Non-simple localizations of finite simple groups
Jos'e L. Rodr'iguez, J'er^ome Scherer, and Antonio Viruel *
Abstract
Often a localization functor (in the category of groups) sends a finite*
* simple
group to another finite simple group. We study when such a localization al*
*so induces
a localization between the automorphism groups and between the universal c*
*entral
extensions. As a consequence we exhibit many examples of localizations of*
* finite
simple groups which are not simple.
Introduction
A group homomorphism ' : H ! G is said to be a localization if and only if ' in*
*duces a
bijection
'* : Hom (G, G) ~=Hom (H, G) (0.*
*1)
This is an ad hoc definition which comes from [Cas , Lemma 2.1]. More details o*
*n local-
izations can be found there or in the introduction of [RST ], where we exclusiv*
*ely study
localizations H ,! G, with both H and G simple groups. Due to the tight links *
*with
homotopical localizations much effort has been dedicated to analyze which algeb*
*raic prop-
erties are preserved under localization. An exhaustive survey about this proble*
*m is nicely
exposed in [Cas ] by Casacuberta. For example, if H is abelian and ' : H ! G i*
*s a
localization, then G is again abelian. Similarly, nilpotent groups of class at*
* most 2 are
preserved (see [Lib2, Theorem 3.3]), but the question remains open for arbitrar*
*y nilpotent
groups. Finiteness is not preserved, as shown by the example An ! SO(n - 1) (th*
*is is
________________________________
*The first and third authors were partially supported by EC grant HPRN-CT-19*
*99-00119 and CEC-
JA grant FQM-213, the first and second authors were partially supported by the *
*Swiss National Science
Foundation, the first author by DGIMCYT grant BFM2001-2031, and the third autho*
*r by MCYT-
FEDER grant BFM2001-1825.
1
the main result in [Lib1]). In the present paper we focus on simplicity of fini*
*te groups and
answer negatively a question posed both by Libman in [Lib2] and Casacuberta in *
*[Cas ]
about preservation of simplicity. In these papers it was also asked whether per*
*fectness is
preserved. This is not the case either, as we show with totally different metho*
*ds in [RSV ].
Our main result here is that if H ,! G is a localization with H simple then *
*G need not
be simple in general, see Corollary 1.7. There is for example a localization ma*
*p from the
Mathieu group M11to the double cover of the Mathieu group M12. This is achieved*
* by a
thorough analysis of the effect of a localization on the Schur multiplier, whic*
*h encodes the
information about the universal central extension. More precisely we prove the *
*following:
Theorem 1.5 Let i : H ,! G be an inclusion of two non-abelian finite simple gro*
*ups and
j : H~ ! G~ be the induced homomorphism on the universal central extensions. A*
*ssume
that G does not contain any non-trivial central extension of H as a subgroup. *
*Then
i : H ,! G is a localization if and only if j : ~H! ~Gis a localization.
We only consider non-abelian finite simple groups since the localization of *
*a cyclic
group of prime order is either trivial or itself ([Cas , Theorem 3.1]). Natural*
*ly the second
part of the paper deals with the effect of a localization on the outer automorp*
*hism group,
which roughly speaking is dual to the Schur multiplier as it encodes the inform*
*ation about
the üs per-groupö f all automorphisms.
Theorem 2.4 Let i : H ,! G be a localization between two non-abelian finite sim*
*ple
groups. It extends then to a monomorphism j : Aut (H) ,! Aut (G), which we ass*
*ume
induces an isomorphism Out(H) ~= Out(G). Then j : Aut(H) ,! Aut(G) is a localiz*
*ation.
The converse does not hold: There exists a localization Aut (L3(2)) ,! S8, *
*but the
induced morphism L3(2) ,! A8 fails to be one.
Acknowledgments: We would like to thank Jon Berrick and Jacques Th'evenaz f*
*or
helpful comments.
