Finite simple groups and localization Jose L. Rodriguez, Jer^ome Scherer and Jacques Thevenaz * Abstract The purpose of this paper is to explore the concept of localization, wh* *ich comes from homotopy theory, in the context of finite simple groups. We give an * *easy criterion for a finite simple group to be a localization of some simple su* *bgroup and we apply it in various cases. Iterating this process allows us to conn* *ect many simple groups by a sequence of localizations. We prove that all sporadic * *simple groups (except possibly the Monster) and several groups of Lie type are co* *nnected to alternating groups. The question remains open whether or not there are * *several connected components within the family of finite simple groups. In some ca* *ses, we also consider automorphism groups and universal covering groups and we sho* *w that a localization of a finite simple group may not be simple. Introduction The concept of localization plays an important role in homotopy theory. The int* *roduction by Bousfield of homotopical localization functors in [2] and more recently its * *populariza- tion by Farjoun in [7] has led to the study of localization functors in other c* *ategories. Special attention has been set on the category of groups Gr, as the effect of a* * homotopi- cal localization on the fundamental group is often best described by a localiza* *tion functor L : Gr ! Gr. A localization functor is a pair (L; j) consisting of a functor L : Gr ! Gr * *together with a natural transformation j : Id ! L, such that L is idempotent, meaning th* *at the two morphisms jLG; L(jG ) : LG ! LLG coincide and are isomorphisms. A group homomorphism ' : H ! G is called in turn a localization if there exists a local* *ization ________________________________ *The first author was partially supported by DGESIC grant PB97-0202 and the * *Swiss National Science Foundation. 1 functor (L; j) such that G = LH and ' = jH : H ! LH (but we note that the funct* *or L is not uniquely determined by '). In this situation, we often say that G is a l* *ocalization of H. A very simple characterization of localizations can be given without men* *tioning localization functors: A group homomorphism ' : H ! G is a localization if and * *only if ' induces a bijection '* : Hom (G; G) ~=Hom (H; G) (0.* *1) as mentionned in [3, Lemma 2.1]. In the last decade several authors (Casacubert* *a, Far- joun, Libman, Rodriguez) have directed their efforts towards deciding which alg* *ebraic properties are preserved under localization. An exhaustive survey about this pr* *oblem is nicely exposed in [3] by Casacuberta. For example, any localization of an abeli* *an group is again abelian. Similarly, nilpotent groups of class at most 2 are preserved, bu* *t the ques- tion remains open for arbitrary nilpotent groups. Finiteness is not preserved, * *as shown by the example An ! SO(n - 1) (this is the main result in [16]). In fact, it h* *as been shown in [8] that any non-abelian finite simple group has arbitrarily large loc* *alizations (under certain set-theoretical assumptions). In particular it is not easy to de* *termine all possible localizations of a given object. Thus we restrict ourselves to the stu* *dy of finite groups and wonder if it would be possible to understand the finite localization* *s of a given finite simple group. This paper is a first step in this direction. Libman [17] observed recently that the inclusion of alternating groups An ,!* * An+1 is a localization if n 7. His motivation was to find a localization where new torsi* *on elements appear (e.g. A10 ,! A11 is such a localization since A11 contains elements of o* *rder 11). In these examples, the groups are simple, which simplifies considerably the ver* *ification of formula (0.1). It suffices to check if Aut(G) ~=Hom (H; G) - {0}: This paper is devoted to the study of the behaviour of injective localizatio* *ns with respect to simplicity. We first give a criterion for an inclusion of a simple * *group in a finite simple group to be a localization. We then find several infinite famili* *es of such localizations, for example L2(p) ,! Ap+1 for any prime p 13 (cf. Proposition 2* *.3). Here L2(p) = P SL2(p) is the projective special linear group. It is striking to noti* *ce that the three conditions that appear in our criterion for an inclusion of simple groups* * H ,! G to be a localization already appeared in the literature. For example the main * *theorem of [15] states exactly that J3 ,! E6(4) is a localization (see Section 3). Sim* *ilarly the main theorem in [21] states that Sz(32) ,! E8(5) is a localization. Hence the l* *anguage of localization theory can be useful to shortly reformulate some rather technical * *properties. 2 By Libman's result, the alternating groups An, for n 7, are all connected b* *y a sequence of localizations. We show that A5 ,! A6 is also a localization. A more* * curious way allows us to connect A6 to A7 by a zigzag of localizations: A6 ,! T ,! Ru - L2(13) ,! A14- . .