Invariants and Cohomology of Groups
Alejandro Ademy R. James Milgramy
Department of Mathematics Department of Applied Homotopy
University of Wisconsin Stanford University
Madison, Wisc., 53076 Stanford Calif., 94305
adem@math.wisc.edu milgram@gauss.stanford.edu
This paper is dedicated to the memory of Jose Adem (19211991)
x1. Introduction
Let G be a finite group and p a prime number dividing 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, wellknown local methods ([Q1], [W1]) w*
*hich allow
the use of the cohomology rings of the normalizers of elementary abelian psubg*
*roups of
G to determine the cohomology of G.
These methods are quite effective, reducing the question to studying the co*
*homology
rings and cohomology maps induced from a lattice of proper subgroups of G (usua*
*lly each
much smaller than G). However, the methods run into difficulties as soon as G n*
*ormalizes
one of the elementary psubgroups. Thus, from a calculational point of view, ex*
*tensions
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 we need to reca*
*ll a few
definitions. For a finite group K let Ap(K) denote the geometric realization*
* of the
partially ordered set of elementary abelian psubgroups of K. For any isimplex*
* oei in this
Kcomplex, 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 1 0 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.
1
where the [oei] run over the simplexes of Ap(K)=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 (K; 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 cohomology of 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 discussed in x4.
After we have proved 2.2 in x3 we give a detailed computer assisted determi*
*nation 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 nonconjugate copies of A5 A8. The ring of invariants for th*
*e first
is given in [AM1] and for the second its invariant 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) whe*
*re 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 of x3. In all cases the subscript denotes*
* the dimension
of the generator and the Di are the generators for the Dickson algebra CGL4(F2).
Remark: The ring in (1) above occurs in studying the Janko groups J2 and J3. *
*In
particular, in J2, the normalizer of one of the two conjugacy classes of involu*
*tions has the
form 21+4oA5, and the quotient semidirect product (Z=2)4oA5, 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 meth*
*ods 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 invariants has 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 sporadi*
*c 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
2
work of [BC]. We also discuss some of the relations between the Mathieu group M*
*12 and
the exceptional Chevalley groups G2 and G2(q). Further details will appear in *
*[AM2],
[AM3]. Unless otherwise indicated, Fp coefficients will be assumed throughout.
This paper is dedicated to the memory of Jose Adem. He was an elegant mathe*
*mati
cian whose understanding of classical questions in algebra led to farreaching *
*contributions
in algebraic topology. As an individual he was an example of that rare combina*
*tion of
style and substance. To one of us he was a valued friend whose support when he*
* was
just starting out was crucially appreciated and whose ideas and comments were v*
*ery very
important in his early work. To the other he was a lifelong example and inspira*
*tion.
3
x2 Invariants from local methods
We consider a a fixed extension
ss
1! H! G! K! 1
and let p be a prime dividing the order of G. The image of the restriction map
(iGH)*: H*(G)! H*(H)
lies in the ring of invariants H*(H)K , but i* is neither injective nor surject*
*ive in general.
However, if (K; p) = 1 then using the transferPmap tr*: H*(H)! H*(G) togethe*
*r with
the fact that i*tr*: H*(H)! H*(H) is the sum k2K k* which is valid when H is*
* nor
mal in G we see that im(i*) is exactly H*(H)K , moreover, using the general fac*
*t that
tr*i*: H*(G)! H*(G) is just multiplication by G: H we have
H*(G) ~=H*(H)K :
The goal of this section is to see how this situation changes when Sylp(K) 6= 1*
* and to
develop a systematic method for evaluating this discrepancy.
