Q00)| since _ sends the first subgroup injectively into the second, and |CS(Q00)| |CS(Q)| since Q* * is fully centralized in NF (P ) and the pairs (Q, P ) and (Q00, P ) are F-conjugate. Thi* *s shows that Q is fully centralized in F. Lemma 2.4 reduces the problem of proving P-saturation, for an F-conjugacy cla* *ss P, to the case where P = {P } and P is normal in F. This case is handled in the* * next lemma. Lemma 2.5. Let F be a fusion system over a p-group S. Assume that P C S is norm* *al in F, and that F is S>P -generated and S>P -saturated. Assume furthermore that * *either P is not F-centric, or OutS (P ) \ Op(Out F(P )) 6= 1. Then F is S P -saturated. Proof.Define P *= {x 2 S | cx 2 Op(Aut F(P ))} . It follows from the definition that P *C S, and we claim that P *is strongly cl* *osed in F. Assume that x 2 P *is F-conjugate to y 2 S. Since P is normal in F, the* *re exists _ 2 Hom F(, ) which satisfies _(P ) = P and _(x) = y. In p* *articular, _ Ocx O_-1 = cy. It follows that y 2 P *, since cx 2 Op(Aut F(P )). Note also that P * CS(P )P . Hence by the assumption OutS(P )\Op(Out F(P )) * *6= 1 if P is F-centric, or by definition if P is not F-centric, P P *in all cases. Since F is assumed to be S>P -saturated, we need only to prove conditions (I)* *P and (II)P. We first prove that these conditions follow from the following statement: _ * (**)each ' 2 AutF (P ) extends to some ' 2 AutF (P ). Since P is normal in F, it is the only subgroup in its F-conjugacy class, and h* *ence it is fully centralized and fully normalized. It is also clear that P *is fully no* *rmalized in F, since P *C S. Hence AutS (P *) 2 Sylp(Aut F(P *)) by (I)>P . The restriction* * map SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 13 from Aut F(P *) to Aut F(P ) is surjective by (**), and so Aut S(P ) 2 Sylp(Aut* * F(P )). Therefore condition (I)P holds. Next we prove condition (II)P: that each automorphism ' 2 AutF (P ) extends t* *o a morphism defined on N'. By (**), ' extends to some _ 2 AutF (P *). Consider t* *he groups of automorphisms * fi K = Ø 2 AutS(P ) fiØ|P = cx some x 2 N' * fi * K0 = Ø 2 AutF (P ) fiØ|P = IdP C AutF (P ). By definition, for all x 2 N', we have (_cx_-1)|P = Ø|P for some Ø 2 AutS(P *).* * In other words, as subgroups of Aut(P *), -1 fi -1 * -1 _K_-1 _cx_ fix 2 N' . (_K0_ ) AutS(P ) . (_K0_ ). In general, if S 2 Sylp(G), H C G, and P SH is a p-subgroup, then there is * *x 2 H such that P xSx-1. Applied to this situation (with G = AutF (P *), S = AutS(P* * *), H = _K0_-1, and P = _K_-1), we see that there is Ø 2 K0 such that (_Ø)K(_Ø)-1 = (_Ø_-1)(_K_-1)(_Ø_-1)-1 AutS(P *). Also, P *is fully centralized in F by (I)>P_, since P *is fully normalized._So * *by (II)>P , _Ø 2 AutF (P *) extends to a morphism ' defined on NKS(P *) N', and '|P = _|P* * = ' since Ø|P = IdP. In order to finish the proof, it remains to prove (**). Since any ' 2 AutF (P* * ) is a composite of automorphisms of P which extend to strictly larger subgroups, it s* *uffices to show (**) when ' itself extends to e'2 IsoF(Q1, Q2), where Qi P . Note that e'(Q1 \ P *) = Q2 \ P * (1) since P *is strongly closed in F. We show (**) by induction on the index [P *:P *\ Q1] = [P *:P *\ Q2]. If this* * index _ def * * * is 1, i.e., if Q1 P *, then e'(P *) = P *by (1), and hence ' = e'|P* lies in* * AutF (P ) and extends '. Now assume Q1 P *, let Q3 be any subgroup F-conjugate to Q1 and Q2 and fully normalized in F, and fix ' 2 IsoF(Q2, Q3). Upon replacing e'by _ and by _ Oe', * *we are reduced to proving the result when the target group is fully normalized. So* * assume Q2 is fully normalized (and hence, by (I)>P , fully centralized). This time, consider the groups of automorphisms fi K = Ø 2 AutF (Q2) fiØ|P 2 Op(Aut F(P )) fi K0 = Ø 2 AutF (Q2) fiØ|P = IdP . Both K and K0 are normal subgroups of AutF (Q2). Also, K=K0 is a p-group, since there is a monomorphism K=K0 ! Op(Aut F(Q2)). So any two Sylow p-subgroups of K are conjugate by an element of K0. Now, Aut P*(Q1) is a p-subgroup of Aut F(Q1), all of whose elements restrict * *to elements of Op(Aut F(P )). Hence e'AutP*(Q1)'e-1is a p-subgroup of K. Since Q2 * *is fully normalized, AutS(Q2) 2 Sylp(Aut F(Q2)), and hence AutP*(Q2) = K \ AutS(Q2) is a Sylow p-subgroup of K. Thus there is Ø 2 K0 such that Ø'eAutP*(Q1)'e-1Ø-1 AutP*(Q2). 14 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER In particular, NP*Q1(Q1) Nffle'. Since Q2 is fully centralized, condition (II* *)>P now im- plies that Ø'eextends to a morphism e'02 Hom F(Q01, NS(Q2)), where Q01= NP*Q1(Q* *1). Furthermore, e'0|P = e'|P since Ø 2 K0. By assumption, P *Q1 Q1, and so Q01= NP*Q1(Q1) Q1. Also, Q01is generated by Q1 and Q01\ P *since Q1 Q01 P *Q1. Hence Q01\ P * Q1 \ P *. This shows that [P *:P *\ Q01] < [P *:P *\ Q1], and so (**) now follows by the induction hypothesis. We are now ready to prove Theorem 2.2. Proof of Theorem 2.2.We are given a set H of subsets of S, closed under F-conju* *gacy, such that F is H-generated and H-saturated, and such that condition (*)each F-conjugacy class of subgroups of S which are F-centric but not in H co* *ntains at least one subgroup P such that OutS(P ) \ Op(Out F(P )) 6= 1. holds. We will prove, by induction on the number of F-conjugacy classes of subg* *roups of S not in H, that F is saturated. If H contains all subgroups, then we are d* *one. Otherwise, let P be any F-conjugacy class of subgroups of S which is maximal am* *ong those not in H. We will show that F is also (H [ P)-saturated. Since F is cle* *arly (H [ P)-generated, the result then follows by the induction hypothesis. By Lemma 2.4, for any fully normalized subgroup P 2 P, the normalizer fusion system NF (P ) is S>P -saturated, and AutF(P ) is generated by restrictions of * *morphisms in NF (P ) between subgroups of NS(P ) which strictly contain P . Let F0 be the