-.A7 where T is the Tits group, and Ru the Rudvalis group. This yields to the conce* *pt of rigid component of a simple group. The idea is that among all inclusions H ,! G* *, those that are localizations deserve our attention because of the "rigidity condition* *" imposed by (0.1): Any automorphism of G is completely determined by its restriction to * *H. So, we say that two groups H and G lie in the same rigid component if H and G can be connected by a zigzag of inclusions which are all localizations. Many finite simple groups can be connected to the alternating groups. Here * *is our main result: Theorem The following finite simple groups all lie in the same rigid component: (i)All alternating groups An (n 5). (ii)The Chevalley groups L2(q) where q is a prime power 5. (iii)The Chevalley groups U3(q) where q is a prime power, q 6= 5. (iv)The Chevalley groups G2(p) where p is an odd prime such that (p + 1; 3) = * *1. (v)All sporadic simple groups, except possibly the Monster. (vi)The Chevalley groups L3(3), L3(5), L3(11), L4(3), U4(2), U4(3), U5(2), U6(* *2), S4(4), S6(2), S8(2), D4(2), 2D4(2), 2D5(2), 3D4(2), D4(3), G2(2)0, G2(4), G2(5), * *G2(11), E6(4), F4(2), and T = 2F4(2)0. The proof is an application of the localization criteria which are given in * *Sections 1 and 2, but requires a careful checking in the ATLAS [4], or in the more complet* *e papers about maximal subgroups of finite simple groups (e.g. [12], [19], [24]). We do * *not know if the Monster can be connected to the alternating group, see Remark 6.7. We finally exhibit an example due to Viruel showing that if H ,! G is a loca* *lization with H simple then G need not be simple. There is a localization map from the M* *athieu group M11 to the double cover of the Mathieu group M12. This answers negativel* *y a 3 question posed by Libman in [17] and also by Casacuberta in [3] about the prese* *rvation of simplicity. In our context this also implies that the rigid component of a s* *imple group may contain a non-simple group. It is still an open problem to know how many rigid components of finite simp* *le groups there are, even though our main theorem seems to suggest that there is only one* *. We note that the similar question for non-injective localizations has a trivial answer * *(see Section 1). Acknowledgments: We would like to thank Antonio Viruel and Jean Michel for h* *elpful comments. 1 A localization criterion In Theorem 1.4 below we list necessary and sufficient conditions for an inclusi* *on H ,! G between two non-abelian finite simple groups to be a localization. These condit* *ions are easier to deal with if the groups H and G satisfy some extra assumptions, as we* * show in the corollaries after the theorem. The proof is a variation of that of Corollar* *y 4 in [8]. We note here that we only deal with injective group homomorphisms because no* *n- injective localizations abound. For example, for any two finite groups G1 and * *G2 of coprime orders, G1 x G2 ! G1 and G1 x G2 ! G2 are localizations. So the analogo* *us concept of rigid component defined using non-injective localizations has no int* *erest, since obviously any two finite groups are in the same component. If the inclusion i : H ,! G is a localization, then so is the inclusion H0 ,* *! G for any subgroup H0 of G which is isomorphic to H. This shows that the choice of the su* *bgroup H among isomorphic subgroups does not matter. Let c : G ! Aut (G) be the natural injection of G defined as c(g) = cg, where cg : G ! G denotes the inner automorphism given by x 7! gxg-1. We shall always identify in this way a simple group G with a subgroup of Aut (G), without writi* *ng the map c. However, we use c explicitly in the following two easy results. Lemma 1.1 Let G be a non-abelian simple group. Then the following diagram co* *mmutes: ____ff___//_ G G c|| |c| fflffl| fflffl| Aut(G) _cff//_Aut(G) for any automorphism ff 2 Aut(G) . 4 Proof. This is a trivial check. * * 2 Lemma 1.2 Let H be a non-abelian simple subgroup of a finite simple group G.* * Suppose that the inclusion i : H ,! G extends to an inclusion i : Aut (H) ,! Aut (G), i* *.e., the following diagram commutes ____i____//_ H G c|| |c| fflffl| fflffl| Aut(H) __i_//_Aut(G) Then every automorphism ff : H ! H extends to an automorphism i(ff) : G ! G. Proof. We have to show that the following square commutes: __ff//_ H" _ H" _ i|| i|| fflffl|fflffl| G _i(ff)//_G To do so we consider this square as the left-hand face of the cubical diagram _______c___//_ H ">_ Aut(H)"L | >>> L_cffLLL | ff>> | LLL | >OEOE> | LL%% | ________|c_______// | H" Aut (H) | _ | |" _ | | | | | | | | fflffl| ||c fflffl| | G >_____|____//_Aut(G) | >> | LLLc | >> | LLi(ff)|LL i(ff)>>| LLL | OEfflffl|OE> %%|fflffl G _________c_______//Aut(G) The top and bottom squares commute by Lemma 1.1. The front and back squares are* * the same and commute by assumption. The right-hand square commutes as well because * *i is a homomorphism. This forces the left-hand square to commute and we are done. * * 2 Remark 1.3 As shown by the preceding lemma, it is stronger to require that th* *e inclusion i : H ,! G extends to an inclusion i : Aut (H) ,! Aut (G) than to require that * *every automorphism of H extends to an automorphism of G. In general we have an exact sequence 1 ! CAut(G)(H) -! NAut(G)(H) -! Aut(H) 5 so the second condition is equivalent to the fact that this is a short exact se* *quence. However, in the presence of the condition CAut(G)(H) = 1, which plays a central* * role in this paper, we find that NAut(G)(H) ~=Aut (H). Thus any automorphism of H exten* *ds to a unique automorphism of G, and this defines a homomorphism i : Aut (H) ,! Aut * *(G) extending the inclusion H ,! G. Therefore, if the condition CAut(G)(H) = 1 hol* *ds, we have a converse of the above lemma and both conditions are equivalent. We will * *use the first in the statements of the following results, even though it is the stronge* *r one. It is indeed easier to check in the applications. Theorem 1.4 Let H be a non-abelian simple subgroup of a finite simple group G* * and let i : H ,! G be the inclusion. Then i is a localization if and only if the follo* *wing three conditions are satisfied: 1.The inclusion i : H ,! G extends to an inclusion i : Aut(H) ,! Aut(G) . 