Finite simple groups and localization

José L. Rodríguez, Jérôme Scherer and Jacques Thévenaz ^{∗}

Abstract

The purpose of this paper is to explore the concept of localization, which 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 subgroup and we apply it in various cases. Iterating this process allows us to connect 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 connected to alternating groups. The question remains open whether or not there are several connected components within the family of finite simple groups. In some cases, we also consider automorphism groups and universal covering groups and we show 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 introduction by Bousfield of homotopical localization functors in [2] and more recently its popularization by Farjoun in [7] has led to the study of localization functors in other categories. Special attention has been set on the category of groups Gr, as the effect of a homotopical localization on the fundamental group is often best described by a localization functor L : Gr → Gr.

A localization functor is a pair (L, η) consisting of a functor L : Gr → Gr together with a natural transformation η : Id → L, such that L is idempotent, meaning that the two morphisms η_{LG}, L(η_{G}) : LG → LLG coincide and are isomorphisms. A group homomorphism φ : H → G is called in turn a localization if there exists a localization functor (L, η) such that G = LH and φ = η_{H} : H → LH (but we note that the functor L is not uniquely determined by φ). In this situation, we often say that G is a localization of H. A very simple characterization of localizations can be given without mentioning localization functors: A group homomorphism φ : H → G is a localization if and only if φ induces a bijection

∗The first author was partially supported by DGESIC grant PB97-0202 and the Swiss National Science Foundation.

φ^{∗} : Hom(G, G) ≅ Hom(H, G)  (0.1)

as mentionned in [3, Lemma 2.1]. In the last decade several authors (Casacuberta, Farjoun, Libman, Rodríguez) have directed their efforts towards deciding which algebraic properties are preserved under localization. An exhaustive survey about this problem is nicely exposed in [3] by Casacuberta. For example, any localization of an abelian group is again abelian. Similarly, nilpotent groups of class at most 2 are preserved, but the question remains open for arbitrary nilpotent groups. Finiteness is not preserved, as shown by the example A_{n} → SO(n − 1) (this is the main result in [16]). In fact, it has been shown in [8] that any non-abelian finite simple group has arbitrarily large localizations (under certain set-theoretical assumptions). In particular it is not easy to determine all possible localizations of a given object. Thus we restrict ourselves to the study of finite groups and wonder if it would be possible to understand the finite localizations 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 A_{n} ↪ A_{n+1} is a localization if n ≥ 7. His motivation was to find a localization where new torsion elements appear (e.g. A_{10} ↪ A_{11} is such a localization since A_{11} contains elements of order 11). In these examples, the groups are simple, which simplifies considerably the verification 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 localizations 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 families of such localizations, for example L_{2}(p) ↪ A_{p+1} for any prime p ≥ 13 (cf. Proposition 2.3). Here L_{2}(p) = P SL_{2}(p) is the projective special linear group. It is striking to notice 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 J_{3} ↪ E_{6}(4) is a localization (see Section 3). Similarly the main theorem in [21] states that Sz(32) ↪ E_{8}(5) is a localization. Hence the language of localization theory can be useful to shortly reformulate some rather technical properties.

By Libman’s result, the alternating groups A_{n}, for n ≥ 7, are all connected by a sequence of localizations. We show that A_{5} ↪ A_{6} is also a localization. A more curious way allows us to connect A_{6} to A_{7} by a zigzag of localizations:

A_{6} ↪ T ↪ Ru ↩ L_{2}(13) ↪ A_{14} ↩ ⋯ ↩ A_{7}

where T is the Tits group, and Ru the Rudvalis group. This yields to the concept 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 A_{n} (n ≥ 5).

(ii) The Chevalley groups L_{2}(q) where q is a prime power ≥ 5.

(iii) The Chevalley groups U_{3}(q) where q is a prime power, q ≠ 5.

(iv) The Chevalley groups G_{2}(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 L_{3}(3), L_{3}(5), L_{3}(11), L_{4}(3), U_{4}(2), U_{4}(3), U_{5}(2), U_{6}(2), S_{4}(4), S_{6}(2), S_{8}(2), D_{4}(2), ^{2}D_{4}(2), ^{2}D_{5}(2), ^{3}D_{4}(2), D_{4}(3), G_{2}(2)′, G_{2}(4), G_{2}(5), G_{2}(11), E_{6}(4), F_{4}(2), and T = ^{2}F_{4}(2)′.