1 Preservation of simplicity
We first need to fix some notation. Let Mult(G) = H2(G; Z) ~=H2(G; Cx) denote *
*the
Schur multiplier of the finite simple group G and Mult(G) ,! G~!!G be its unive*
*rsal
2
central extension. In particular the only non-trivial endomorphisms of G~ are *
*automor-
phisms. This is due to the fact that the only proper normal subgroups of ~Gare *
*contained
in Mult(G) and Hom(G, ~G) = 0 since the universal central extension is not spli*
*t. For
more details, a good reference is [Wei , Section 6.9]. Recall also that a group*
* G is perfect
if it is equal to its commutator subgroup. Equivalently G is perfect if H1(G; Z*
*) = 0. If
moreover H2(G; Z) = 0 we say that G is superperfect. Hence for a perfect group*
* G we
have that ~G= G if and only if G is superperfect.
Is simplicity preserved under localization? We next show that the answer is *
*affirmative
if H is maximal in G. By Cp we denote a cyclic group of order p.
Proposition 1.1 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. First notice that H cannot be normal in G. Indeed if H is normal, the ma*
*ximality
of H implies that the quotient G=H does not have any non-trivial proper subgrou*
*p.
Hence G=H ~= Cp for some prime p. But then G has a subgroup of order p and the*
*re
is an endomorphism of G factoring through Cp, whose restriction to H is trivial*
*. This
contradicts the assumption that the inclusion H ,! G is a localization.
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 just showed t*
*hat
the latter case is impossible. 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 o H, but
H ,! N o H cannot be a localization since both the identity of G and the projec*
*tion
onto H extend the inclusion H ,! G . Therefore there are no normal proper non-t*
*rivial
subgroups in G. 2
We indicate next (in Corollary 1.7) a generic situation where the localizati*
*on of a
simple group can be non-simple (it will actually be the universal cover of a si*
*mple group).
To achieve this we study when a localization of finite simple groups induces a *
*localization
of the universal covers.
Proposition 1.2 Let H and G be non-abelian finite simple groups. Assume that a*
*ny ho-
momorphism between the universal central extensions ~H! ~Gsends Mult(H) to Mult*
*(G).
Then p : ~G!!G and q : ~H!!H induce an isomorphism F : Hom (H~, ~G) -'! Hom (H,*
* G).
3
Proof. First notice that p and q induce indeed a map F : Hom (H~, ~G) ! Hom (H*
*, G)
by our assumption that any homomorphism H~ ! ~Gsends the center to the center. *
*We
show now that F is surjective. Let ff : H ! G. Using the k-invariants kH : K(H,*
* 1) !
K(Mult(H), 2) and kG : K(G, 1) ! K(Mult(G), 2) classifying the universal centra*
*l ex-
tensions, construct the commutative diagram
K(H, 1) _____ff____//K(G, 1)
kH || kG||
fflffl| fflffl|
K(Mult(H), 2) H___//_K(Mult(G), 2)
2(ff)*
Taking vertical fibres gives precisely a map K(H~, 1) ! K(G~, 1) induced by som*
*e mor-
phism fi : ~H! ~Gwith F (fi) = ff. Let us show now that F is also injective by *
*indicating
an equivalent construction. Given a morphism ff : H ! G, construct the pull-bac*
*k Pffof
p along ff. Then Pff!!H is a central extension, so that there exists a unique c*
*ompatible
morphism H~ ! Pff. The composite H~ ! Pff! ~Gis hence the unique morphism whose
image under F is ff. *
* 2
Corollary 1.3 Let G be a non-abelian finite simple group and denote by p : G~!*
*!G its
universal central extension. Then we have an isomorphism F : Aut(G~) -'! Aut(G).
Proof. We have to check that any homomorphism ~G! ~Gsends the center to the cen*
*ter.
As the only morphism which is not an automorphism is the trivial one, this is a*
* clear
consequence of the fact that the image of the center is contained in the center*
* of the image.
The proposition tells us that we have an isomorphism F : Hom (G~, ~G) -'! Hom *
*(G, G),
therefore also one F : Aut(G~) -'! Aut(G). *
* 2
One should be warned that this result does not imply that an automorphism of*
* the
universal central extension always induce the identity on the center (of course*
* all inner
automorphisms do so). For example let G = L3(7) = A2(7), so ~L3(7) = SL3(7) and
Mult(L3(7)) = Z(SL3(7)) ~=Z=3 is generated by the diagonal matrix D whose coeff*
*icients
are 2's. There is an outer "graph automorphismö f order 2 given by the transpo*
*se of the
inverse. It sends a matrix A to tA-1, so the image of D is D-1.