2.Any subgroup of G which is isomorphic to H is conjugate to H in Aut(G). 3.The centralizer CAut(G)(H) = 1. Proof. If i is a localization, all three conditions have to be satisfied. By * *Lemma 1.2, condition (1) claims that the composite of an automorphism of H with i can be e* *xtended to an automorphism of G (see also Remark 1.3). Condition (2) claims that the in* *clusion of a subgroup of G isomorphic to H can be extended, while condition (3) says th* *at there exists a unique extension for i, namely the identity. Assume now that all three conditions are satisfied. For any given homomorph* *ism ' : H ! G , we have to find a unique homomorphism : G ! G such that O i = '. The trivial homomorphism G ! G obviously extends the trivial homomorphism from H to G. It is unique since H is in the kernel of , which must be equal to G by si* *mplicity. Hence, we can suppose that ' is not trivial. Since H is simple we have that '(H* *) G and H ~='(H). By (2) there is an automorphism ff 2 Aut(G) such that cff('(H)) = H, or equi* *valently by Lemma 1.1, ff('(H)) = H. Therefore the composite map ' ff|'(H) H -! '(H) - ! H is some automorphism fi of H. By condition (1) this automorphism of H extends t* *o an 6 automorphism i(fi) : G ! G. That is, the following square commutes: fi H"____//__H" _ i|| i|| fflffl|fflffl| G _i(fi)//_G The homomorphism = ff-1i(fi) extends ' as desired. We prove now it is unique. Suppose that 0 : G ! G is a homomorphism such that 0O i = '. Then, since G is simple, 02 Aut(G) . The composite -10is an element in the centralizer CAut(G)(H* *), which is trivial by (3). This finishes the proof of the theorem. * * 2 Remark 1.5 As already mentionned in Remark 1.3, conditions (1) and (3) imply * *that Aut (H) = NAut(G)(H). By condition (2), the cardinal of the orbit of H under th* *e conju- gation action of Aut(G) is equal to the number k of conjugacy classes of H in G* * multiplied by the cardinal of the orbit of H under the conjugation action of G. That is, __|_Aut(G)_|_ | G | = k . _________; | NAut(G)(H) | | NG (H) | Condition (3) is thus equivalent to the following one, which is sometimes easie* *r to verify: 3'.The number of conjugacy classes of subgroups of G isomorphic to H is equal* * to |_Out(G)_|_ | NG (H) | . _________: | Out(H) | | H | We obtain immediately the following corollaries. Using the terminology in [2* *0, p.158], recall that a group is complete if it has no outer-automorphism and trivial cen* *tre. The first corollary describes the situation when the groups involved are complete. Corollary 1.6 Let H be a non-abelian simple subgroup of a finite simple group * *G and let i : H ,! G be the inclusion. Assume that H and G are complete groups. Then * *i is a localization if and only if the following two conditions are satisfied: 1.Any subgroup of G which is isomorphic to H is conjugate to H. 2.CG (H) = 1. 2 The condition CG (H) = 1 is here equivalent to NG (H) = H. This is often eas* *ier to check. It is in particular always the case when H is a maximal subgroup of G. T* *his leads us to the next corollary. 7 Corollary 1.7 Let H be a non-abelian simple subgroup of a finite simple group * *G and let i : H ,! G be the inclusion. Assume that H is a maximal subgroup of G. Then* * i is a localization if and only if the following three conditions are satisfied: 1.The inclusion i : H ,! G extends to an inclusion i : Aut(H) ,! Aut(G) . 2.Any subgroup of G which is isomorphic to H is conjugate to H in Aut(G). |Out (G)| 3.The number of conjugacy classes of H in G is equal to _________. |Out (H)| Proof. Since H is a maximal subgroup of G, NG (H) = H. The corollary is now a direct consequence of Theorem 1.4 taking into account Remark 1.5 about the numb* *er of conjugacy classes of subgroups of G isomorphic to H. * * 2 2 Localization in alternating groups We describe in this section a method for finding localizations of finite simple* * groups in alternating groups. Let H be a simple group and K a subgroup of index n. The (l* *eft) action of H on the cosets of K in H defines a permutation representation H ! Sn* * as in [20, Theorem 3.14, p.53]. The degree of the representation is the number n of cosets* *. As H is simple, this homomorphism is actually an inclusion H ,! An. Recall that Aut(An)* * = Sn if n 7. Theorem 2.1 Let H be a non-abelian finite simple group and K a maximal subgro* *up of index n 7. Suppose that the following two conditions hold: 1.The order of K is maximal (among all maximal subgroups). 2.Any subgroup of H of index n is conjugate to K. Then the permutation representation H ,! An is a localization. Proof. We show that the conditions of Theorem 1.4 are satisfied, starting with * *condi- tion (1). Since K is maximal, it is self-normalizing and therefore the action o* *f H on the cosets of K is isomorphic to the conjugation action of H on the set of conjugat* *es of K. By our second assumption, this set is left invariant under Aut(H) . Thus the ac* *tion of H extends to Aut(H) and this yields the desired extension Aut(H) ! Sn = Aut(An). 