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 complete 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 localization with H simple then G need not be simple. There is a localization map from the Mathieu group M_{11} to the double cover of the Mathieu group M_{12}. This answers negatively a question posed by Libman in [17] and also by Casacuberta in [3] about the preservation of simplicity. In our context this also implies that the rigid component of a simple group may contain a non-simple group.

It is still an open problem to know how many rigid components of finite simple 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 helpful comments.

1 A localization criterion

In Theorem 1.4 below we list necessary and sufficient conditions for an inclusion H ↪ G between two non-abelian finite simple groups to be a localization. These conditions 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 Corollary 4 in [8].

We note here that we only deal with injective group homomorphisms because noninjective localizations abound. For example, for any two finite groups G_{1} and G_{2} of coprime orders, G_{1} × G_{2} → G_{1} and G_{1} × G_{2} → G_{2} are localizations. So the analogous concept of rigid component defined using non-injective localizations has no interest, 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 H′ ↪ G for any subgroup H′ of G which is isomorphic to H. This shows that the choice of the subgroup H among isomorphic subgroups does not matter.

Let c : G → Aut(G) be the natural injection of G defined as c(g) = c_{g}, where c_{g} : G → G denotes the inner automorphism given by x ↦ gxg^{−1}. We shall always identify in this way a simple group G with a subgroup of Aut(G), without writing 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 commutes:

\xymatrix{G \ar[r]^{α} \ar[d]_{c} & G \ar[d]^{c} \\ Aut(G) \ar[r]_{c_{α}} & Aut(G)}

for any automorphism α ∈ Aut(G).

Proof. This is a trivial check. □

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

\xymatrix{H \ar[r]^{i} \ar[d]_{c} & G \ar[d]^{c} \\ Aut(H) \ar[r]_{i} & Aut(G)}

Then every automorphism α : H → H extends to an automorphism i(α) : G → G.

Proof. We have to show that the following square commutes:

\xymatrix{H \ar[r]^{α} \ar@{^{(}->}[d]_{i} & H \ar@{^{(}->}[d]^{i} \\ G \ar[r]_{i(α)} & G}

To do so we consider this square as the left-hand face of the cubical diagram

[figure: H; c; α; Aut(H); c_{α}; H; c; Aut(H); G; i(α); c; Aut(G); c_{i(α)}; 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. □

Remark 1.3 As shown by the preceding lemma, it is stronger to require that the 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 → C_{Aut(G)}(H) ⟶ N_{Aut(G)}(H) ⟶ Aut(H)

so the second condition is equivalent to the fact that this is a short exact sequence. However, in the presence of the condition C_{Aut(G)}(H) = 1, which plays a central role in this paper, we find that N_{Aut(G)}(H) ≅ Aut(H). Thus any automorphism of H extends 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 C_{Aut(G)}(H) = 1 holds, 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 stronger 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 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).

3. The centralizer C_{Aut(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 extended to an automorphism of G (see also Remark 1.3). Condition (2) claims that the inclusion of a subgroup of G isomorphic to H can be extended, while condition (3) says that there exists a unique extension for i, namely the identity.

Assume now that all three conditions are satisfied. For any given homomorphism φ : H → G, we have to find a unique homomorphism Φ : G → G such that Φ ∘ i = φ. The trivial homomorphism G → G obviously extends the trivial homomorphism from H to G. It is unique since H is in the kernel of Φ, which must be equal to G by simplicity. Hence, we can suppose that φ is not trivial. Since H is simple we have that φ(H) ≤ G and H ≅ φ(H).

By (2) there is an automorphism α ∈ Aut(G) such that c_{α}(φ(H)) = H, or equivalently by Lemma 1.1, α(φ(H)) = H. Therefore the composite map

H ⟶^{φ} φ(H) ⟶^{α|_{φ(H)}} H

is some automorphism β of H. By condition (1) this automorphism of H extends to an automorphism i(β) : G → G. That is, the following square commutes:

\xymatrix{H \ar[r]^{β} \ar@{^{(}->}[d]_{i} & H \ar@{^{(}->}[d]^{i} \\ G \ar[r]_{i(β)} & G}

The homomorphism Φ = α^{−1}i(β) extends φ as desired. We prove now it is unique. Suppose that Φ′ : G → G is a homomorphism such that Φ′ ∘ i = φ. Then, since G is simple, Φ′ ∈ Aut(G). The composite Φ^{−1}Φ′ is an element in the centralizer C_{Aut(G)}(H), which is trivial by (3). This finishes the proof of the theorem. □

Remark 1.5 As already mentionned in Remark 1.3, conditions (1) and (3) imply that Aut(H) = N_{Aut(G)}(H). By condition (2), the cardinal of the orbit of H under the conjugation 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,

\frac{| Aut(G) |}{| N_{Aut(G)}(H) |} = k · \frac{| G |}{| N_{G}(H) |},

Condition (3) is thus equivalent to the following one, which is sometimes easier to verify:

3’. The number of conjugacy classes of subgroups of G isomorphic to H is equal to

\frac{| Out(G) |}{| Out(H) |} · \frac{| N_{G}(H) |}{| H |}.