Proposition 1.4 Let G be a finite simple group. Then, the universal cover G~!!*
*G is a
localization.
4
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 center 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
Theorem 1.5 Let i : H ,! G be an inclusion of two non-abelian finite simple g*
*roups and
j : H~ ! G~ be the induced homomorphism on the universal central extensions. A*
*ssume
that G does not contain any non-trivial central extension of H as a subgroup. *
*Then
i : H ,! G is a localization if and only if j : ~H! ~Gis a localization.
Proof. The map i : H ,! G is a localization if and only if it induces an isomor*
*phism
Hom (H, G) ~= Hom (G, G). Let us analyze the behavior of morphisms ' : H~ ! G*
*~.
By composing with q : ~G! G we get a morphism H~ ! G. As G does not contain any
subgroup isomorphic to a central extension of H, we see that '(Mult(H)) Mult(*
*G). We
deduce now by Proposition 1.2 that the universal central extensions induce isom*
*orphisms
Hom (H~, ~G) ~= Hom (H, G) as well as Hom (G~, ~G) ~= Hom (G, G). Both isomo*
*rphisms
are compatible, so i : H ,! G is a localization if and only if j : H~ ! G~ indu*
*ces an
isomorphism Hom (H~, ~G) ~=Hom (G~, ~G). *
* 2
Remark 1.6 We do not know how to remove the assumption on the centers in Prop*
*osi-
tion 1.2. There exist indeed morphisms between covers of finite simple groups w*
*hich do
not send the center to the center. One example is given in [CCN , p.34] by the*
* inclusion
A~5,! U3(5). A larger class of examples is obtained as follows: Let H be a fini*
*te simple
group of order k and H~ its universal central extension of order n = |Mult(H)| *
*. k. The
regular representations H ,! Sk and H~ ,! Sn actually lie in Ak and An because *
*the
groups are perfect. Therefore An contains both H and H~ as subgroups. However*
* we
do not know of a single example of a localization H ,! G which does not satisfy*
* this
assumption and it is rather easy to check in practice.
Question: Let i : H ,! G be a localization. Is it possible that some subgroup o*
*f G be
isomorphic to a non-trivial central extension of H? If the answer is no, we wou*
*ld get a more
general version of Theorem 1.5. This would form a perfectly dual result to Theo*
*rem 2.4
if the extra assumption that i induces an isomorphism H2(i; Z) : Mult(H) ! Mult*
*(G)
has to be used.
5
Beware that in general the induced morphism on the universal central extensi*
*ons given
by the above theorem is not an inclusion. For example L2(11) ,! U5(2) is a loca*
*lization
by the main theorem in [RST ]. However U5(2) is superperfect and the universal*
* cen-
tral extension SL2(11) of L2(11) is not a subgroup of U5(2). Nevertheless ther*
*e is a
localization SL2(11) ! U5(2). The dual situation when H is superperfect leads *
*to our
counterexamples.
Corollary 1.7 Let i : H ,! G be an inclusion of two non-abelian finite simple *
*groups
and assume that H is superperfect. Let also j : H = ~H ,! ~Gdenote the induced *
*homo-
morphism on the universal central extensions. Then i : H ,! G is a localization*
* if and
only if j : H ,! ~Gis a localization.
Proof. There are no non-trivial central extensions of H so Theorem 1.5 applies.*
* 2
Example 1.8 The inclusion M11,! M~12of the Mathieu group M11into the double c*
*over
of the Mathieu group M12is a localization. This follows from the above proposit*
*ion. Note
that M11 is not maximal in M~12(the maximal subgroup is M11x C2), so this does *
*not
contradict Proposition 1.1. The following inclusions are localizations: Co2 ,!*
* Co1 and
Co3 ,! Co1 by [RST , Section 4]. As the smaller group is superperfect we get lo*
*calizations
Co2 ,! ~Co1 and Co3 ,! ~Co1.