8 To check condition (2) of Theorem 1.4, let H0be a subgroup of An which is is* *omorphic to H and denote by ff : H ! H0 an isomorphism. Let J be the stabilizer of a poi* *nt in {1; : :;:n} under the action of H0. Since the orbit of this point has cardinal* *ity n, the index of J is at most n, hence equal to n by our first assumption. Thus H0* * acts transitively. So H has a second transitive action via ff and the action of H0.* * For this action, the stabilizer of a point is a subgroup of H of index n, hence conjugat* *e to K by assumption. So K is also the stabilizer of a point for this second action and t* *his shows that this action of H is isomorphic to the permutation action of H on the coset* *s of K, that is, to the first action. It follows that the permutation representation H * *-ff!H0,! An is conjugate in Sn to H ,! An. Finally, since H is a transitive subgroup of Sn with maximal stabilizer, the* * centralizer CSn(H) is trivial by [6, Theorem 4.2A (vi)] and thus condition (3) of Theorem 1* *.4 is satisfied. * * 2 Among the twenty-six sporadic simple groups, twenty have a subgroup which sa* *tisfies the conditions of Theorem 2.1. Corollary 2.2 The following inclusions are localizations: M11,! A11 , M22,! A22 , M23,! A23 , M24,! A24 , J1 ,! A266, J2 ,! A100, J3 ,! * *A6156, J4 ,! A173067389, HS ,! A100 , McL ,! A275 , Co1 ,! A98280, Co2 ,! A2300, Co3 ,* *! A276, Suz ,! A1782, He ,! A2058, Ru ,! A4060, F i22,! A3510, F i23,! A31671, HN ,! A1* *140000, Ly ,! A8835156. Proof. In each case, it suffices to check in the ATLAS [4] that the conditions * *of The- orem 2.1 are satisfied. It is necessary to check the complete list of maximal * *subgroups in [13] for the Fischer group F i23 and [14] for the Janko group J4. * * 2 We obtain now two infinite families of localizations. The classical project* *ive special linear groups L2(q) = P SL2(q) of type A1(q), as well as the projective special* * unitary groups U3(q) = P SU3(q) of type 2A2(q), are almost all connected to an alternat* *ing group by a localization. Recall that the notation L2(q) is used only for the simple * *projective special linear groups, that is if the prime power q 4. Similarly the notation* * U3(q) is used for q > 2. 9 Proposition 2.3 (i) The permutation representation L2(q) ,! Aq+1 induced by th* *e action of SL2(q) on the projective line is a localization for any prime power q 62 {4;* * 5; 7; 9; 11}. (ii) The permutation representation U3(q) ,! Aq3+1induced by t* *he ac- tion of SU3(q) on the set of isotropic points in the projective plane is a loca* *lization for any prime power q 6= 5. Proof. We prove both statements at the same time. The group L2(q) acts on the projective line, whereas U3(q) acts on the set of isotropic points in the proje* *ctive plane. In both cases, let B be the stabilizer of a point for this action (Borel subgro* *up). Let us also denote by G either L2(q) or U3(q), where q is a prime power as specified a* *bove, and r is q + 1, or q3+ 1 respectively. Then B is a subgroup of G of index r by [11,* * Satz II-8.2] and [11, Satz II-10.12]. By [11, Satz II-8.28], which is an old theorem of Galois when q is a prime, * *L2(q) has no non-trivial permutation representation of degree less than r if q 62 {4; 5; * *7; 9; 11}. The same holds for U3(q) by [5, Table 1] if q 6= 5. Thus B satisfies condition (1) * *of Theorem 2.1. It remains to show that condition (2) is also satisfied. The subgroup B is t* *he normalizer of a Sylow p-subgroup U, and B = UT , where T is a complement of U in B. If N d* *enotes the normalizer of T in G, we know that G = UNU (Bruhat decomposition). We are n* *ow ready to prove that any subgroup of G of index r is conjugate to B. Let H be s* *uch a subgroup. It contains a Sylow p-subgroup, and we can thus assume it actually co* *ntains U. Since G is generated by U and N, the subgroup H is generated by U and N \ H. As* *sume H contains an element x 2 N - T . The class of x in the Weyl group N=T ~= C2 i* *s a generator and we have G = < U; xUx-1 > (see for example [10, Theorem 2.3.8 (e)]* *). But both U and its conjugate xUx-1 are contained in H. This is impossible because H* * 6= G, so N \ H = T \ H. It follows that H is contained in < U; T > = B. But H and B h* *ave the same order and therefore H = B. * * 2 Remark 2.4 This proof does not work for the action of Ln+1(q) on the n-dimens* *ional projective space if n 2, because there is a second action of the same degree, * *namely the action on the set of all hyperplanes in (Fq)n+1. Thus there is another conjugac* *y class of subgroups of the same index, so condition 1 does not hold. 10 3 Proof of the main theorem In order to prove our main theorem, we have to check that any group of the list* * is connected to an alternating group by a zigzag of localizations. When no specifi* *c proof is indicated for an inclusion to be a localization, it means that all the necessar* *y information for checking conditions (1)-(3) of Theorem 1.4 is available in the ATLAS [4]. (i) Alternating groups. The inclusions An ,! An+1, for n 7, studied by Libman in [17, Example 3.4] * *are localizations by Corollary 1.7, with Out(An) ~=C2 ~=Out(An+1). The inclusion A5* * ,! A6 is a localization as well, since Out (A6) = (C2)2, Out (A5) = C2, and there are* * indeed two conjugacy classes of subgroups of A6 isomorphic to A5 with fusion in Aut(A6* *). The inclusion A6 ,! A7 is not a localization, but we can connect these two groups v* *ia a zigzag of localizations, for example as follows: A6 ,! T ,! Ru - L2(13) ,! A14 where T denotes the Tits group, Ru the Rudvalis group and the last arrow is a l* *ocalization by Proposition 2.3. (ii) Chevalley groups L2(q). By Proposition 2.3, all but five linear groups L2(q) are connected to an alt* *ernating group. The group L2(4) ~= L2(5) is A5, and L2(9) ~= A6, which are connected to* * all alternating groups by the argument above. We connect L2(7) to A28 via a chain o* *f two localizations L2(7) ,! U3(3) = G2(2)0,! A28 where we use Theorem 2.1 for the second map. Similarly, we connect L2(11) to A2* *2 via the Mathieu group M22: L2(11) ,! M22,! A22: (iii) Chevalley groups U3(q). For q 6= 5, we have seen in Proposition 2.3 (ii) that U3(q) ,! Aq3+1is a loc* *alization. Recall that U3(2) is not simple. We do not know if U3(5) is connected to the al* *ternating groups. 