We obtain immediately the following corollaries. Using the terminology in [20, p.158], recall that a group is complete if it has no outer-automorphism and trivial centre. 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. C_{G}(H) = 1. □

The condition C_{G}(H) = 1 is here equivalent to N_{G}(H) = H. This is often easier to check. It is in particular always the case when H is a maximal subgroup of G. This leads us to the next corollary.

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).

3. The number of conjugacy classes of H in G is equal to \frac{|Out(G)|}{|Out(H)|}.

Proof. Since H is a maximal subgroup of G, N_{G}(H) = H. The corollary is now a direct consequence of Theorem 1.4 taking into account Remark 1.5 about the number of conjugacy classes of subgroups of G isomorphic to H. □

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 (left) action of H on the cosets of K in H defines a permutation representation H → S_{n} 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 ↪ A_{n}. Recall that Aut(A_{n}) = S_{n} if n ≥ 7.

Theorem 2.1 Let H be a non-abelian finite simple group and K a maximal subgroup 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 ↪ A_{n} is a localization.

Proof. We show that the conditions of Theorem 1.4 are satisfied, starting with condition (1). Since K is maximal, it is self-normalizing and therefore the action of H on the cosets of K is isomorphic to the conjugation action of H on the set of conjugates of K. By our second assumption, this set is left invariant under Aut(H). Thus the action of H extends to Aut(H) and this yields the desired extension Aut(H) → S_{n} = Aut(A_{n}).

To check condition (2) of Theorem 1.4, let H′ be a subgroup of A_{n} which is isomorphic to H and denote by α : H → H′ an isomorphism. Let J be the stabilizer of a point in {1, …, n} under the action of H′. Since the orbit of this point has cardinality ≤ n, the index of J is at most n, hence equal to n by our first assumption. Thus H′ acts transitively. So H has a second transitive action via α and the action of H′. For this action, the stabilizer of a point is a subgroup of H of index n, hence conjugate to K by assumption. So K is also the stabilizer of a point for this second action and this shows that this action of H is isomorphic to the permutation action of H on the cosets of K, that is, to the first action. It follows that the permutation representation H ⟶^{α} H′ ↪ A_{n} is conjugate in S_{n} to H ↪ A_{n}.

Finally, since H is a transitive subgroup of S_{n} with maximal stabilizer, the centralizer C_{S_{n}}(H) is trivial by [6, Theorem 4.2A (vi)] and thus condition (3) of Theorem 1.4 is satisfied. □

Among the twenty-six sporadic simple groups, twenty have a subgroup which satisfies the conditions of Theorem 2.1.

Corollary 2.2 The following inclusions are localizations: M_{11} ↪ A_{11} , M_{22} ↪ A_{22} , M_{23} ↪ A_{23} , M_{24} ↪ A_{24} , J_{1} ↪ A_{266} , J_{2} ↪ A_{100} , J_{3} ↪ A_{6156} , J_{4} ↪ A_{173067389} , HS ↪ A_{100} , McL ↪ A_{275} , Co_{1} ↪ A_{98280} , Co_{2} ↪ A_{2300} , Co_{3} ↪ A_{276} , Suz ↪ A_{1782} , He ↪ A_{2058} , Ru ↪ A_{4060} , F i_{22} ↪ A_{3510} , F i_{23} ↪ A_{31671} , HN ↪ A_{1140000} , Ly ↪ A_{8835156} .

Proof. In each case, it suffices to check in the ATLAS [4] that the conditions of Theorem 2.1 are satisfied. It is necessary to check the complete list of maximal subgroups in [13] for the Fischer group F i_{23} and [14] for the Janko group J_{4}. □

We obtain now two infinite families of localizations. The classical projective special linear groups L_{2}(q) = P SL_{2}(q) of type A_{1}(q), as well as the projective special unitary groups U_{3}(q) = P SU_{3}(q) of type ^{2}A_{2}(q), are almost all connected to an alternating group by a localization. Recall that the notation L_{2}(q) is used only for the simple projective special linear groups, that is if the prime power q ≥ 4. Similarly the notation U_{3}(q) is used for q > 2.