We get many other examples of this type using [RST , Corollary 2.2]. All sp*
*oradic
groups appearing in this corollary 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 alte*
*rnating
group as localization (as Mult(An) is cyclic of order 2 for n > 7).
Remark 1.9 The inclusion F i23 ,! B of the Fischer group into the baby monste*
*r is a
localization by [RST , Section 3 (vi)]. This yields a localization F i23,! ~B. *
*As the double
cover ~Bis a maximal subgroup of the Monster M, it would be nice to know if ~B,*
*! M is
a localization. This would connect the Monster to the rigid component of the al*
*ternating
groups (in [RST ] we were able to connect all other sporadic groups to an alter*
*nating group
by a zigzag of localizations).
2 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), similarly to the dual phenomenon observed i*
*n Theo-
6
rem 1.5. This generalizes the observation made by Libman (cf. [Lib2, Example 3.*
*4]) that
the localization An ,! An+1 extends to a localization Sn ,! Sn+1 if n 7. Thi*
*s result
could be the starting point for determining the rigid component (as defined in *
*[RST ]) of
the symmetric groups, but we will not go further in this direction.
Lemma 2.1 Let G be a non-abelian finite simple group. Then any proper norma*
*l sub-
group of Aut (G) contains G. In particular any endomorphism of Aut (G) is eith*
*er an
isomorphism, or contains G in its kernel.
Proof. Let N be a normal subgroup of Aut(G) and assume that it does not contain*
* G.
Since N \ G is a normal subgroup of G, it must be the trivial subgroup. Hence *
*the
composite N ,! Aut(G)!! Out (G) is injective. The orbit under conjugation by G *
*of an
automorphism in Aut (G) is reduced to a point if and only if the automorphism i*
*s the
identity. Thus N has to be trivial. *
* 2
Lemma 2.2 Let G be a non-abelian finite simple group. Then any non-abelian *
*simple
subgroup of Aut(G) is contained in G.
Proof. Let H be a non-abelian simple subgroup of Aut (G). The kernel G of the
projection Aut(G)!! Out (G) contains H because Out (G) is solvable (this is the*
* Schreier
conjecture, whose proof depends on the classification of finite simple groups, *
*see [GLS ,
Theorem 7.1.1]). *
* 2
Lemma 2.3 Let i : H ,! G be a localization between two non-abelian finite si*
*mple groups.
Then it extends in a unique way to a monomorphism j : Aut(H) ,! Aut(G) .
Proof. Since i is a localization, it extends to a monomorphism j : Aut(H) ,! A*
*ut(G)
by Theorem 1.4 in [RST ]. The uniqueness is given by [RST , Remark 1.3]. Indeed*
*, given
a commutative square
" i
H Ø________//G
| |
| |
fflffl|" fflffl|
Aut(H) Ø_j_//_Aut(G)
Lemma 1.2 in [RST ] implies that, for any ff 2 Aut (H), j(ff) is an automorphis*
*m of G
extending ff. But there exists a unique automorphism fi : G ! G such that fi O *
*i = i O ff
because i is a localization. *
* 2
7
Theorem 2.4 Let i : H ,! G be a localization between two non-abelian finite s*
*imple
groups. It extends then to a monomorphism j : Aut (H) ,! Aut (G), which we ass*
*ume
induces an isomorphism Out(H) ~= Out(G). Then j : Aut(H) ,! Aut(G) is a localiz*
*ation.
Proof. As i is a localization it extends to a unique inclusion j : Aut(H) ,! A*
*ut(G) by
the above lemma. Let ' : Aut(H) ! Aut(G) be any homomorphism. We have to show
that there is a unique ff : Aut(G) ! Aut(G) such that ff O i = '. If ff : Aut(G*
*) ! Aut(G)
is not an isomorphism, it factorizes through some quotient Q of Out (G) by Lemm*
*a 2.1.
The assumption that j induces an isomorphism on the outer automorphism groups i*
*mplies
then that the restriction of ff to Aut(H) is trivial if and only if ff is triv*
*ial. Therefore if
' is trivial, we conclude that the unique such ff is the trivial homomorphism.