11 (iv) Chevalley groups G2(p). When p is an odd prime such that (p + 1; 3) = 1, we will see in Proposition * *4.2 that U3(p) ,! G2(p) is a localization. We can conclude by (iii), since 5 is not a pr* *ime in the considered family. (v) Sporadic simple groups. By Corollary 2.2, we already know that twenty sporadic simple groups are con* *nected with some alternating group. We now show how to connect all the other sporadic * *groups, except the Monster for which we do not know what happens (see Remark 6.7). For the Mathieu group M12, we note that the inclusion M11 ,! M12 is a locali* *zation because there are two conjugacy classes of inclusions of M11 in M12 (of index 1* *1) with fusion in Aut(M12) (cf. [4, p.33]). We conclude by Corollary 1.7. The list of all maximal subgroups of F i024is given in [19] and one sees tha* *t He ,! F i024 is a localization. Looking at the complete list of maximal subgroups of the Baby Monster B in [* *24], we see that F i23,! B is a localization, as well as T h ,! B, HN ,! B, and L2(11) * *,! B (see Proposition 4.1 in [24]). This connects Thompson's group T h and the Baby Monst* *er (as well as the Harada-Norton group HN) to the Fischer groups and also to the Cheva* *lley groups L2(q). Finally we consider the O'Nan group O0N. By [23, Proposition 3.9] we see th* *at M11,! O0N is a localization. (vi) Other Chevalley groups. Suzuki's construction of the sporadic group Suz provides a sequence of graph* *s whose groups of automorphisms are successively Aut(L2(7)), Aut(G2(2)0), Aut(J2), Aut(* *G2(4)) and Aut (Suz) (see [9, p.108-9]). Each one of these five groups is an extension* * of C2 by the appropriate finite simple group. All arrows in the sequence L2(7) ,! G2(2)0,! J2 ,! G2(4) ,! Suz (3.* *1) are thus localizations by Corollary 1.7 because they are actually inclusions of* * the largest maximal subgroup (cf. [4]). This connects the groups G2(2)0 and G2(4) to alte* *rnating groups since we already know that Suz is connected to A1782by Corollary 2.2. Al* *terna- tively, note that G2(4) - L2(13) ,! A14 are localizations, using Proposition 2.* *3 for the second one. 12 The Suzuki group provides some more examples of localizations: L3(3) ,! Suz * *by [22, Section 6.6], and U5(2) ,! Suz by [22, Section 6.1]. We also have localizations A9 ,! D4(2) = O+8(2) ,! F4(2) - 3D4(2) which connect these Chevalley groups (see Proposition 4.3 for the last arrow). * *We are able to connect three symplectic groups since A8 ,! S6(2) and S4(4) ,! He are locali* *zations, as well as S8(2) ,! A120by Theorem 2.1. This allows us in turn to connect more * *Chevalley groups as U4(2) ,! S6(2), and O-8= 2D4(2),! S8(2) are all localizations. Each of the following localizations involves a linear group and connects som* *e new group to the component of the alternating groups: L2(11) ,! U5(2), L3(3) ,! T , L2(7) ,! L3(11), and L4(3) ,! F4(2). The localization U3(3) ,! G2(5) connects G2(5) and thus L3(5) by Proposition* * 4.1 below. Likewise, since we just showed above that L3(11) belongs to the same rig* *id com- ponent, then so does G2(11). Next M22,! U6(2) and A12,! O-10= 2D5(2) are also localizations. In the last three localizations, connecting the groups U4(3), E6(4), and D4(* *3), the orders of the outer-automorphism groups is larger than 2. Nevertheless, Theore* *m 1.4 applies easily. There is a localization A7 ,! U4(3). There are four conjugacy* * classes of subgroups of U4(3) isomorphic to A7, all of them being maximal. We have Out(U4(* *3)) ~= D8 acting transitively on those classes and S7 is contained in Aut(U4(3)) (see * *[4, p.52]). We have also a localization J3 ,! E6(4). Here Out(E6(4)) ~=D12and there are * *exactly six conjugacy classes of J3 in E6(4) which are permuted transitively by D12. Th* *is is exactly the statement of the main theorem of [15]. Finally D4(2) ,! D4(3) is a localization. Here Out(D4(2)) ~=S3 and Out(D4(3)* *) ~=S4. There are four conjugacy classes of subgroups of D4(3) isomorphic to D4(2). 4 Other localizations In this section, we give further examples of localizations between simple group* *s. We start with three infinite families of localizations. Except the second family, we do * *not know if the groups belong to the rigid component of alternating groups. Proposition 4.1 Let p be an odd prime with (3; p - 1) = 1. Then there is a loc* *alization L3(p) ,! G2(p). 