Proposition 2.3 (i) The permutation representation L_{2}(q) ↪ A_{q+1} induced by the action of SL_{2}(q) on the projective line is a localization for any prime power q ∉ {4, 5, 7, 9, 11}.

(ii) The permutation representation U_{3}(q) ↪ A_{q^{3}+1} induced by the ac-

tion of SU_{3}(q) on the set of isotropic points in the projective plane is a localization for any prime power q ≠ 5.

Proof. We prove both statements at the same time. The group L_{2}(q) acts on the projective line, whereas U_{3}(q) acts on the set of isotropic points in the projective plane. In both cases, let B be the stabilizer of a point for this action (Borel subgroup). Let us also denote by G either L_{2}(q) or U_{3}(q), where q is a prime power as specified above, and r is q + 1, or q^{3} + 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, L_{2}(q) has no non-trivial permutation representation of degree less than r if q ∉ {4, 5, 7, 9, 11}. The same holds for U_{3}(q) by [5, Table 1] if q ≠ 5. Thus B satisfies condition (1) of Theorem 2.1.

It remains to show that condition (2) is also satisfied. The subgroup B is the normalizer of a Sylow p-subgroup U, and B = UT, where T is a complement of U in B. If N denotes the normalizer of T in G, we know that G = UNU (Bruhat decomposition). We are now ready to prove that any subgroup of G of index r is conjugate to B. Let H be such a subgroup. It contains a Sylow p-subgroup, and we can thus assume it actually contains U. Since G is generated by U and N, the subgroup H is generated by U and N ∩ H. Assume H contains an element x ∈ N − T. The class of x in the Weyl group N/T ≅ C_{2} is 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 ≠ G, so N ∩ H = T ∩ H. It follows that H is contained in < U, T > = B. But H and B have the same order and therefore H = B. □

Remark 2.4 This proof does not work for the action of L_{n+1}(q) on the n-dimensional 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 (𝔽_{q})^{n+1}. Thus there is another conjugacy class of subgroups of the same index, so condition 1 does not hold.

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 specific proof is indicated for an inclusion to be a localization, it means that all the necessary information for checking conditions (1)-(3) of Theorem 1.4 is available in the ATLAS [4].

(i) Alternating groups.

The inclusions A_{n} ↪ A_{n+1}, for n ≥ 7, studied by Libman in [17, Example 3.4] are localizations by Corollary 1.7, with Out(A_{n}) ≅ C_{2} ≅ Out(A_{n+1}). The inclusion A_{5} ↪ A_{6} is a localization as well, since Out(A_{6}) = (C_{2})^{2}, Out(A_{5}) = C_{2}, and there are indeed two conjugacy classes of subgroups of A_{6} isomorphic to A_{5} with fusion in Aut(A_{6}). The inclusion A_{6} ↪ A_{7} is not a localization, but we can connect these two groups via a zigzag of localizations, for example as follows:

A_{6} ↪ T ↪ Ru ↩ L_{2}(13) ↪ A_{14}

where T denotes the Tits group, Ru the Rudvalis group and the last arrow is a localization by Proposition 2.3.

(ii) Chevalley groups L_{2}(q).

By Proposition 2.3, all but five linear groups L_{2}(q) are connected to an alternating group. The group L_{2}(4) ≅ L_{2}(5) is A_{5}, and L_{2}(9) ≅ A_{6}, which are connected to all alternating groups by the argument above. We connect L_{2}(7) to A_{28} via a chain of two localizations

L_{2}(7) ↪ U_{3}(3) = G_{2}(2)′ ↪ A_{28}

where we use Theorem 2.1 for the second map. Similarly, we connect L_{2}(11) to A_{22} via the Mathieu group M_{22}:

L_{2}(11) ↪ M_{22} ↪ A_{22}.

(iii) Chevalley groups U_{3}(q).

For q ≠ 5, we have seen in Proposition 2.3 (ii) that U_{3}(q) ↪ A_{q^{3}+1} is a localization. Recall that U_{3}(2) is not simple. We do not know if U_{3}(5) is connected to the alternating groups.

(iv) Chevalley groups G_{2}(p).

When p is an odd prime such that (p + 1, 3) = 1, we will see in Proposition 4.2 that U_{3}(p) ↪ G_{2}(p) is a localization. We can conclude by (iii), since 5 is not a prime in the considered family.

(v) Sporadic simple groups.