fusion system over S0 def=NS(P ) generated by the restriction o* *f NF (P ) to S>P , that is, the smallest fusion system over S0 for which morphisms between subgroups in S>P are the same as those in NF (P ). Then AutF0(P ) = AutF (P ), * *and F0 is S>P -saturated and S>P -generated. Also, by the assumption (*), either P * *is not centric in F (hence not centric in F0), or Out S(P ) \ Op(Out F0(P )) 6= 1. The* *n F0 is S P -saturated by Lemma 2.5, and so F is (H [ P)-saturated by Lemma 2.4 again. We end this section with a description of a example which shows why the assum* *ption (*) in Theorem 2.2 (Out S(P ) \ Op(Out F(P )) 6= 1 if P is not centric) is need* *ed. We use the following standard notation: if k is a finite field, and n 1, then Ln(k)* * denotes the semidirect product of SLn(k) with the group of field automorphisms of k. T* *his group has an obvious action on the vector space kn and on the projective space * *P(kn). It is not hard to see that L2(F4) ~=S5: via its permutation action on the five* * points in P(F42). Let = F24o S5, where S5 acts on F24via the above isomorphism. Note that c* *an be identified with the subgroup of L3(F4) generated by matrices with bottom row (0, 0, 1) and the field automorphism. Therefore acts faithfully on P = F34. We are going to define a fusion system F over S = P oS0, where S0= <(1 2), (4* * 5)> S5 . Consider the following subgroups of S: Q1 = P o <(1 2)>, Q2 = P o <(4 5* *)>, and Q3 = P o <(1 2)(4 5)>. We regard all of these groups, including , as subgr* *oups of P o . To define the morphisms in the fusion system F, let x 2 O2( ) ~= F24be the el* *e- ment of order two which centralizes S0, and consider the subgroups R1 = , R2 = , and R02= xR2x-1. Set Out F(S) = 1, Aut F(Q1) = Aut PR1(Q1), SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 15 AutF (Q2) = Aut PR02(Q2), and Aut F(Q3) = Aut S(Q3). All other morphisms in the fusion system are restrictions of the ones just described. Note in particular * *that OutF (Q1) ~= S3, Out F(Q2) ~= S3, and Aut F(P ) = = . The last equal- ity holds since =P =

= S5; and can* *not be a splitting of =P in since any splitting containing S0 must be P -conjugate to* * the given S5 ; so \ P 6= 1, and P since P is irreducible as * *an S5-representation. Consider the set of subgroups H = {S, Q1, Q2, Q3}. It follows from the above* * de- scription of morphisms in F that the subgroups in H are the only F-centric, F-r* *adical subgroups. Also, F is H-generated by construction, and one can check that F is * *H- saturated. But F is not saturated, since axiom (I)P fails: Aut S(P ) =2Syl2(Aut* * F(P )) since Aut S(P ) ~=C22and Aut F(P ) ~= . (One can also show that (II)P fails.) * * Note that OutS(P ) \ O2(Out F(P )) = S0\ O2( ) = 1, so Condition (*) in Theorem 2.2 * *does not hold. 3. Expanding and restricting the classifying space: quasicentric subgroups The goal of this section is to show how the centric linking system of a p-loc* *al finite group (S, F, L) can be extended to a larger category or restricted to a smaller* * one without changing the homotopy type of the nerve of L. One motivation for doing this is a problem which frequently occurs when tryin* *g to construct maps between p-local finite groups. A functor between fusion systems * *need not send centric subgroups to centric subgroups, in which case it cannot be lif* *ted to a functor between associated centric linking systems. One could try to get around* * this by extending the linking systems to include all subgroups as objects. There is * *in fact a natural extension of the linking system to a category whose objects are all sub* *groups of S, but in general the homotopy type of the p-completed nerve is not preserve* *d by this extension. We introduce here the collection of F-quasicentric subgroups, which contains * *the centric subgroups and supports an associated linking system Lq with properties * *anal- ogous to those of the centric one. The important fact proved in this section i* *s that the nerve of Lq is homotopy equivalent to |L|. Moreover, any full subcategory o* *f Lq whose object set contains all subgroups which are centric and radical also has * *nerve homotopy equivalent to |L|. Definition 3.1. Let F be a saturated fusion system over a p-group S. A subgroup P S is called F-quasicentric if for each P 0which is fully centralized in F * *and F-conjugate to P , the centralizer system CF (P 0) is the fusion system of the * *p-group CS(P 0). Equivalently, when F is a saturated fusion system over S, a subgroup P S is F-quasicentric if there is no Q CS(P 0) S such that P 0is F conjugate to P * *, and such that there is 1 6= ff 2 AutF (QP 0) of order prime to p with ff|P0 = IdP0.* * Note that the set of F-quasicentric subgroups of S is closed under F-conjugation and over* *groups. If F is a saturated fusion system, then Fq denotes the full subcategory whose o* *bjects are the F-quasicentric subgroups of S. One way to extend the centric linking system of a p-local finite group (S, F,* * L) to a category containing other subgroups of S as objects is provided by [BLO2 , x7]* *. There, 16 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER a (discrete) category LS,f(X) is associated to any triple (X, S, f), where X is* * a space, S is a p-group, and f :BS ---! X is a map. We recall this construction in the c* *ase where f is the natural inclusion of BS into X = |L|^p(f = |`S|^pas defined in t* *he next paragraph). As we will see, LS,f(|L|^p) is then an extension of L contain* *ing all subgroups of S as objects. Let (S, F, L) be a p-local finite group, and let ß :L --! Fc be the projectio* *n functor. For each subgroup P S, let B(P ) be the category with one object oP and with EndB(P)(oP) = P , and identify BP = |B(P )|. We let ~gdenote the morphism in B(* *P ) corresponding to g 2 P . Let `P :B(P ) -----! L be the