Let us assume that ' is not trivial. If it is an injection, the image of the*
* composite
'
_ : H ,! Aut(H) -! Aut(G) actually lies in G by Lemma 2.2 and because H ,! G *
*is
a localization, there is a unique automorphism ff of G making the appropriate d*
*iagram
commute. Conjugation by ff on Aut (G)is the unique extension we need. Indeed in*
* the
following diagram all squares are commutative and so is the top triangle:
__________i__________//_
H| MMM qqG|
| MM_MMM 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)
The map _ is also a localization so that the bottom triangle commutes as well by
Lemma 2.3. If ' is not injective, then H Ker ' by Lemma 2.1. In that case the*
* image of
' in Aut(G) is some quotient of Out(H) . As j induces an isomorphism Out(H) ~= *
*Out(G),
' clearly extends to a unique endomorphism of Aut(G) . *
* 2
The assumption that j induces the isomorphism between the outer automorphism
groups cannot be dropped. In fact even when Out (H) and Out (G)are cyclic of or*
*der 2,
a localization H ,! G does not always induce one Aut (H) ,! Aut (G). For examp*
*le
i : L3(3) ,! G2(3) is a localization (see [RST , Proposition 4.2]), but Aut(L3(*
*3)) is actually
contained in G2(3). Thus j : Aut(L3(3)) ,! Aut(G2(3)) cannot be a localization *
*for the
good and simple reason that the composite Aut (L3(3)) ,! Aut (G2(3))!! Out(G2(3*
*)) ~=
8
C2 ,! Aut(G2(3)) is trivial. The same phenomenon occurs again for i : He ,! F i*
*024. Still,
many examples can be directly derived from this theorem, as it is often easy to*
* check that
j must induce an isomorphism Out(H) ~=Out(G) .
Corollary 2.5 Let i : H ,! G be a localization between two non-abelian finite *
*simple
groups. Assume that H is a maximal subgroup of G and that both Out(H) and Out(G*
*) are
cyclic groups of order p for some prime p. Then j : Aut(H) ,! Aut(G) is a local*
*ization.
Proof. We only have to show that j itself induces the isomorphism Out(H) ~=Out*
*(G) .
Because these outer automorphism groups are cyclic of prime order, the induced *
*morphism
must be either trivial or an isomorphism. Being trivial means that any automorp*
*hism of
H is sent by j to an inner automorphism of G, which can happen only if Aut (H) *
*is a
subgroup of G. 2
Directly from the corollary we deduce that Sn ,! Sn+1 and SL2(p) ,! Sp+1 are
localizations (by [RST , Proposition 2.3(i)] L2(p) ,! Ap+1 is a localization). *
* Suzuki's
chain of groups L2(7) ,! G2(2)0,! J2 ,! G2(4) ,! Suz (see [Gor , p.108-9]) also*
* extends
to localizations of their automorphism groups
Aut(L2(7)) ,! Aut(G2(2)0) ,! Aut(J2) ,! Aut(G2(4)) ,! Aut(Suz).
Remark 2.6 The converse of the above theorem is false. There exists for exam*
*ple an
inclusion Aut(L3(2)) ,! S8 which is actually a localization (Condition (0.1) ca*
*n be checked
quickly with the help of MAGMA). However the induced morphism L3(2) ,! A8 fails*
* to
be a localization: There are two conjugacy classes of subgroups of A8 isomorphi*
*c to L3(2),
which are not conjugate in S8.
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* groups
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[Lib2]Libman, A., Cardinality and nilpotency of localizations of groups and G-m*
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[RST] Rodr'iguez, J. L., Scherer, J., and Th'evenaz, J., Finite simple groups *
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Jos'e L. Rodr'iguez
Departamento de Geometr'ia, Topolog'ia y Qu'imica Org'anica, Universidad de Alm*
*er'ia,
E-04120 Almer'ia, Spain, e-mail: jlrodri@ual.es
J'er^ome Scherer
Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193 Bella*
*terra,
Spain, e-mail: jscherer@mat.uab.es
Antonio Viruel
Departamento de Geometr'ia y Topolog'ia, Universidad de M'alaga, E-04120 M'alag*
*a, Spain,
e-mail: viruel@agt.cie.uma.es
10