13 Proof. We first treat the case p = 3. Then Out (L3(3)) = C2 = Out (G2(3)). B* *y [18, Table 1, p.300] we know there are two conjugacy classes of Aut(L3(3)) in G2(3),* * which are both maximal. They fuse in Aut (G2(3)) by a graph automorphism by [12, Proposi* *tion 2.2]. Also NAut(G2(3))(L3(3)) = Aut(L3(3)) by the same proposition, so that con* *dition (3) of Theorem 1.4 is satisfied. There are also only two copies of L3(3) in G2(3), * *as one can see in [12, Theorem A] that the only subgroups of type L3(3) in G2(3) are those* * contained in Aut(L3(3)). The case p 6= 3 is simpler as there is only one conjugacy class of Aut(L3(p)* *) in G2(p) by [18, Table 1, p.300]. In this case G2(p) is complete. * * 2 Proposition 4.2 Let p be an odd prime with (3; p + 1) = 1. Then there is a loc* *alization U3(p) ,! G2(p). Proof. The proof is similar to that of the preceding proposition. Apply also [1* *2, Propo- sition 2.2]. * * 2 Proposition 4.3 Let p be any prime. Then there is a localization 3D4(p) ,! F4(* *p). Proof. We have Out (F4(2)) = C2 while for an odd prime p, F4(p) is complete. * *On the other hand Out ( 3D4(p)) = C3. By [18, Proposition 7.2] there are exactly * *(2; p) conjugacy classes of 3D4(p) in F4(p), fused by an automorphism if p = 2. The in* *clusion 3D4(p),! Aut ( 3D4(p)) ,! F4(p) given by [18, Table 1, p.300] is thus a locali* *zation by Theorem 1.4. 2 For sporadic simple groups, we have seen various localizations in the proof * *of the main theorem. We give here further examples involving sporadic groups. We start with the five Mathieu groups. Recall that the Mathieu groups M12 a* *nd M22 have C2 as outer-automorphism groups, while the three other Mathieu groups * *are complete. The inclusions M11,! M23and M23,! M24are localizations by Corollary 1* *.6. We have already seen in Section 3 that the inclusion M11 ,! M12 is a localizati* *on. The inclusion M12,! M24 is also a localization. Indeed Aut(M12) is the stabilizer * *in M24of a pair of dodecads, the stabilizer of a single dodecad is a copy of M12. Up to co* *njugacy, these are the only subgroups of M24isomorphic to M12and thus the formula (3') in Rema* *rk 1.5 about the number of conjugacy classes of M12in M24is satisfied. Similarly M22,!* * M24is also a localization, because Aut(M22) can be identified as the stabilizer of a * *duad in M24 14 whereas M22 is the pointwise stabilizer. In short we have the following diagram* *, where all inclusions are localizations: " M11"O___//_M23 | _ "|_ | | fflffl|O fflffl|"_ M12 ___//_M24oo_?M22:_ We consider next the sporadic groups linked to the Conway group Co1. Inside * *Co1 sits Co2 as stabilizer of a certain vector OA of type 2 and Co3 as stabilizer of ano* *ther vector OB of type 3. These vectors are part of a triangle OAB and its stabilizer is th* *e group HS, whereas its setwise stabilizer is Aut(HS) = HS:2. The Conway groups are com* *plete, the smaller ones are maximal simple subgroups of Co1 and there is a unique conj* *ugacy class of each of them in Co1 as indicated in the ATLAS [4, p.180]. Hence Co2 ,!* * Co1 and Co3 ,! Co1 are localizations by Corollary 1.6. Likewise the inclusions HS ,! Co* *2 and McL ,! Co3 are also localizations: They factor through their group of automorph* *isms, since for example Aut(McL) is the setwise stabilizer of a triangle of type 223 * *in the Leech lattice, a vertex of which is stabilized by Co3. Finally, M22 ,! HS is a locali* *zation for similar reasons, since any automorphism of M22 can be seen as an automorphism o* *f the Higman-Sims graph (cf. [1, Theorem 8.7 p.273]). We get here the following diagr* *am of localizations: " M22" McL O___//_Co3 | _ "|_ | | fflffl|O " O " fflffl| HS ____//_Co2___//_Co1 Some other related localizations are M23,! Co3, M23,! Co2 and M11,! HS. We move now to the Fischer groups and Janko's group J4. The inclusion T ,! F* * i22is a localization (both have C2 as outer automorphism groups) as well as M12,! F i* *22, and A10,! F i22. Associated to the second Fischer group, we have a chain of localiz* *ations A10,! S8(2) ,! F i23: By [13, Theorem 1] the inclusion A12 ,! F i23 is also a localization. Finally * *M11,! J4 and M23,! J4 are localizations by Corollaries 6.3.2 and 6.3.4 in [14]. Let us now end this section with a list without proofs of a few inclusions w* *e know to be localizations. We start with two examples of localizations of alternating gr* *oups: A12,! HN, and A7 ,! Suz by [22, Section 4.4]. 15 Finally we list a few localizations of Chevalley groups: L2(8) ,! S6(2), L2(13) ,! G2(3), L2(32) ,! J4 (by [14, Proposition 5.3.1]), U3(* *3) ,! S6(2), 3D4(2) ,! T h, G2(5) ,! Ly, E6(2) ,! E7(2), and E6(3) ,! E7(3). The incl* *usion E6(q) ,! E7(q) is actually a localization if and only if q = 2 or q = 3 by [18,* * Table 1]. The main theorem in [21] states that Sz(32) ,! E8(5) is a localization. There is on* *e conjugacy class of Sz(32), and Out(Sz(32)) ~=C5. 5 Localizations between automorphism groups The purpose of this section is to show that a localization H ,! G can often be * *extended to a localization Aut(H) ,! Aut(G). This generalizes the observation made by Libma* *n (cf. [17, Example 3.4]) that the localization An ,! An+1 extends to a localization S* *n ,! Sn+1 if n 7. This result could be the starting point for determining the rigid comp* *onent of the symmetric groups, but we will not go further in this direction. Lemma 5.1 Let G be a finite simple group with Out(G) ~= Cp, where p is a pri* *me. Then G is the only proper normal subgroup of Aut (G) and any non-trivial endomorphis* *m of Aut (G) is either an isomorphism, or has G as kernel. * * 2 Lemma 5.2 Let H ,! G be an inclusion of simple groups. Then any subgroup of * *Aut(G) isomorphic to H is contained in G. Proof. Let H0 be a subgroup of Aut(G) isomorphic to H. The kernel of the projec* *tion Aut (G)!!Out (G) contains H0 because H0 is simple, while Out(G) is solvable (th* *is is the Schreier conjecture, whose proof depends on the classification of finite simple* * groups, see [10, Theorem 7.1.1]). * * 2 Theorem 5.3 Let H ,! G be a localization between two finite simple groups. S* *uppose that Out (H) ~= Out(G) ~= Cp, where p is a prime. Then Aut (H) ,! Aut (G) is a* *lso a localization. Proof. The idea is similar to the proof of Theorem 1.4. Let ' : Aut (H) ! Aut* * (G) be any homomorphism. Let us assume that ' is not trivial. If it is an injecti* *on, the ' composite OE : H ,! Aut (H) -! Aut(G) actually lies in G by Lemma 5.2 and beca* *use H ,! G is a localization by Theorem 1.4, there is a unique automorphism ff of G* * making the appropriate diagram commute. Conjugation by ff on Aut(G) is the unique exte* *nsion 16 we need. Indeed in the following diagram all squares are commutative and so is * *the top triangle: __________i__________//_ H| MMM qqG| | MMOEMMM ffqqqqq| | MMM qqqq | || M&&Mxxqq || | G | | | | | | | fflffl| || fflffl| Aut(H) ________|_i_____//_Aut(G) LLL | ss L'LL | cffsss LLL | sss L%%fflffl|yyss Aut (G) Conjugation by ff of an automorphism of H is an automorphism of OE(H). Therefo* *re cff(Aut (H)) Aut (OE(H)) and thus cff(Aut (H)) and '(Aut (H)) coincide because* * they cff-1 OE both are equal to Aut (OE(H)). The composite Aut (OE(H)) -! Aut (H) - ! Aut(* *OE(H)) is conjugation by some automorphism fi of OE(H) since Aut (OE(H)) is complete b* *y [20, Theorem 7.14]. We have thus shown that ' = cfiO cffO i. In particular OE is the* * restriction of ff to H composed with fi. But by construction OE = ff |H and so fi has to be* * trivial. By Lemma 5.1, the only other case is when ker' = H. In that case Cp is a sub* *group of Out(G) . Thus it clearly extends to a unique endomorphism of Aut(G) . * * 2 Many examples can be directly derived from previous examples, such as Sn ,! * *Sn+1 and SL2(p) ,! Sp+1. Suzuki's chain of groups (3.1), as well as M22 ,! HS, also * *extend to localizations of their automorphism groups. The converse of the above theorem is false, as shown by the following exampl* *e. There exists an inclusion Aut (L3(2)) ,! S8 which is actually a localization (Conditi* *on (0.1) can be checked for example with the help of MAGMA). However the induced morphism L3(2) ,! A8 fails to be a localization: There are two conjugacy classes of subg* *roups of A8 isomorphic to L3(2), which are not conjugate in S8. 6 Further results It was asked in [16] and also in [3] whether simple groups are preserved under * *localization, i.e. if H ,! G is a localization and H is simple, is G necessary simple? We n* *ext show that the answer is affirmative if H is maximal in G. However, without this assu* *mption G need not be simple, as illustrated by Proposition 6.5, where we show that under* * certain 17 conditions a localization H ,! G induces a localization H ,! "Gfrom H to the un* *iversal cover G" of G. This result was elaborated on an observation made by A. Viruel * *(cf. Example 6.6 below). Proposition 6.1 Let G be a finite group and let H be a maximal subgroup which * *is simple. If the inclusion H ,! G is a localization, then G is simple. Proof. Let N be a normal subgroup of G. As H is simple, N \ H is either equal t* *o {1} or H. If N \ H = H, as H is maximal, then either N = G or N = H, and we show that the latter case is impossible. By maximality of H, the quotient G=H does not ha* *ve any non-trivial proper subgroup, so G=H ~=Cp for some prime p. Then G has a subgrou* *p of order p and there is an endomorphism of G factoring through Cp, whose restricti* *on to H is trivial. This contradicts the assumption that the inclusion H ,! G is a loca* *lization. If N \ H = {1}, then either N = {1} or NH = G as H is maximal. The second ca* *se cannot occur because it would imply that G = N o H, but H ,! N o H cannot be a localization since both the identity of G and the projection onto H extend the * *inclusion H ,! G . 2 We indicate now a generic situation where the localization of a simple group* * can be non-simple (it will actually be a double cover of a simple group). We first nee* *d to recall some basic facts. Let Mult(G) be the Schur multiplier of a finite simple group * *G. It is well known that the universal cover "G!!G induces an exact sequence 1 ! S ! Aut(G") ! Aut(G) ! 1; (6.* *2) where S is the subgroup of automorphisms of "Gwhich induce the identity on G. T* *hus, if S = 1, then Aut(G") ~=Aut (G). Lemma 6.3 Let G be a non-abelian finite simple group with Mult(G) ~= C2. Th* *en Aut (G") ~=Aut (G). Proof. We show that any automorphism ff of G" which induces the identity on G * *is itself the identity. Such an automorphism induces an automorphism on Mult(G) ~=* * C2. The only automorphism of C2 is the identity, so we have to determine the set of* * automor- phisms of "Ginducing the identity on both G and Mult(G). This set is in bijecti* *on with Hom (G; Mult(G)), which is trivial since G is simple and Mult(G) abelian. * * 2 18 Proposition 6.4 Let G be a finite simple group with Schur multiplier Mult(G). * *Suppose that S = 1 in (6.2). Then, the universal cover G"!!G is a localization. In part* *icular, if Mult(G) ~=C2, we have that "G! G is a localization. Proof. We have to show that "G! G induces a bijection Hom (G; G) ~=Hom (G"; G)* * or equivalently, Aut (G) ~= Hom (G"; G) \ {0}. This follows easily since the only* * non-trivial proper normal subgroups of "Gare contained in its centre Mult(G). Thus any non-* *trivial homomorphism "G! G can be decomposed as the canonical projection "G! G followed by an automorphism of G. 2 Proposition 6.5 Let i : H ,! G be an inclusion of two finite simple groups. Su* *ppose that the Schur multipliers of H and G have orders 1 and 2 respectively and let j : H* * = "H,! "G be the induced homomorphism. Then i : H ,! G is a localization if and only if j* * : H ,! "G is a localization. Proof. Suppose that i : H ,! G is a localization and let ' : H ! "Gbe a non-tri* *vial homomorphism. We have to show that this homomorphism extends to an automorphism ' of "G. The composite H -! G"!!G extends to a unique automorphism of G, since* * i is a localization. Now, as S = 1 in (6.2) above, can be lifted to a unique autom* *orphism of "G, which is the desired automorphism. The proof of the other implication is* * similar. 