Invariants and Cohomology of Groups
Alejandro Ademy R. James Milgramy
Department of Mathematics Department of Applied Homotopy
University of Wisconsin StanfordUniversity
Madison, Wisc., 53076 Stanford Calif., 94305
adem@math.wisc.edu milgram@gauss.stanford.edu
This paper is dedicated to the memory of Jose Adem (1921-1991)
x1. Introduction
Let G be a finite group and p a prime numb erdividing its order. The effec*
*tive
calculation of its cohomology ring H (G;Fp) becomes difficult as soon as the p-*
*rank of G
is larger than one. There are,however, well-known local methods ([Q1],[W1]) whi*
*ch allow
the use of the cohomology rings of the normalizers of elementary abelian p-subg*
*roups of
G to determine the cohomology of G.
These methods are quite effective,reducing the question to studying the coh*
*omology
rings and cohomology maps induced from alattice of proper subgroups of G (usual*
*ly each
much smaller than G).However, the methods run into difficulties as soon as G no*
*rmalizes
one of the elementary p-subgroups. Thus,from a calculational point of view, ext*
*ensions
of the form
1! V! G! K! 1
with V = (Z=p)n are of great importance in group cohomology. In particular we h*
*ave an
induced restriction map
H (G; Fp)! H (V ; Fp)K
indicating that rings of invariants play an important part in many calculations*
*. Indeed,
existing results for the symmetric groups[AMM1] and the general linear groups [*
*Q2] rely
heavily on determining rings of invariants.
In this paper we will describe a cohomological decomposition for group exte*
*nsions
where rings of invariants play a significant role. To state it weneed to recal*
*l a few
definitions. Fora finite group K let jAp(K)j denote the geometric realization *
*of the
partially ordered set of elementaryab elian p-subgroups of K. For any i-simplex*
* oeiin this
K-complex, we denote its stablizer by Koeiand its orbit representative by [oei]*
*. We then
have
Theorem!2.2:Let
!! 1! H ! G! K! 1
be!an!extension of finite groups. Then, with Fp coefficients we have an isomorp*
*hism
! 0 1 0 1
! M M
!! H (G) @ H (ss1 (Koei))A @ H (H)KoeiA
! [oei]; i odd [oei]; i even
!! 0 10 1
!! = H (H)K @ M H (ss1 (Koei))A@ M H (H)KoeiA
!! [oei]; i even [oei];i odd
!
y Partially supported by grants from the N.S.F.
where the [oei]run over the simplexes of jAp(K)j=K. *
* ==
The point of this expression is that it measures very precisely how far H (*
*G) differs
from H (H)K ;note in particular the extreme case (jKj;p) = 1 when H (G) = H (H)*
*K .
The proof of this theorem relies on a splitting for group cohomology due to P. *
*Webb [W1]
together with some techniques from equivariant cohomology.
As particular examples we note that there are a number of extensions of the*
* form
(Z=2)n ! G! Am
where n = 3 or 4 and Am is the alternating group on m letters which are critica*
*l in studying
the cohomology of some of the sporadic groups. More precisely, there are repres*
*entations
An! GL4(F2)= A8
for n = 5; 6; 7 (two distinct representations of A5) which give rise to rings o*
*f invariants
involved in the cohomologyof the sporadic simple groups O0N, M22, M23, and McL *
*which
are critical ingredients in the determination of their cohomology rings ([AM2],*
* [AM3]).
Many of these examples are discussedin x4.
After we have proved 2.2 in x3 we give a detailed computerassisted determin*
*ation of
the rings of invariants mentioned above. Some of the invariant subrings of GL4(*
*F2) are
determined in [AM1]. Here we determine most of the remaining rings which play a*
* role in
the structure of the sporadic groups.The main result along these lines is
Theorem :
(1) There are two non-conjugate copies of A5 aeaeA8. The ring of invariants for*
* the first
is given in [AM1] and for the second itsinvariant subring in C = F2[x1; : :*
*;:x4] is
F2[v2; v3; v4; v5](1; b10).
(2) CA6 = F2[fl3; fl5; D8; D12](1; fl9; b15; fl9b15). Here, fl3= Sq1(oe2) where*
* oe2 is the sym-
metric monomial in x1; : :;:x4, fl5 =Sq2(fl3) and fl9= Sq4(fl5).
(3) CA7 = F2[D8; D12; D14; D15](1; d18; d21; d22; d23;d24; d27; d45).
This is a summary of the main results ofx3. In all cases the subscript denotes *
*the dimension
of the generator and the Diare the generators for the Dickson algebra C GL4(F2).
Remark: The ring in (1) above occurs in studying the Janko groups J2and J3. In
particular, in J2, the normalizerof one of the two conjugacy classes of involut*
*ions has the
form 21+4oA5, and the quotient semi-direct product (Z=2)4o A5,is given via this*
* action.
The other invariant subrings are needed in studying the cohomology of the Mathi*
*eu groups
M22, M23, and the group O0N .
There is every indication that similar rings of invariants play an essentia*
*l role in
other cohomology calculations,hence it is important to develop systematic metho*
*ds for
computing them. In this regard Peter Kropholler informs us that he has general *
*results
on the rings F2[x1; : :x:2n]SP2n(F2)and F2[x1; : :;:xn]On(F2). For Sp2n(F2), he*
* shows that
the ring of invariantshas the form
F2[fl3; fl5; fl9; : :;:fl2n2+1; D22n1 ; : :;:D22n2n](1; fl2n1 +1)
which explains most of the elements in part (2) of our theorem.
In x4 we apply (2.2) to decompose the mod (2)cohomology of certain sporadic*
* simple
groups explicitly exhibiting the contribution of the invariants discussed above*
*. In partic-
ular we give a complete discussion of the Mathieu group M11, replacing the much*
* longer