Consider the partially ordered set Ap(K) of pelementary abelian subgroups *
*of K,
and let X = Ap(K) be the geometric realization of the associated category con*
*sisting of
objects in Ap(K) and maps (proper) inclusions. X is a finite cell complex with *
*a cellular
Kaction induced by conjugation. An nsimplex oe 2 X corresponds to a flag
F(oe) = {0} 6= (Z=p)i1( . .(.(Z=p)in+1:
Let Koedenote the stabilizer of F(oe). Then the action of K on Ap(K) extends t*
*o a
simplicial action on X and the fixing subgroup of K on the simplex oe is exactl*
*y Koe. We
recall a theorem due to P. Webb [W1]
Theorem 2.1: There exist projective FpKmodules P and Q so that
0 1 0 1
M M
Fp @ Fp[K=Koei]A P ~=K @ Fp[K=Koei]A Q
[oei]; i odd [oei]; i even
where the [oei] range over orbit representatives in (X=K)(i). *
* 
The virtual projective module P  Q is called the Steinberg module of K at *
*p as it
generalizes the usual projective module arising from a Tits building [W2].
Returning to our extension let Goe= ss1 (Koe) G. It is reasonable to ass*
*ume
that the Goeshould play a part in any cohomological splitting for H*(G) arising*
* from K;
however, the representation theoretic discrepancy in 2.1 (expressed as P  Q) g*
*ives rise to
an "error term" involving invariants. The precise result is
Theorem 2.2: Let
1! H! G! K! 1
4
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 1 0 1
~=H*(H)K @ M H*(ss1 (Koei))A @ M H*(H)KoeiA :
[oei]; i even [oei]; i odd
Proof: Let EG denote a free, contractible GCW complex with mod (p) cellular co*
*chain
complex C*(EG). H acts freely on EG so EG=H ' BH. Also BH, in this representati*
*on,
inherits a free Kaction realizing BH as a principal K cover of BG = EG=G. Ten*
*sor
formula 2.1 with C*(EG=H). This yields a corresponding Kisomorphism of cochain
complexes,
8 0 1 9
< M =
C*(EG=H) : Fp @ Fp[K=Koei]A P
[oei]; i odd ;
8 0 1 9
< M =
~=K C*(EG=H) @ Fp[K=KoeA Q :
: [oe i ;
i]; i even
Apply the functor H*(K;  ) to this equation where we regard the tensor product*
* of two
FpKmodules as a FpKmodule via the usual rule k(a b) = k(a) k(b) when k 2 K.
Note that for any subgroup L K we have
H*(K; C* Fp[K=L]) ~=H*(L; C*=L):
On the other hand we also have the following result
Lemma 2.3: For any subgroup L K, we have
H*(L; C*(EG=H)) ~=H*(ss1 (L)):
Proof of 2.3: The left hand side computes the cohomology of
ffi ffi 1
EL xL EG=H = (EL x EG=H) L ' (EG=H) L ' Bss (L):

At this point the only terms which remain to be understood are H*(K; C*(EG=*
*H)
P ) and H*(K; C*(EG=H) Q). For this step we need
Lemma 2.4: If P is a projective FpK module and C* is any FpK cochain complex, t*
*hen
H*(K; C* P ) ~=[H*(C) P ]K :
Proof of 2.4: There is a spectral sequence with
Ep;q2= Hp(K; Hq(C* P )) ) Hp+q(K; C* P ):
5
However, as P is projective we have
ae
Hp(K; Hq(C P )) ~=Hp(K; Hq(C) P ) ~= 0(Hq(C) P )K forfpo>r0,p = 0.
Thus, the spectral sequence collapses at E2 and 2.4 follows. *
* 
We return to the proof of 2.2. Take 2.1, tensor with H*(H), and apply Kinv*
*ariants
to it. From the discussion above we obtain
0 1
M K
H*(H)K @ H*(H)KoeiA (H*(H) P )
[oei]; i odd
0 1
~=@ M H*(H)KoeiA (H*(H) Q) K ;
[oei]; i even
or, stated in virtual terms,
iX j
(H*(H) St(K)) K ~= (1)iH*(H)Koei  H*(H)K :
When we substitute this expression 2.2 follows. *
* 
Remark: Instead of Ap(K) we could have used any other Kcomplex for which 2.1
holds. These include Sp(K) (the poset of nontrivial psubgroups) or the Tits*
* buildings
if K is of Chevalley type, and for M22 the results of [RSY] give another lattic*
*e of subgroups
distinct from these which also satisfy 2.1. See Example 4.4 for further details.