2 Example 6.6 This example was communicated to us by Antonio Viruel. The inclus* *ion M11 ,! M"12of the Mathieu group M11 into the double cover of the Mathieu group * *M12 is a localization. This follows from the above proposition. Note that M11 is no* *t maximal in M"12(the maximal subgroup is M11x C2), so this does not contradict Propositi* *on 6.1. Since Mult(An) ~= C2, we get many other examples of this type using Corollary 2* *.2. All sporadic groups appearing in this list which have trivial Schur multiplier * *(that is M11; M23; M24; J1; J4; Co2; Co3; He; F i23; HN, and Ly) admit the double cover * *of an al- ternating group as localization. Here are some more examples of localizations w* *hich give rise to similar examples: Co2 ,! Co1, Co3 ,! Co1, G2(2)0,! J2, F i23,! B. Remark 6.7 The latter example F i23 ,! B produces a localization F i23 ,! "Ba* *nd the double cover B" is a maximal subgroup of the Monster M. It would be nice to kn* *ow if B" ,! M is a localization, which would connect the Monster to the rigid compone* *nt of the alternating groups. 19 References [1]Beth, T., Jungnickel, D., and Lenz, H., Design Theory, Cambridge University* * Press, 1986. [2]Bousfield, A. K., Constructions of factorization systems in categories, J. * *Pure Appl. Algebra 9 (1976/77), no. 2, 207-220. [3]Casacuberta, C., On structures preserved by idempotent transformations of g* *roups and homotopy types, in: Crystallographic Groups and Their Generalizations (* *Kor- trijk, 1999), Contemp. Math. 262, Amer. Math. Soc., Providence, 2000, 39-68. [4]Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., and Wilson, R.A., A* *tlas of finite groups. Maximal subgroups and ordinary characters for simple groups,* * Oxford: Clarendon Press. XXXIII, 1985. [5]Cooperstein, B. N., Minimal degree for a permutation representation of a cl* *assical group, Israel J. Math. 30 (1978), no. 3, 213-235. [6]Dixon, J. D, and Mortimer, B., Permutation groups, Springer, GTM 163, (1996* *). [7]Farjoun, E. D., Cellular spaces, null spaces and homotopy localization, Lec* *ture Notes in Math., 1622, Springer-Verlag, Berlin, 1996. [8]G"obel, R., Rodriguez, J. L., and Shelah, S., Large localizations of finite* * simple groups, preprint, 1999. [9]Gorenstein, D., Finite simple groups. An introduction to their classificati* *on, The University Series in Mathematics. New York - London: Plenum Press. X, 1982. [10]Gorenstein, D., Lyons, R., and Solomon, R., The classification of the finit* *e simple groups. Number 3. Part I. Chapter A. Almost simple K-groups, Mathematical S* *ur- veys and Monographs, 40.3. American Mathematical Society, Providence, RI, 1* *998 [11]Huppert, B., Endliche Gruppen I, Springer-Verlag, Berlin, 1967. [12]Kleidman, P. B., The maximal subgroups of the Chevalley groups G2(q) with q* * odd, the Ree groups 2G2(q), and their automorphism groups, J. Algebra 117 (1988)* *, no. 1, 30-71. [13]Kleidman, P. B., Parker, R. A., and Wilson, R. A., The maximal subgroups of* * the Fischer group Fi23, J. London Math. Soc. (2) 39 (1989), no. 1, 89-101. [14]Kleidman, P. B., and Wilson, R. A., The maximal subgroups of J4, Proc. Lond* *on Math. Soc. (3) 56 (1988), no. 3, 484-510. [15]Kleidman, P. B., and Wilson, R. A., J3 < E6(4) and M12< E6(5), J. London Ma* *th. Soc. (2) 42 (1990), no. 3, 555-561. 20 [16]Libman, A., A note on the localization of finite groups, J. Pure Appl. Alge* *bra 148 (2000), no. 3, 271-274. [17]Libman, A., Cardinality and nilpotency of localizations of groups and G-mod* *ules, Israel J. Math. 117 (2000), 221-237. [18]Liebeck, M. W., and Saxl, J., On the orders of maximal subgroups of the fin* *ite exceptional groups of Lie type, Proc. London Math. Soc. (3) 55 (1987), no. * *2, 299- 330. [19]Linton, S. A., and Wilson, R. A., The maximal subgroups of the Fischer grou* *ps Fi24 and Fi024, Proc. London Math. Soc. (3) 63 (1991), no. 1, 113-164. [20]Rotman, J. J., An introduction to the theory of groups, Fourth edition. Gra* *duate Texts in Mathematics 148, Springer-Verlag, New York, 1995. [21]Saxl, J., Wales, D. B., and Wilson, R. A., Embedings of Sz(32) in E8(5), Bu* *ll. London Math. Soc. 32 (2000), no. 2, 196-202. [22]Wilson, R. A., The complex Leech lattice and maximal subgroups of the Suzuk* *i group, J. Algebra 84 (1983), no. 1, 151-188. [23]Wilson, R. A., The maximal subgroups of the O'Nan group., J. Algebra 97 (19* *85), no. 2, 467-473. [24]Wilson, R. A., The maximal subgroups of the Baby Monster, I, J. Algebra 211* * (1999), no. 1, 1-14. Jose L. Rodriguez: Departamento de Geometria, Topologia y Quimica Organica, Universidad de Almeria, E-04120 Almeria, Spain, e-mail: jlrodri@ual.es Jer^ome Scherer and Jacques Thevenaz: Institut de Mathematiques, Universite de Lausanne, CH-1015 Lausanne, Switzerlan* *d, e-mail: jerome.scherer@ima.unil.ch, jacques.thevenaz@ima.unil.ch 21