By Corollary 2.2, we already know that twenty sporadic simple groups are connected 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 M_{12}, we note that the inclusion M_{11} ↪ M_{12} is a localization because there are two conjugacy classes of inclusions of M_{11} in M_{12} (of index 11) with fusion in Aut(M_{12}) (cf. [4, p.33]). We conclude by Corollary 1.7.

The list of all maximal subgroups of F i′_{24} is given in [19] and one sees that He ↪ F i′_{24} is a localization.

Looking at the complete list of maximal subgroups of the Baby Monster B in [24], we see that F i_{23} ↪ B is a localization, as well as T h ↪ B, HN ↪ B, and L_{2}(11) ↪ B (see Proposition 4.1 in [24]). This connects Thompson’s group T h and the Baby Monster (as well as the Harada-Norton group HN) to the Fischer groups and also to the Chevalley groups L_{2}(q).

Finally we consider the O’Nan group O′N. By [23, Proposition 3.9] we see that M_{11} ↪ O′N is a localization.

(vi) Other Chevalley groups.

Suzuki’s construction of the sporadic group Suz provides a sequence of graphs whose groups of automorphisms are successively Aut(L_{2}(7)), Aut(G_{2}(2)′), Aut(J_{2}), Aut(G_{2}(4)) and Aut(Suz) (see [9, p.108-9]). Each one of these five groups is an extension of C_{2} by the appropriate finite simple group. All arrows in the sequence

L_{2}(7) ↪ G_{2}(2)′ ↪ J_{2} ↪ G_{2}(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 G_{2}(2)′ and G_{2}(4) to alternating groups since we already know that Suz is connected to A_{1782} by Corollary 2.2. Alternatively, note that G_{2}(4) ↩ L_{2}(13) ↪ A_{14} are localizations, using Proposition 2.3 for the second one.

The Suzuki group provides some more examples of localizations: L_{3}(3) ↪ Suz by [22, Section 6.6], and U_{5}(2) ↪ Suz by [22, Section 6.1]. We also have localizations

A_{9} ↪ D_{4}(2) = O_{8}^{+}(2) ↪ F_{4}(2) ↩ ^{3}D_{4}(2)

which connect these Chevalley groups (see Proposition 4.3 for the last arrow). We are able to connect three symplectic groups since A_{8} ↪ S_{6}(2) and S_{4}(4) ↪ He are localizations, as well as S_{8}(2) ↪ A_{120} by Theorem 2.1. This allows us in turn to connect more Chevalley groups as U_{4}(2) ↪ S_{6}(2), and O_{8}^{−} = ^{2}D_{4}(2) ↪ S_{8}(2) are all localizations.

Each of the following localizations involves a linear group and connects some new group to the component of the alternating groups: L_{2}(11) ↪ U_{5}(2), L_{3}(3) ↪ T, L_{2}(7) ↪ L_{3}(11), and L_{4}(3) ↪ F_{4}(2).

The localization U_{3}(3) ↪ G_{2}(5) connects G_{2}(5) and thus L_{3}(5) by Proposition 4.1 below. Likewise, since we just showed above that L_{3}(11) belongs to the same rigid component, then so does G_{2}(11).

Next M_{22} ↪ U_{6}(2) and A_{12} ↪ O_{10}^{−} = ^{2}D_{5}(2) are also localizations.

In the last three localizations, connecting the groups U_{4}(3), E_{6}(4), and D_{4}(3), the orders of the outer-automorphism groups is larger than 2. Nevertheless, Theorem 1.4 applies easily. There is a localization A_{7} ↪ U_{4}(3). There are four conjugacy classes of subgroups of U_{4}(3) isomorphic to A_{7}, all of them being maximal. We have Out(U_{4}(3)) ≅ D_{8} acting transitively on those classes and S_{7} is contained in Aut(U_{4}(3)) (see [4, p.52]).

We have also a localization J_{3} ↪ E_{6}(4). Here Out(E_{6}(4)) ≅ D_{12} and there are exactly six conjugacy classes of J_{3} in E_{6}(4) which are permuted transitively by D_{12}. This is exactly the statement of the main theorem of [15].

Finally D_{4}(2) ↪ D_{4}(3) is a localization. Here Out(D_{4}(2)) ≅ S_{3} and Out(D_{4}(3)) ≅ S_{4}. There are four conjugacy classes of subgroups of D_{4}(3) isomorphic to D_{4}(2).

4 Other localizations

In this section, we give further examples of localizations between simple groups. 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 localization L_{3}(p) ↪ G_{2}(p).