We now give several examples where 2.2 is useful, starting with familiar ex*
*tensions.
The following notation will be useful from here on: we set Vn = (Z=2)n and we r*
*egard Vn
as the ndimensional vector space over the field F2.
Example 2.5: The symmetric group on 4 letters, 4, is given as the extension
1! K (~=Z=2 x Z=2)! 4! SL2(F2) (~=3)! 1
where K is the Klein group. The poset space A2(3) consists of three copies of*
* Z=2 so
X(3) consists of a single point with isotropy group Z=2, and we have
H*(4) H*(V2)Z=2 ~=H*(V2)SL2(F2) H*(D8):
Now, H*(Vn)n = F2[x1; : :;:xn]n = F2[w1; w2; : :;:wn], where wi denotes the i*
*th sym
metric power of the monomials xi, the well known symmetric algebra, and
F2[x1; : :;:xn]GLn(F2)= F2[D2n1; D2n2n2; : :;:D2n1];
the well known Dickson algebra. Here the D2n2i are the Dickson invariants. C*
*onse
quently, passing to Poincare series we have
*
* 2 + t3
P4 = ______1_______(1+_t2)(11_t3)(1_t)2__1______(1=_t)(11+t2)t_*
*+_t(1: t2)(1  t3)
6
Remark: Since H*(D8) is detected by abelian subgroups the same is true of H*(4)
and we have that H*(4) is CohenMacaulay, i.e., freely and finitely generated o*
*ver a
polynomial subalgebra, in this case the subalgebra of elements which restrict t*
*o the Dickson
algebras F2[D2; D3] in both nonconjugate copies of (Z=2)2 4.
Example 2.6: Example 2.5 generalizes to any extension of the form
1! V2n! G! SL2(F2n)! 1
by using the Tits building for SL2(F2n). Once more X consists only of isolated*
* points
all in the same orbit and stabilizer the Borel group which, in this case, is gi*
*ven as the
semidirect product
B ~=(Z=2)n o (Z=2n  1):
Hence 2.2 gives
n1
H*(G) H*(V2n)B ~=H*(V2n)SL2(F2n) H*(Syl2(G))Z=2 :
From the geometry of the constructions above it is direct to see that this expr*
*ession is
more than merely a formal isomorphism. It corresponds to an exact sequence
n1 * B
0! H*(G)! H*(V2n)SL2(F2n) H*(Syl2(G))Z=2 ! H (V2n) !0
which identifies H*(G) as those elements 2 H*(Syl2(G))Z=2n1 which satisfy the*
* con
dition
(iSyl2(G)V2n)*() 2 H*(V2n)SL2(F2n):
Indeed, we could have stated Theorem 2.2 in terms of a split long exact seq*
*uence
of the type above. This follows from the version of 2.1 which is proved in [W3*
*]. In
practice, however, we use 2.2 for Poincare series calculations, usually preferr*
*ing double
coset decompositions to determine the ring structure of the resulting cohomolog*
*y groups.
In the special case SL2(F4) ~=A5, the invariants H*(V4)A5 are studied in [A*
*M1] as
they play an important role in the calculation of H*(L3(4)) and the cohomology *
*of the
sporadic simple groups J2, J3, M22, and O0N as discussed in the introduction.
Example 2.7: There is a nonsplit extension of the form
1! (Z=2)3! E! GL3(F2)! 1
where the action of GL3(F2) on (Z=2)3 is the usual one. Indeed, Alperin [A] has*
* given a
complete discussion of extensions of the form (Z=2r)3 . GL3(F2) where the actio*
*n is the
usual one when restricted to (Z=2)3 (Z=2r)3, and has shown that there are exac*
*tly two,
the first which is split and one other which is nonsplit. The group E is the b*
*asic nonsplit
example and isparticularly interesting because Syl2(E) ~=Syl2(M12), (compare 4.*
*2).