functor which sends oP to P , and sends a morphism ~g(for g 2 P ) to ffi* *P(g) 2 AutL(P ). This induces natural maps |`P|^p:BP --! |L|^p. For each ' 2 Hom L(P,* * Q), we can view ß(') 2 Hom F(P, Q) as a functor B(P ) ! B(Q). Let j' :`P ----! `Q Oß(') be the natural transformation of functors given by ' `P(oP) = P -------! Q = `Q ß(')(oP) . This defines an explicit homotopy |j'|: BP x I --! |L|^pbetween |`P|^pand |`Q|^* *pO B'. If for each F-centric subgroup P S, we choose a morphism 'P 2 Mor L(P, S) which is sent to the inclusion of P in S by the projection functor to F, we obt* *ain a fixed collection of natural transformations j'P,fandiinduced homotopies |j'P|: * *BP x I --! |L|^pfrom |`P|^pto the restriction |`S|^pfiBP. Write f = |`S|^pfor short. LS,f(|L|^p) is defined as the category whose obje* *cts are the subgroups of S, and where morphisms are Mor LS,f(|L|^p)(P, Q) fi = (', [H]) fi' 2 Hom (P, Q), [H] 2 Mor i(Map(BP,|L|^p))(f|BP , f|BQ OB* *') . Here, ß denotes the fundamental groupoid functor. A functor ,L :L -----! LS,f(|L|^p) is also defined as follows. On objects, ,L is the inclusion, and for each ' 2 M* *or L(P, Q), ,L(') = ßP,Q('), [H'] , where H' is the homotopy BP x I --! |L|^pdefined by 8 ><|j'P|(x, 1 - 3t) 0 t 1_3 H'(x, t) = |j'|(x, 3t - 1) 1_ t 2_ >: 32 3 |j'Q|(B'(x), 3t - 2)_3 t 1 . By [BLO2 , Proposition 7.3], ,L defines an equivalence of categories to the * *full sub- category LcS,f(|L|^p) LS,f(|L|^p) whose objects are the F-centric subgroups o* *f |L|^p. In this sense, we say that LS,f(|L|^p) is an extension of L. Definition 3.2. Let (S, F, L) be any p-local finite group. Let Lq LS,f(|L|^p)* * be the full subcategory whose objects are the F-quasicentric subgroups of S. Let ß :Lq --! Fq be the functor which sends an F-quasicentric subgroup to itself, and which send* *s a mor- phism (', [H]) to '. For each object P in Lq, define the distinguished monomorp* *hism ffiP :P .CS(P ) ------! Aut Lq(P ) SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 17 by sending g 2 P .CS(P ) to (cg, [Hg]), where cg is conjugation by g restricted* * to P and |''g| f ^ Hg is the homotopy BP xI ---! BS ---! |L|p induced by the natural transformation ''g Id --! cg which sends the unique object of B(P ) to the morphism ~gof B(S). We call Lq the associated quasicentric linking system to the p-local finite g* *roup (S, F, L). Note that the functor ,L factors through L --! Lq, also denoted ,L. * *For any F-centric subgroup P S and any g 2 P , [HffiP(g)] = [Hg], where ffiP :P --! * *AutLq(P ) is the distinguished monomorphism for the centric linking system L, and Hbgis t* *he homotopy defined above. Hence ,L is compatible with the projection functors of * *L and Lq to the fusion system F, and with the distinguished homomorphisms. In this wa* *y, we can think of L as a subcategory of Lq (with ,L as inclusion functor), and re* *gard Lq as an extension of the centric linking system L. There is also a homotopy theoretic characterization of F-quasicentric subgrou* *ps. If we define a map f :X ! Y to be quasicentric if the homotopy fibre of the map f]: Map (X, X)IdX--! Map (X, Y )f is homotopically discrete, then it turns out* * that P S is F-quasicentric in (S, F, L) if and only if the natural map f|BP :BP --* *! |L|^p is quasicentric. Proposition 3.3. For any p-local finite group (S, F, L) and any P S, the foll* *owing are equivalent: (a)P is F-quasicentric. (b)There is a fully centralized subgroup P 0 S which is F-conjugate to P and s* *uch that Map (BP, |L|^p)f|BP' Map (BP 0, |L|^p)f|BP0' BCS(P 0). (c)The homotopy fibre of the map Map (BP, BP )IdBP--! Map (BP, |L|^p)f|BP is h* *o- motopically discrete. (d)Map (BP, |L|^p)f|BPis an Eilenberg-MacLane space K(ß, 1). Proof.((a))(b)) follows by definition of F-quasicentric and [BLO2 , Theorem 6.* *3]. ((b))(c)) and ((c))(d)) follow from the long exact sequence of homotopy groups of the relevant fibration because Map (BP, BP )IdBP' BZ(P ). Finally we prove that ((d))(a)). Let P 0be a fully centralized subgroup of S * *which is F-conjugate to P . According to [BLO2 , Theorem 6.3], we have that |CL(P 0* *)|^p' Map (BP 0, |L|^p)f|BP0' Map (BP, |L|^p)f|BP' K(ß, 1). In particular ß ~=ß1(|CL(* *P 0)|^p) is a finite p-group, and then the fusion system CF (P 0) coincides with the fus* *ion system of ß (see [BLO2 , Theorem 7.3]). From the definition, and the description of mapping spaces in [BLO2 , Theore* *m 4.6], we see easily that associated quasicentric linking systems satisfy the same pro* *perties as were used to define associated centric linking systems to a saturated fusion* * system. Proposition 3.4. Let (S, F, L) be any p-local finite group, and let Lq be the a* *ssociated quasicentric linking system. This satisfies the following conditions. (A) ß is the identity on objects and surjective on morphisms. For each pair of * *objects P, Q 2 Lq such that P is fully centralized, CS(P ) acts freely on Mor Lq(P, * *Q) by composition (upon identifying CS(P ) with ffiP(CS(P )) AutLq(P )), and ß i* *nduces a bijection ~= Mor Lq(P, Q)=CS(P ) ------! Hom F(P, Q). 