Proof. We first treat the case p = 3. Then Out(L_{3}(3)) = C_{2} = Out(G_{2}(3)). By [18, Table 1, p.300] we know there are two conjugacy classes of Aut(L_{3}(3)) in G_{2}(3), which are both maximal. They fuse in Aut(G_{2}(3)) by a graph automorphism by [12, Proposition 2.2]. Also N_{Aut(G_{2}(3))}(L_{3}(3)) = Aut(L_{3}(3)) by the same proposition, so that condition (3) of Theorem 1.4 is satisfied. There are also only two copies of L_{3}(3) in G_{2}(3), as one can see in [12, Theorem A] that the only subgroups of type L_{3}(3) in G_{2}(3) are those contained in Aut(L_{3}(3)).

The case p ≠ 3 is simpler as there is only one conjugacy class of Aut(L_{3}(p)) in G_{2}(p) by [18, Table 1, p.300]. In this case G_{2}(p) is complete. □

Proposition 4.2 Let p be an odd prime with (3, p + 1) = 1. Then there is a localization U_{3}(p) ↪ G_{2}(p).

Proof. The proof is similar to that of the preceding proposition. Apply also [12, Proposition 2.2]. □

Proposition 4.3 Let p be any prime. Then there is a localization ^{3}D_{4}(p) ↪ F_{4}(p).

Proof. We have Out(F_{4}(2)) = C_{2} while for an odd prime p, F_{4}(p) is complete. On the other hand Out(^{3}D_{4}(p)) = C_{3}. By [18, Proposition 7.2] there are exactly (2, p) conjugacy classes of ^{3}D_{4}(p) in F_{4}(p), fused by an automorphism if p = 2. The inclusion 3D_{4}(p) ↪ Aut(^{3}D_{4}(p)) ↪ F_{4}(p) given by [18, Table 1, p.300] is thus a localization by Theorem 1.4. □

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 M_{12} and M_{22} have C_{2} as outer-automorphism groups, while the three other Mathieu groups are complete. The inclusions M_{11} ↪ M_{23} and M_{23} ↪ M_{24} are localizations by Corollary 1.6. We have already seen in Section 3 that the inclusion M_{11} ↪ M_{12} is a localization. The inclusion M_{12} ↪ M_{24} is also a localization. Indeed Aut(M_{12}) is the stabilizer in M_{24} of a pair of dodecads, the stabilizer of a single dodecad is a copy of M_{12}. Up to conjugacy, these are the only subgroups of M_{24} isomorphic to M_{12} and thus the formula (3’) in Remark 1.5 about the number of conjugacy classes of M_{12} in M_{24} is satisfied. Similarly M_{22} ↪ M_{24} is also a localization, because Aut(M_{22}) can be identified as the stabilizer of a duad in M_{24} whereas M_{22} is the pointwise stabilizer. In short we have the following diagram, where all inclusions are localizations:

\xymatrix{M_{11} \ar@{^{(}->}[r] \ar@{^{(}->}[d] & M_{23} \ar@{^{(}->}[d] \\ M_{12} \ar@{^{(}->}[r] & M_{24} & M_{22} . \ar@{^{(}->}[l]}

We consider next the sporadic groups linked to the Conway group Co_{1}. Inside Co_{1} sits Co_{2} as stabilizer of a certain vector OA of type 2 and Co_{3} as stabilizer of another vector OB of type 3. These vectors are part of a triangle OAB and its stabilizer is the group HS, whereas its setwise stabilizer is Aut(HS) = HS.2. The Conway groups are complete, the smaller ones are maximal simple subgroups of Co_{1} and there is a unique conjugacy class of each of them in Co_{1} as indicated in the ATLAS [4, p.180]. Hence Co_{2} ↪ Co_{1} and Co_{3} ↪ Co_{1} are localizations by Corollary 1.6. Likewise the inclusions HS ↪ Co_{2} and McL ↪ Co_{3} are also localizations: They factor through their group of automorphisms, 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 Co_{3}. Finally, M_{22} ↪ HS is a localization for similar reasons, since any automorphism of M_{22} can be seen as an automorphism of the Higman-Sims graph (cf. [1, Theorem 8.7 p.273]). We get here the following diagram of localizations:

\xymatrix{M_{22} \ar@{^{(}->}[d] & McL \ar@{^{(}->}[r] & Co_{3} \ar@{^{(}->}[d] \\ HS \ar@{^{(}->}[r] & Co_{2} \ar@{^{(}->}[r] & Co_{1}}

Some other related localizations are M_{23} ↪ Co_{3}, M_{23} ↪ Co_{2} and M_{11} ↪ HS.

