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
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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