18 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER (B) For each F-quasicentric subgroup P S and each g 2 P , ß sends ffiP(g) 2 Aut Lq(P ) to cg 2 AutF (P ). (C) For each f 2 Mor Lq(P, Q) and each g 2 P , the following square commutes in* * Lq: f P ______! Q | | ffiP(g)| |ffiQ(i(f)(g)) # # f P ______! Q. Proof.The proof follows the lines of the proof of Theorem 7.5 in [BLO2 ]. We are now ready to state the main result of this section: Theorem 3.5. Let (S, F, L) be a p-local finite group and let Lq be the associat* *ed quasicentric linking system. Let L0 Lq be any full subcategory which contain* *s all F-radical F-centric subgroups of S. Then the inclusions of L0 and L in Lq indu* *ce homotopy equivalences |L0| ' |Lq| ' |L|. Theorem 3.5 is an immediate consequence of Proposition 3.11 below. The rest o* *f the section is directed towards the proof of that proposition. We first prove some * *lemmas that will provide us with a better understanding of morphism sets in Lq. Lemma 3.6. Fix a p-local finite group (S, F, L), and let ß :Lq --! Fq be the pr* *o- jection. Fix F-quasicentric subgroups P, Q, R in S. Let ' 2 Mor Lq(P, R) and * *_ 2 Mor Lq(Q, R) be any pair of morphisms such that Im(ß(')) Im(ß(_)). Then there* * is a unique morphism Ø 2 Mor Lq(P, Q) such that ' = _ OØ. _ Proof.By definition of a_fusion system, there is a unique morphism Ø 2 Hom F(P,* * Q)_ such that ß(') = ß(_)OØ . Let Ø02 Mor Lq(P, Q) be any morphism such that ß(Ø0) * *= Ø. Choose a fully centralized group P 0in the F-conjugacy class of P and a particu* *lar ff 2 IsoLq(P 0, P ). Then by (A), there is a unique element g 2 CS(P 0) such t* *hat ' Off = _ OØ0Off OffiP0(g), and we can define Ø = Ø0Off OffiP0(g) Off-1. If Ø1 2 Mor Lq(P, Q) is any other morphism such that ' = _ OØ1, then ß(Ø) = ß* *(Ø1), hence by (A) again, there is a unique element h 2 CS(P 0) such that ØOff = Ø1Of* *fOffiP0(h); and since _OØ1Off = _OØOff = _OØ1OffOffiP0(h), and the action of CS(P 0) on Mor* *Lq(P 0, Q) is free, we obtain h = 1 and then Ø = Ø1. Lemma 3.7. Fix a p-local finite group (S, F, L). Let Lq be the associated quasi* *centric linking system and let ß :Lq --! Fq be the projection. Fix a choice of an incl* *usion morphism 'P 2 Mor Lq(P, S) for each F-quasicentric subgroup P S such that ß('* *P) = incl2 Hom (P, S) and 'S = IdS. Then, there are unique injections ffiP,Q:NS(P, Q) -----! Mor Lq(P, Q) , for all F-quasicentric subgroups P, Q S, such that: (a)ß(ffiP,Q(g)) = cg 2 Hom (P, Q), for all g 2 NS(P, Q), (b)ffiP,S(1) = 'P and ffiP,P(g) = ffiP(g), for all g 2 P . CS(P ), (c)ffiQ,R(h) OffiP,Q(g) = ffiP,R(hg), for all g 2 NS(P, Q) and h 2 NS(Q, R). Proof.This follows easily from Proposition 3.4 and Lemma 3.6 (see [BLO2 , Prop* *osi- tion 1.11]). SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 19 For the rest of the section, whenever we are given a p-local finite group (S,* * F, L), we assume that we have chosen morphisms 'P 2 Mor Lq(P, S), for each object P , * *such that ß('P) is the inclusion. Then for each P Q in Lq, we let 'QP2 Mor Lq(P, Q* *) be the unique morphism such that 'P = 'Q O'QP(Lemma 3.6). If ' 2 Mor Lq(P, Q), and P 0 P and Q0 Q are quasicentric subgroups such that ß(')(P 0) Q0, then we 0 0 0 write '|QP02 Mor Lq(P , Q ) for the "restrictionö f ': the unique morphism suc* *h that 0 P Q0 'QQ0O'|QP0= ' O'P0 (Lemma 3.6 again). We also write '|P0 = '|P0 when the target* * group Q0is clear from the context. Lemma 3.8. Fix a saturated fusion system F over a p-group S, and let Q S be an F-quasicentric subgroup. Let P S be such that Q C P , and let ', '02 Hom F(P,* * S) be such that '|Q = '0|Q. Then there is x 2 CS('(Q)) such that '0= cx O'. Proof.We first reduce this to the case where Q is fully centralized and '0is th* *e inclusion of P in S. Upon replacing P by '0(P ) and Q by '(Q) = '0(Q), we can assume that '0 = inclSPand '|Q = IdQ. By Lemma 2.3 (condition (IIA) holds), there is a ful* *ly normalized subgroup Q0in the F-conjugacy class of Q, and a morphism fi :NS(Q) ! NS(Q0) in F which sends Q to Q0. Now replace P , Q, and ' by fi(P ), Q0, and fi* * O'Ofi-1. The idea of the proof is to show that for some x 2 CS(Q), we can extend ' Ocx* * to _ __ __ _ __ __ some ' 2 Hom F(P , S), for some P P , such that '|_Q= Id_Qwhere Q Q C P. The lemma then follows by downward induction on |Q|. Recall that the lemma holds wh* *en Q is F-centric by [BLO2 , Proposition A.8]. By definition of an F-quasicentric subgroup, '|CP(Q)is conjugation by some el* *ement x 2 CS(Q). So after composing with cx,_we can_assume_that '|CP(Q).Q= Id. We are thus done if CP(Q).Q Q by taking P = P and Q = CP(Q).Q. Assume now that CP(Q) Q. Set K = AutP (Q). As in [BLO2 , Appendix A], we write NKS(Q) = {x 2 NS(Q) | cx 2 K}, and let NKF(Q) be the fusion system over NKS(Q) whose morphisms are defined (for P, P 0 NKS(Q)) by Hom NKF(Q)(P, P 0) = 0 fi 0 ' 2 Hom F(P, P ) fi_|P = ', _|Q 2 K, some _ 2 Hom F(P Q, P Q) . Then P , '(P ), and CS(Q) are all contained in NKS(Q). If Q is not fully K-norm* *alized in F, then there is some _ 2 Hom F(NKS(Q), S) such that _(Q) is fully _K_-1-normal* *ized in F (see [BLO2 , Proposition A.2(b)]); and upon replacing all of these subgro* *ups by their images under _, we are reduced to the case where Q is fully K-normalized * *in F. The fusion system NKF(Q) is saturated by [BLO2 , Proposition A.6]; and upon replacing F by NKF(Q) we can assume that S = NKS(Q) = P .CS(Q) and F = NKF(Q). In particular, each ff 2 Hom F(R, R0) extends to a morphism in Hom F(RQ, R0Q) w* *hose restriction to Q is conjugation by some element of P . Fix _ 2 Hom F(P, S) such that _(P ) is fully normalized in F. Since _|Q is co* *njuga- tion by an element g 2 P , we can replace _ by _ Oc-1g, and thus arrange that _* *|Q = Id. If _ and _ O'-1 are both conjugation by some element of CS(Q), then so is '; so* * it suffices to prove the result under the assumption that '(P ) is fully normalize* *d in F. Now, (CS(Q).Q)=Q is a nontrivial normal subgroup of NS(Q)=Q = S=Q. So there is an element x 2 CS(Q)r Q such that 1 6= xQ 2 Z(S=Q). Then x 2 NS(P ), and acts 20 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER via the identity on Q and on P=Q. Thus cx 2 Ker AutF (P ) ----! AutF(Q) x Aut(P=Q) , a normal p-subgroup of Aut F(P ) (see [Go , Corollary 5.3.3]). Also, Aut S('(P* * )) 2 Sylp(Aut F('(P ))) since '(P ) is fully normalized. In particular, 'cx'-1 2 Aut* *S('(P )) (after replacing ' by ' O, where , 2 Aut F('(P )) if necessary). Thus, x 2 N' * *and _ __ Q N'._By (II), ' extends to ' 2 Hom F (N', S). Now set P = N' \ NS(Q) = N' and Q = C_P(Q).Q. __ _ By construction, x 2 Q rQ. Since