We move now to the Fischer groups and Janko’s group J_{4}. The inclusion T ↪ F i_{22} is a localization (both have C_{2} as outer automorphism groups) as well as M_{12} ↪ F i_{22}, and A_{10} ↪ F i_{22}. Associated to the second Fischer group, we have a chain of localizations

A_{10} ↪ S_{8}(2) ↪ F i_{23}.

By [13, Theorem 1] the inclusion A_{12} ↪ F i_{23} is also a localization. Finally M_{11} ↪ J_{4} and M_{23} ↪ J_{4} 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 we know to be localizations. We start with two examples of localizations of alternating groups: A_{12} ↪ HN, and A_{7} ↪ Suz by [22, Section 4.4].

Finally we list a few localizations of Chevalley groups: L_{2}(8) ↪ S_{6}(2), L_{2}(13) ↪ G_{2}(3), L_{2}(32) ↪ J_{4} (by [14, Proposition 5.3.1]), U_{3}(3) ↪ S_{6}(2), ^{3}D_{4}(2) ↪ T h, G_{2}(5) ↪ Ly, E_{6}(2) ↪ E_{7}(2), and E_{6}(3) ↪ E_{7}(3). The inclusion E_{6}(q) ↪ E_{7}(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) ↪ E_{8}(5) is a localization. There is one conjugacy class of Sz(32), and Out(Sz(32)) ≅ C_{5}.

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 Libman (cf. [17, Example 3.4]) that the localization A_{n} ↪ A_{n+1} extends to a localization S_{n} ↪ S_{n+1} if n ≥ 7. This result could be the starting point for determining the rigid component 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) ≅ C_{p}, where p is a prime. Then G is the only proper normal subgroup of Aut(G) and any non-trivial endomorphism of Aut(G) is either an isomorphism, or has G as kernel. □

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 H′ be a subgroup of Aut(G) isomorphic to H. The kernel of the projection Aut(G)→ →Out(G) contains H′ because H′ is simple, while Out(G) is solvable (this is the Schreier conjecture, whose proof depends on the classification of finite simple groups, see [10, Theorem 7.1.1]). □

Theorem 5.3 Let H ↪ G be a localization between two finite simple groups. Suppose that Out(H) ≅ Out(G) ≅ C_{p}, where p is a prime. Then Aut(H) ↪ Aut(G) is also a localization.

Proof. The idea is similar to the proof of Theorem 1.4. Let φ : Aut(H) → Aut(G) be any homomorphism. Let us assume that φ is not trivial. If it is an injection, the composite ϕ : H ↪ Aut(H) ⟶^{φ} Aut(G) actually lies in G by Lemma 5.2 and because H ↪ G is a localization by Theorem 1.4, there is a unique automorphism α of G making the appropriate diagram commute. Conjugation by α on Aut(G) is the unique extension we need. Indeed in the following diagram all squares are commutative and so is the top triangle:

[figure: H; i; ϕ; G; α; G; Aut(H); i; φ; Aut(G); c_{α}; Aut(G)]

Conjugation by α of an automorphism of H is an automorphism of ϕ(H). Therefore c_{α}(Aut(H)) ⊂ Aut(ϕ(H)) and thus c_{α}(Aut(H)) and φ(Aut(H)) coincide because they both are equal to Aut(ϕ(H)). The composite Aut(ϕ(H)) ⟶^{c_{α^{−1}}} Aut(H) ⟶^{ϕ} Aut(ϕ(H)) is conjugation by some automorphism β of ϕ(H) since Aut(ϕ(H)) is complete by [20, Theorem 7.14]. We have thus shown that φ = c_{β} ∘ c_{α} ∘ i. In particular ϕ is the restriction of α to H composed with β. But by construction ϕ = α |_{H} and so β has to be trivial.

By Lemma 5.1, the only other case is when ker φ = H. In that case C_{p} is a subgroup of Out(G). Thus it clearly extends to a unique endomorphism of Aut(G). □

Many examples can be directly derived from previous examples, such as S_{n} ↪ S_{n+1} and SL_{2}(p) ↪ S_{p+1}. Suzuki’s chain of groups (3.1), as well as M_{22} ↪ HS, also extend to localizations of their automorphism groups.

The converse of the above theorem is false, as shown by the following example. There exists an inclusion Aut(L_{3}(2)) ↪ S_{8} which is actually a localization (Condition (0.1) can be checked for example with the help of MAGMA). However the induced morphism L_{3}(2) ↪ A_{8} fails to be a localization: There are two conjugacy classes of subgroups of A_{8} isomorphic to L_{3}(2), which are not conjugate in S_{8}.