Using the Tits building for GL3(F2) with parabolics P1 ~= 4, P2 ~= 4 and B =
P1 \ P2 ~=D8 we obtain
H*(E) H*(Syl2(E)) H*(V3)P1 H*(V3)P2
~=H*(V3)GL3(F2) H*(ss1 (P1)) H*(ss1 (P2)) H*(V3)D8:
7
Example 2.8: Consider the group A6 ( Sp4(F2) ~=6. The resulting action of A6 on*
* V4
allows us to define a semidirect product
G = V4 o A6
which is a 2local subgroup of the third Mathieu group M22. In this case 2.1 r*
*educes
modulo projectives to
F2 F2[A6=D8] ~=A6F2[A6=Q1] F2[A6=Q2]
where Q1 ~=Q2 ~=4 are the two distinct copies of 4 contained in A6. We have
H*(G) H*(Syl2(G)) H*(V4)Q1 H*(V4)Q2
~=H*(V4)A6 H*(V4 o Q1) H*(V4 o Q2) H*(V4)D8:
The invariants H*(V4)A6 will be determined in x4.
8
x3. The invariant subrings for subgroups of GL4(F2)
The classical isomorphism of GL4(2) with A8 is given explicitly in [D, pp.2*
*90292] by
setting
0 1 1 1 1 1 0 0 1 0 1 1 0 1
0 1 1 1
E1 = B@01 01 00 10CA; E2 = B@00 01 10 00CA; E3 = @ 0 1 0 1 A ;
0 1 0 1 1 0 1 0 1 1 0 0
0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 1 1 1
3:1 E4 = B@00 10 01 00CA; E5 = B@01 10 00 10CA; E6 = B@00 01 10 00CA;
0 1 0 1 0 0 0 1 1 1 1 0
verifying the relations
E31= E2i+1= (EiEi+1)3 = (EiEj)2 = 1; (i; j = 1; : :;:6; j > i + 1);
and setting up the correspondence
E1 ~ (123); E2 ~ (12)(34); E3 ~ (12)(45):
If we make a change of basis
0 01 0 01 0 11 0 01
e1 = B@01CA; e2 = B@10CA; e3 = B@11CA; e4 = B@11CA;
1 1 0 1
then the first four matrices take the simpler form
0 0 1 0 0 1 0 1 0 0 0 1
E1 ~ B@10 10 01 01CA; E2 ~ B@01 10 01 00CA;
0 0 1 0 0 1 0 1
0 0 1 1 0 1 0 0 1 0 0 1
E3 ~ B@10 00 00 11CA; E4 ~ B@11 00 00 01CA:
0 0 1 0 0 1 1 0
Consequently, if we set i = 01 11 , J = 01 10 , I = 10 01 , we obtain
E1 ~ i0 0i2 ; E2 ~ II 0I ; E3 ~ J0 IJ ; E4 ~ JI 0J :
9
It follows that E3E4E3 = 0 I and the three matrices (E ; E ; E E E ) are ou*
*r usual
I 0 1 2 3 4 3
generators for SL2(F4) = A5. Finally adding E4 gives generators for the commut*
*ator
subgroup of the symplectic group Sp4(F2) ~=6, Sp4(F2)0= A6.
We studied certain of these subgroups in [AM1]. In particular we studied th*
*e first 4
and SL2(F4) there. The essential step was to use cohomology with F4 as coeffici*
*ent ring
so that the elements of order three could be diagonalized. In particular the ma*
*in process
was as follows. Set A = i3a + i23b, B = i23a + i3b, C = i3c + i23d, D = i23c + *
*i3d where
a; b; c; d 2 Hom(F42; F2) are dual to e1, e2, e3, e4 respectively. Then the ac*
*tion of our
generators above (or more accurately their duals) is given by
E*1(A) = i3A; E*1(B) = i23B; E*1(C) = i23C;
E*2(A) = A; E*2(B) = B; E*2(C) = A + C; E*2(D) = B + D;
and setting o = E3E4E3, then
o*(A) = C; o*(B) = D;
Finally, in terms of this basis the action of the new generator is given by the*
* formula
E*4(A) = B; E*4(B) = A;
Incidently, if we set E12 = 10 11 then we see that with respect to the new ba*
*sis,
E5 ~ E12;I E0 ; E6 ~ J 0 :
12 0 J
Note that o = E3E4E3 ~ (12)(46) under the correspondence with A8 so the span
~= 5. This copy of 5 is obtained by extending the action of SL*
*2(F4)
via the Galois automorphism of F4. It also embeds in the full symplectic group *
*Sp4(F2).