Q is F-quasicentric, '|C_P(Q)is conjugation* * by _ _ -1 some element g 2 CS(Q). So we can replace ' by ' O(cg) , and thus arrange that _ __ __ __ '|_Q= Id_Q. Since Q Q and Q C P, this finishes the induction step. The next lemma can be thought of as a "liftingö f the last one to quasicentr* *ic linking systems. It says that all inclusions in Lq are epimorphisms in the categorical * *sense. Lemma 3.9. Fix a p-local finite group (S, F, L), and let Lq be the associated q* *uasi- centric linking system. Assume Q P S and R S are F-quasicentric, and let ', '02 Mor Lq(P, R) be two morphisms such that ' O'PQ= '0O'PQ. Then ' = '0. Proof.Since there is always a subnormal series Q = Q0 C Q1 C . .C.Qk = P , it suffices to prove the lemma when Q is normal in P . So we assume this from now * *on. It will be convenient, throughout the proof, to write bff= ß(ff) 2 Mor (F) fo* *r any ff 2 Mor (Lq). By Lemma 3.6, ' = '0if and only if 'SRO' = 'SRO'02 Mor Lq(P, S),* * and similarly replacing ' (resp. '0) by ' O'PQ(resp. '0O'PQ). We can thus replace R* * by any other subgroup of S which contains the images of b'and b'0, and in particular a* *ssume that R NS('b(Q)). The proof itself will be divided in two steps: the first dealing with a restr* *icted case, and the second reducing the general case to that in Step 1. Step 1: Assume first that Q = b'(Q) and is fully normalized, and that P is ful* *ly centralized. Set '0 = ' O'PQ= '0O 'PQ. By condition (II) in Definition 1.3 (and* * since Q = b'0(Q) is fully centralized), there is b_2 Hom F(P .CS(Q), S) such that b_|* *Q = b'0. Choose any '002 Mor Lq(P, S) such that b'00= _b|P. Thus b'00|Q = b'0, so there* * is a unique element a 2 CS(Q) such that '00O'PQ= '0 Offi(a). By Lemma 3.8, there is some x 2 CS('b(Q)) such that cx Ob'O'SR= b'00. Since P* * is fully centralized, by Proposition 3.4(A), there is y 2 CS(P ) such that ffi(x) O' = '00Offi(y) = ffi(_b(y)) O'00. It follows that '00= ffi(z) O', where z = b_(y)-1.x 2 CS('b0(Q)). Hence ffi(z) O'0 = '00O'PQ= ' O'PQOffi(a) = ffi(_b(a)) O'0. Since '0 = 'SQO! for some ! 2 AutLq(Q), upon composing with !-1, this shows that ffiQ,S(_b(a)) = ffiQ,S(z), and hence that z = b_(a). After making a similar argument involving '0, we now have ffi(_b(a)) O' = '00= ffi(_b(a)) O'0, and this shows that ' = '0. SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 21 Step 2: (General case.) We first reduce the problem to the case in which P is f* *ully centralized. We choose an isomorphism , 2 Mor Lq(P, P 0) such that b,(P ) = P 0* *is fully centralized. Upon replacing P by P 0, ' by 'O,-1, and '0by '0O,-1 we are now re* *duced to the case where P is fully centralized in F. Set Q0= b'(Q) = b'0(Q) for short; we now reduce the problem to the case in wh* *ich Q = Q0 and is fully normalized. Let Q00be any fully normalized subgroup in the * *F- conjugacy class of Q (and of Q0). By Lemma 2.3 (condition (IIB) holds), there * *are morphisms fi 2 Mor Lq(NS(Q), NS(Q00)) and fi02 Mor Lq(NS(Q0), NS(Q00)) such that bfi(Q) = bfi0(Q0) = Q00. Set P 00= bfi(P ), and let fi0 2 IsoLq(P, P* * 00) be the 00) restriction of fi (i.e., by Lemma 3.6 the unique morphism such that 'NS(QP00Ofi* *0 = fi O'NS(Q)P). Set 00) -1 0 0 NS(Q00) 0 -1 00 00 _ = fi0O 'NS(QR O' Ofi0 , _ = fi O 'R O' Ofi0 2 Mor Lq(P , NS(Q )). 00 0 P00 P * *0 P Then _ = _0if and only if ' = '0, and _ O'PQ00= _ O'Q00if and only if ' O'Q = '* * O'Q. Note that P 00is F-conjugated to P and the following inequality holds: |CS(P )| = |CNS(Q)(P )| |CNS(Q00)(P 00)| = |CS(P 00)|. Since P is fully centralized, it follows that |CS(P )| = |CS(P 00)| and P 00is * *also fully centralized. Thus, upon replacing (Q, P, R) by (Q00, P 00, NS(Q00)), ' by _, and '0 by _0,* * we are reduced to the case where Q = '(Q) is fully normalized and P is fully centraliz* *ed. An immediate consequence of Lemmas 3.6 and 3.9 is: Corollary 3.10. Fix a p-local finite group (S, F, L), and let Lq be the associa* *ted qua- sicentric linking system. Then all morphisms in Lq are monomorphisms and epimor- phisms in the categorical sense. Proof.By the uniqueness in Lemma 3.6, _ OØ = _ OØ0in Lq implies Ø = Ø0. Hence a* *ll morphisms in Lq are monomorphisms. Since each morphism in Lq is the composite of an isomorphism followed by an i* *n- clusion, it suffices to prove that inclusions 'PQare epimorphisms, and it clear* *ly suffices to do this when Q C P . So assume P 0 S and ', '0 2 Mor Lq(P, R) are such that ' O'PQ= '0O'PQ. Then 'SP0O' = 'SP0O'0by Lemma 3.9, and so ' = '0by Lemma 3.6. We are now ready to prove the following proposition, of which Theorem 3.5 is * *an immediate consequence. Proposition 3.11. Let (S, F, L) be a p-local finite group, and let Lq be the qu* *asicentric linking system associated to L. Let L0 Lq be any full subcategory such that O* *b (L0) is closed under F-conjugacy. Let P 2 Ob (Lq) be maximal among those F-quasicent* *ric subgroups not in L0, and let L1 Lq be the full subcategory whose objects are * *the objects in L0 together with all subgroups F-conjugate to P . Assume furthermore* * that P is not F-centric or not F-radical. Then the inclusion of nerves |L0| |L1| * *is a homotopy equivalence. Proof.Throughout the following proof, when working in any linking system, we as* *sume that inclusion morphisms 'QPhave been chosen as in Lemma 3.7. By "extensions" and "restrictionsö f morphisms we mean with respect to these inclusions. Also* *, for 22 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER ' 2 Mor Lq(Q, Q0), we write Im (') = Im (ß(')) Q0 and '(R) = ß(')(R) Q0 if R Q. We must show that the inclusion functor ': L0 ! L1 induces a homotopy equival* *ence |L0| ' |L1|. By Theorem A in [Qu2 ], it will be enough to prove that the underc* *ategory Q#' is contractible (i.e., |Q#'| ' *) for each Q in L1. This is clear when Q i* *s not isomorphic to P (since Q#' has initial