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 next show that the answer is affirmative if H is maximal in G. However, without this assumption G need not be simple, as illustrated by Proposition 6.5, where we show that under certain conditions a localization H ↪ G induces a localization H ↪ G̃ from H to the universal cover G̃ 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 to {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 have any non-trivial proper subgroup, so G/H ≅ C_{p} for some prime p. Then G has a subgroup of order p and there is an endomorphism of G factoring through C_{p}, whose restriction to H is trivial. This contradicts the assumption that the inclusion H ↪ G is a localization.

If N ∩ H = {1}, then either N = {1} or NH = G as H is maximal. The second case cannot occur because it would imply that G = N ⋊ H, but H ↪ N ⋊ H cannot be a localization since both the identity of G and the projection onto H extend the inclusion H ↪ G. □

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 need 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 G̃→ →G induces an exact sequence

1 → S → Aut(G̃) → Aut(G) → 1,  (6.2)

where S is the subgroup of automorphisms of G̃ which induce the identity on G. Thus, if S = 1, then Aut(G̃) ≅ Aut(G).

Lemma 6.3 Let G be a non-abelian finite simple group with Mult(G) ≅ C_{2}. Then Aut(G̃) ≅ Aut(G).

Proof. We show that any automorphism α of G̃ which induces the identity on G is itself the identity. Such an automorphism induces an automorphism on Mult(G) ≅ C_{2}. The only automorphism of C_{2} is the identity, so we have to determine the set of automorphisms of G̃ inducing the identity on both G and Mult(G). This set is in bijection with Hom(G, Mult(G)), which is trivial since G is simple and Mult(G) abelian. □

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 G̃→ →G is a localization. In particular, if Mult(G) ≅ C_{2}, we have that G̃ → G is a localization.

Proof. We have to show that G̃ → G induces a bijection Hom(G, G) ≅ Hom(G̃, G) or equivalently, Aut(G) ≅ Hom(G̃, G) \ {0}. This follows easily since the only non-trivial proper normal subgroups of G̃ are contained in its centre Mult(G). Thus any non-trivial homomorphism G̃ → G can be decomposed as the canonical projection G̃ → G followed by an automorphism of G. □

Proposition 6.5 Let i : H ↪ G be an inclusion of two finite simple groups. Suppose that the Schur multipliers of H and G have orders 1 and 2 respectively and let j : H = H̃ ↪ G̃ be the induced homomorphism. Then i : H ↪ G is a localization if and only if j : H ↪ G̃ is a localization.

Proof. Suppose that i : H ↪ G is a localization and let φ : H → G̃ be a non-trivial homomorphism. We have to show that this homomorphism extends to an automorphism of G̃. The composite H ⟶^{φ} G̃→ →G extends to a unique automorphism ψ of G, since i is a localization. Now, as S = 1 in (6.2) above, ψ can be lifted to a unique automorphism of G̃, which is the desired automorphism. The proof of the other implication is similar. □

Example 6.6 This example was communicated to us by Antonio Viruel. The inclusion M_{11} ↪ M̃_{12} of the Mathieu group M_{11} into the double cover of the Mathieu group M_{12} is a localization. This follows from the above proposition. Note that M_{11} is not maximal in M̃_{12} (the maximal subgroup is M_{11} × C_{2}), so this does not contradict Proposition 6.1. Since Mult(A_{n}) ≅ C_{2}, 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 M_{11}, M_{23}, M_{24}, J_{1}, J_{4}, Co_{2}, Co_{3}, He, F i_{23}, HN, and Ly) admit the double cover of an alternating group as localization. Here are some more examples of localizations which give rise to similar examples: Co_{2} ↪ Co_{1}, Co_{3} ↪ Co_{1}, G_{2}(2)′ ↪ J_{2}, F i_{23} ↪ B.

Remark 6.7 The latter example F i_{23} ↪ B produces a localization F i_{23} ↪ B̃ and the double cover B̃ is a maximal subgroup of the Monster M. It would be nice to know if B̃ ↪ M is a localization, which would connect the Monster to the rigid component of the alternating groups.

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José L. Rodríguez: Departamento de Geometría, Topología y Química Orgánica, Universidad de Almería, E–04120 Almería, Spain, e-mail: jlrodri@ual.es

Jérôme Scherer and Jacques Thévenaz: Institut de Mathématiques, Université de Lausanne, CH–1015 Lausanne, Switzerland,

e-mail: jerome.scherer@ima.unil.ch, jacques.thevenaz@ima.unil.ch