object (Q, Id) in that case), so it suff* *ices to consider the case Q = P . Since P was arbitrarily chosen in its isomorphism cla* *ss, we can also assume that P is fully normalized. Let 'N :L0 \ NLq(P ) ------! L1 \ NLq(P ) be the restriction of '. Consider the functor i : P #'N ! P #' induced by the i* *nclusions Li\ NLq(P ) ! Li for i = 0, 1. We will first show that |P #'| ' |P #'N | and th* *en that |P #'N | ' *. To prove the first statement, we construct a retraction functor r :P #' ! P #* *'N such '' that rOi = IdP#'N, together with a natural transformation iOr ___! IdP#i. By * *Lemma 2.3 (condition (IIB)), for each P 0 S which is F-conjugate to P , there is a m* *orphism in F from NS(P 0) to NS(P ) which sends P 0isomorphically to P . Hence upon lif* *ting this to the linking system, we can choose a morphism P0 2 Mor Lq(NS(P 0), NS(P )) for each such P 0which restricts to an isomorphism from P 0to P . In particular* *, we set P = IdNS(P). For each nonisomorphism ' 2 Mor Lq(P, Q), set br(') = '(P)(NQ('(P ))) P . * *We can factor ' as ' = j(') Or('), where r(') = 'br(')PO( '(P)|'(P)O') 2 Mor NLq(P)(P, br(')) and j(') = 'Q_QO( '(P)|_Q)-1 2 Mor L0(br('), Q), __ where Q = NQ('(P )). We define the functor r : P #' ! P #'N on objects by setti* *ng ' r(') r P ----! Q = P ----! br(') . For any morphism fi 2 Mor P#L0((Q, '), (Q0, '0)); i.e., for any commutative squ* *are of the form ' P ______! Q | | Id| fi| (2) # # '0 0 P ______! Q , we claim there is a unique morphism br(fi) such that the two squares in the fol* *lowing diagram commute: r(') ''(') P ________! br(')_______! Q | | | Id| br(fi)| fi| (3) # # # r('0) 0 ''('0) 0 P ________! br(' )_______! Q . To see this, note that by commutativity of the square (2), fi sends NQ('(P )) i* *nto NQ0('0(P )). Hence upon defining br(fi) def= '0(P)Ofi O '(P)-1, SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 23 where the three morphisms are replaced by appropriate restrictions, we get br(f* *i) such that the right square in (3) commutes. Since the combination of the two squares commutes by assumption, we obtain that j('0)Obr(fi)Or(') = j('0)Or('0), and the* *refore br(fi) Or(') = r('0) by Lemma 3.6. By the uniqueness of br(fi), it follows tha* *t this '' construction defines a functor, as well as a natural transformation i Or ___! * *IdP#i. Since r Oi = IdP#iN, this finishes the proof that |P #'| ' |P #'N |. It remains to prove that |P #'N | ' *. Set bP= {x 2 NS(P ) | cx 2 Op(Aut F(P ))}. Note that Pb P .CS(P ), and hence Pb P if P is not centric. Moreover, Pb * * P if P is not radical, and thus bP2 L0 in both cases covered by the hypotheses of* * the proposition. Since P is normal in bP, this last is an object in L0 \ NLq(P ). Recall that 'N :L0 \ NLq(P ) ! L1 \ NLq(P ) denotes the inclusion. Let i be * *the functor i : bP#'N ! P #'N which is induced by precomposing with the inclusion '* *PbP2 Mor Lq(P, bP). We show that i induces a homotopy equivalence |P #'N | ' |Pb#'N* * |, by defining a functor r :P #'N ! bP#'N such that r Oi = IdbP#', and such that i Or* * ' IdP#'N (such that there is a natural transformation of functors from the identity to i* *Or). Then |P #'N | ' |Pb#'N |, and the last space is contractible since bP2 L0 \ NLq(P ).* * This will finish the proof. Fix subgroups Q, Q0 NS(P ) containing P , and a morphism ' 2 Mor NLq(P)(Q, Q* *0). Set ff = ß(')|P 2 Aut F(P ) for short. Since P is fully normalized, Aut S(P* * ) 2 Sylp(Aut F(P )), and hence Op(Aut F(P )) AutS(P ). It follows that fi -1 Nffdef=x 2 NS(P ) fiffcxff 2 AutS(P ) bP; and Nff Q since ff extends to ß(') 2 Hom F(Q, Q0). Thus, since P is fully cent* *ralized, ff extends to some '0 2 Hom F (QPb, Q0bP) by condition (II) in Definition 1.3. * * After possibly composing this extension with ffiQPb(x) for some element x 2 CS(P ) * *QPb, we get a lifting b'2 Mor Lq(QPb, Q0bP) such that the following diagram commutes in* * Lq: 'QP ' 0 P ________! Q ________! Q | | 0b 'QPbP#| #'QQP0| b' 0 QPb __________________! Q bP. Pb Q0bP Hence by Lemma 3.9, b'O'QQ = 'Q0 O'. This lifting is unique by Corollary 3.10;* * and it lies in L0 \ NLq(P ), or in L1 \ NLq(P ) if Q = P . The functor r is defined on objects by setting ' 'b r P ----! Q = bP----! QPb . If fi : Q ! Q0is a morphism such that fi O' = '0, then we define r(fi) = bfi. B* *ecause of the uniqueness of the extension bfi, this construction defines a functor. M* *oreover, r Oi = IdbP#'N, and i Or ' IdP#'N, where the homotopy is induced by the natural Pb transformation given by the inclusions 'QQ . 24 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER 4. Constrained fusion systems We now look at a class of saturated fusion systems which have very simple, re* *gular behavior: the constrained fusion systems. The main results here say that constr* *ained fusion systems are always realized as fusion systems of finite groups in a pred* *ictable way, and have unique associated centric linking systems. Let F be an arbitrary saturated fusion system over a p-group S. Recall (Defin* *ition 1.5) that_a subgroup Q C S is normal in F if each ff 2 Hom F (P, P 0) extends t* *o a morphism ff2 Hom F(P Q, P 0Q) which sends Q to itself. If Q and Q0are both norm* *al in F, then clearly QQ0 is normal in F. Hence, there is a unique maximal normal p-subgroup in F, which we denote Op(F) by analogy with the subgroup Op(G) of a finite group G. By Proposition 1.6, Op(F) is contained in the intersection of * *all F- radical subgroups of S. We are interested in the case when Op(F) is itself F-ce* *ntric, or equivalently, when there is a subgroup P C S which is both normal and centri* *c in F. Definition 4.1. A saturated fusion system F over a p-group S is constrained if * *there is some Q C S which is F-centric and normal in F. When G is a finite p0-reduced group, then G is said to be p-constrained if th* *ere exists some normal p-subgroup P C G which is centric in G (i.e., CG(P ) P ). * *(More generally, an arbitrary finite group G is p-constrained if its p0-reduction G=O* *p0(G) is p-constrained.) Our aim is to show that any constrained fusion system is the f* *usion system of a unique p0-reduced p-constrained group G. This will be done by first* * showing that each constrained fusion system has a unique associated centric linking sys* *tem L, and then choosing G to be a certain automorphism group in L. We first show that for any constrained fusion system, the obstruction groups * *to the existence and uniqueness of an associated centric linking system vanish. Fo* *r any saturated fusion system F, let ZF denote the functor on O(Fc) defined by setting ZF (P ) = Z(P ) for all F-centric P S. (See [BLO2 , x3] for details.) Proposition 4.2. Let F be any constrained saturated fusion system over a p-grou* *p S. Then lim-i(ZF ) = 0 for all i >.0 O(Fc) In particular, there is a centric linking system L associated to F which is uni* *que up to isomorphism. Proof.Fix Q C S which is F-centric and normal in F. Let P1, P2, . .,.Pm be F- conjugacy class representatives for all F-centric subgroups P S such thatP * *Q, arranged such that |Pi| |Pi+1| for each i. For i = 0, 1, . .,.m, let Zi ZF* * be the subfunctor ( Z(P ) if P is conjugate to Pj for some j > i Zi(P ) = 0 otherwise. This gives a sequence of subfunctors ZF Z0 Z1 . . . Zm = 0, where for each i = 1, . .,.m, Zi-1=Zivanishes except on subgroups F-conjugate to Pi. Henc* *e by [BLO2 , Proposition 3.2], lim-*(Zi-1=Zi) ~= *(Out F(Pi); Z(Pi)). O(Fc) SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 25 Furthermore, since Pi Q, NPiQ(Pi)=Pi~= OutQ(Pi) is a nontrivial normal p-subgr* *oup of OutF (Pi) (normal by the same argument as the one used in the proof of Propo* *sition 1.6), and so *(Out F(Pi); Z(Pi)) = 0 by [JMO , Proposition 6.1(ii)]. This pro* *ves that lim-*(Zi) = 0 for all i, and in particular that lim-*(Z0) = 0. Thus lim-*(ZF ) ~= lim-*(ZF =Z0), (1) O(Fc) O(Fc) where ZF =Z0 is the quotient functor ( Z(P ) = Z(Q)P if P Q (ZF =Z0)(P ) = (2) 0 if P Q. Now set = Out F(Q) and S0 = Out S(Q) ~=S=Q. Thus S0 2 Sylp( ). Set M = Z(Q), regarded as a Z(p)[ ]-module. Let H0M be the fixed-point functor on OS0(* * ) defined by H0M(P ) = MP . Then H0M is acyclic by [JM , Proposition 5.14] (shown more explicitly in [JMO , Proposition 5.2]). So by (1), we will be done upon s* *howing that lim-*(ZF =Z0) ~= lim-*(H0M). (3) O(Fc) OS0( ) Since Q is normal and centric in F, it is easy to check that OS0( ) is isomorph* *ic to the full subcategory of O(Fc) with objects the subgroups of S containing Q. Under t* *his identification, H0M is the restriction of ZF =Z0 by (2). Isomorphism (3) now fo* *llows since (ZF =Z0)(P ) = 0 for all P Q, and since there are no morphisms in O(Fc)* * from an object in the subcategory to an object not in it. The existence and uniqueness of a centric linking system associated to F now * *follow from [BLO2 , Proposition 3.1]. We are now ready to show that each constrained fusion system is the fusion sy* *stem of a group. The following proposition includes Proposition C. Proposition 4.3. Let F be a constrained saturated fusion system over a p-group * *S. Then there is a unique finite p0-reduced p-constrained group G, containing S as* * a Sylow p-subgroup, such that F = FS(G) as fusion systems over S. Furthermore, if L is* * a centric linking system associated to F, then (a)G ~=AutL (Q) for any subgroup Q C S which is F-centric and normal in F; and (b)L ~=LcS(G). Proof.Using Proposition 4.2, fix a centric linking system L associated to F. L* *et ß :L ___! Fc denote the canonical projection functor. By Lemma 3.7, any choice* * of "inclusion" morphisms 'P 2 Mor L(P, S) determines unique injections ffiP,P0:NS(P, P 0) ------! Mor L(P, P 0), for all F-centric subgroups P, P 0 S, which satisfy the following conditions: (i)ß(ffiP,P0(g)) = cg 2 Hom F(P, P 0) for g 2 NS(P, P 0); (ii)ffiP,P(g) = ffiP(g) 2 AutL(P ) for g 2 P ; (iii)ffiP,P00(hg) = ffiP0,P00(h) OffiP,P0(g) for g 2 NS(P, P 0) and h 2 NS(P 0,* * P 00); and (iv)ffiP,S(1) = 'P. 26 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER 0 Set 'PP = ffiP,P0(1) 2 Hom L(P, P 0) for all P P 0containing Q. We think of t* *hese as the "inclusion morphisms" in L. By construction, 'SP= 'P and 'PP= IdP for all P* * , and 00 P00 P0 0 00 'PP = 'P0 O'P whenever P P P . The proposition follows from the following points, which will be proven in St* *eps 1-2. (1)Assume Q C S is F-centric and normal in F, and G = Aut L(Q). Then G is p0-reduced and p-constrained; and we can identify S with a subgroup of G in * *such a way that S 2 Sylp(G) and F = FS(G). (2)Assume G is p0-reduced and p-constrained, and such that S 2 Sylp(G) and F = FS(G). Then L ~=LcS(G). Also, if Q C S is any subgroup which is F-centric and normal in F, then Q C G, and G ~=AutL (Q). Step 1: Fix Q C S which is F-centric and normal in F, and set G = AutL(Q). Via the injection ffiQ,Q: S = NS(Q) ------! Aut L(Q) = G, we identify S as a subgroup of G. Since Q is fully normalized, S=Z(Q) ~=AutS (Q) 2 Sylp(Aut F(Q)), where AutF (Q) ~=G=Z(Q); and thus S 2 Sylp(G). Let P, P 0 S be any pair of subgroups which contain Q. For any f 2 Mor L(P, * *P 0), there is (by Lemma 3.6) a unique "restrictionö f f to Q: a unique element fl(f* *) 2 G = AutL(Q) such that 'SQOfl(f) = f O'PQ. These restrictions clearly satisfy the fo* *llowing two conditions: (v)fl(f0O f) = fl(f0).fl(f) for any f0 2 Mor L(P 0, P 00), any Q P 00 S; and (vi)fl(ffiP,P0(x)) = x for all x 2 NS(P, P 0). Furthermore, by condition (C) in Definition 1.4, for each g 2 P , ffiS(ß(f)(g)) Of = f OffiP(g) 2 Mor L(P, S). Upon restriction to Q (and applying (v) and (vi)), this gives the relation ffiQ,Q(ß(f)(g)) Ofl(f) = fl(f) OffiQ,Q(g) 2 AutL(Q) = G. In other words, under the identification S = ffiQ,Q(S) AutL(Q) = G, this show* *s that (vii)fl(f) 2 NG(P, P 0) and cfl(f)= ß(f) 2 Hom F(P, P 0). Now, i CG(Q) = Ker AutL (Q) ----! AutF(Q) = Z(Q) : the first equality by (vii) (applied with P = P 0= Q, so fl(f) = f), and the se* *cond by condition (A) in Definition 1.4. Thus Q is centric in G. This also shows t* *hat Op0(G) = 1 (since [Op0(G), Q] = 1), and hence that G is p0-reduced and p-constr* *ained. We must show that F = FS(G). We first show that Hom F(P, P 0) Hom G(P, P 0)* * for each P, P 0 S. Since Q is normal in F, each morphism in Hom F(P, P 0) extends * *to a morphism in Hom F(P Q, P 0Q), and hence it suffices to work with subgroups P, P* * 0 Q. In particular, P and P 0are F-centric in this case. For any ' 2 Hom F(P, S), an* *d any f 2 Mor L(P, S) such that ß(f) = ', fl(f) 2 NG(P, P 0) and ' = cfl(f)2 Hom G(P,* * P 0) by (vii), and thus Hom F(P, P 0) Hom G(P, P 0). SUBGROUP FAMILIES CONTROLLING P-LOCAL FINITE GROUPS 27 Conversely, for any P, P 0 S and any g 2 NG(P, P 0) = NG(P Q, P 0Q), we claim that cg 2 Hom F(P, P 0). Again, we can assume that P, P 0 Q. Now, cg|Q 2 AutF * *(Q) by (vii) (applied with P = P 0= Q and f = g). Since Q = gQg-1 is F-centric, it * *is fully centralized in F, and so cg|Q extends to an F-morphism defined on Ncg|Qdef={x 2 S | cgxg-12 AutS(Q)} P, by condition (II) of Definition 1.3. In particular, cg|Q extends to a morphism* * ' 2 Hom F(P, S) Hom G (P, S) (where the inclusion holds by the previous paragraph* *). Let h 2 NG(P, S) be such that ' = ch. Then ch|Q = '|Q = cg|Q, so h = gx for some x 2 CG(Q), and CG(Q) = Z(Q) as already shown. Since x 2 P , cx 2 Aut F(P ), so cg 2 Hom F(P, S), and cg 2 Hom F(P, P 0) since cg(P ) = gP g-1 P 0. Step 2: Let G be any finite p0-reduced p-constrained group such that S 2 Sylp* *(G) and F = FS(G). Then L ~=LcS(G) by the uniqueness in Proposition 4.2. Let Q C S be any subgroup normal in F = FS(G). Set Q0= Op(G); thus CG(Q0) = Z(Q0) by assumption. Since Q is normal in FS(G), for any g 2 G, cg 2 Aut G(Q0) extends to some cg0 2 Aut G(QQ0); then g-1g0 2 CG(Q0) = Z(Q0), g0 2 NG(QQ0), and so g 2 NG(QQ0). This shows that QQ0 C G, a normal p-subgroup, and hence Q Q0= Op(G). Hence for any g 2 G, cg 2 AutG (Q0) restricts to an automorphism of Q (since Q is normal in FS(G)), so g 2 NG(Q), and this shows that Q C G. In particular, if Q is both F-centric and normal in F, then AutL(Q) ~=AutLcS(G)(Q) ~=NG(Q)=Op(CG(Q)) = G=1 ~=G. It is in general not true, for a constrained fusion system F over a p-group S* * and a finite group G such that S 2 Sylp(G) and F = FS(G), that p-subgroups of S norma* *l in F are also normal in G. For example, if G = A5, p = 2, S 2 Syl2(G), and F = FS(* *G), then F is a constrained fusion system, with O2(F) = S ~= C22. Thus S is normal * *in F, but not in G, in this case. This shows the importance of assuming G is p0-re* *duced and p-constrained. In the given example, the unique 20-reduced 2-constrained g* *roup associated to F is A4. References [Al] J. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222-241 [AB] J. Alperin & M. Brou'e, Local methods in block theory, Annals of Math. 11* *0 (1979), 143-157 [BL] C. Broto, R. Levi, On spaces of self homotopy equivalences of p-completed* * classifying spaces of finite groups and homotopy group extensions, Topology 41 (2002), 229-2* *55 [BLO1]C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of p-completed clas* *sifying spaces of finite groups, Invent. math. 151 (2003), 611-664 [BLO2]C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, J.* * Amer. Math. Soc. 16 (2003), 779-856 [BCGLO2] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Extensions o* *f p-local finite groups (in preparation) [BM] C. Broto & J. Møller, Homotopy finite Chevalley versions of p-compact gro* *ups (in preparation) [Dw1] W. Dwyer, Homology decompositions for classifying spaces of finite groups* *, Topology 36 (1997), 783-804 [Gs] D. Goldschmidt, A conjugation family for finite groups, J. Algebra 16 (19* *70), 138-142 [Go] D. Gorenstein, Finite groups, Harper & Row (1968) [Gr] J. Grodal, Higher limits via subgroup complexes, Annals of Math. 155 (200* *2), 405-457 [HV] J. Hollender & R. Vogt, Modules of topological spaces, applications to ho* *motopy limits and E1 structures, Arch. Math. 59 (1992), 115-129 28 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER [JM] S. Jackowski & J. McClure, Homotopy decomposition of classifying spaces v* *ia elementary abelian subgroups, Topology 31 (1992), 113-132 [JMO] S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-ma* *ps of BG via G- actions, Annals of Math. 135 (1992), 184-270 [LO] R. Levi, & B. Oliver, Construction of 2-local finite groups of a type stu* *died by Solomon and Benson, Geometry & Topology 6 (2002), 917-990 [MacL]S. Mac Lane, Homology, Springer-Verlag (1963, 1975) [Pu1] Ll. Puig, Structure locale dans les groupes finis, Bull. Soc. Math. Franc* *e Suppl. M'em. 47 (1976) [Pu2] Ll. Puig, Unpublished notes (ca. 1990) [Pu3] Ll. Puig, The hyperfocal subalgebra of a block, Invent. math. 141 (2000),* * 365-397 [Qu2] D. Quillen, Algebraic K-theory I, Lecture notes in mathematics 341 (1973)* *, 77-139 [RV] A. Ruiz & A. Viruel, The classification of p-local finite groups over the* * extraspecial group of order p3 and exponent p, Math. Z. (to appear) Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193 Bel- laterra, Spain E-mail address: broto@mat.uab.es Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193 Bel- laterra, Spain E-mail address: natalia@mat.uab.es Department of Mathematics, University of Chicago, Chicago, IL 60637, USA E-mail address: jg@math.uchicago.edu Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K. E-mail address: ran@maths.abdn.ac.uk LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France E-mail address: bob@math.univ-paris13.fr