The 4 subgroups of A6
There are two nonconjugate copies of 4 A6, the respective normalizers of *
*the two
nonconjugate copies of (Z=2)2 there.
The first, generated by (E1; E2; E4) is characterized by the fact that the *
*element E1
of order 3 has no fixed nonzero vectors in (Z=2)4. The second is generated by
S = E1E3E4 ~ (123)(465)
T = E2E3E21E2E1E3E2 ~ (15)(24)
E3 ~ (12)(45)
For this subgroup Dim(((Z=2)4)S) = 2, and defining new coordinates by
0 01 0 01 0 11 0 01
A = B@10CA; B = B@10CA C = B@00CA D = B@01CA
1 1 1 0
10
the original matrices become
0 0 0 1 01 0 0 0 0 11
S ~ B@10 01 00 00CA T ~ B@00 01 10 00CA
0 0 0 1 1 0 0 0
0 0 1 1 11 0 1 1 1 11 0 1 0 0 01
E3 ~ B@11 10 01 11CA= B@11 11 11 11CA+ B@00 01 10 00CA:
1 1 1 0 1 1 1 1 0 0 0 1
The invariant subring for the first 4 is discussed in [AM1], see in particu*
*lar 4.10, so we
don't discuss it further here. We now describe the invariants of the second 4 =*
* .
We set wi equal to the ith symmetric monomial in the variables . N*
*ote that,
while the wi, 1 i 4 are not invariants of this 4 action, they are invariants *
*for the
action of the subgroup A4, since the change of basis above shows that A4 = acts
by the usual permutation of coordinates.
The following result is contained in [H].
Lemma 3.2: Let An act on by permuting coordinates in the usual ma*
*nner,
then
F2[x1; : :;:xn]An = F2[w1; : :;:wn](1; an(n1)=2)
P n1
where an(n1)=2= ff2Anff(x1x22x33: :x:n1).
Now we are able to state
Theorem 3.3: Let w3 = w3 + w1w2, fl4 = w2(w2 + w21), fl8 = w4(w4 + w3w1), 5 =
Sq2(w3 ) = w1w4 + (w2 + w21)w3 , and b6 = a6 + w2(fl4 + w41+ w4) + w1w2w3 . Fin*
*ally, set
b7 = w1b6 + w4w3 . Then
F2[x1; x2; x3; x4]4 = F2[w1; w3; fl4; fl8](1; 5; b7; 5b7)
where the action of 4 is determined by the matrices for S; T; E3 above.
Proof: We begin by noting that E3 normalizes A4 so that it acts on
F2[w1; w2; w3; w4](1; a6);
and the desired invariants are the elements in this ring which are fixed under *
*E3. We have
Lemma 3.4:
E*3(w1) = w1
E*3(w2) = w2 + w21
E*3(w3 )= w3
E*3(w4) = w4 + w1w3
E*3(a6) = a6 + w21(fl4 + w41+ w4) + w3w3 :
Proof of 3.4: Since E3 = E*3we have E*3(xi) = xj(i)+ w1 for appropriate j(i). C*
*onse
quently E3(w1) = 5w1 = w1. Similarly E3(xixj) = xaxb+ (xa + xb)w1+ w21, and sum*
*ming
11
over all pairs i; j we get all pairs a; b. Similarly E3(w3) = w3 + w31, and the*
* above result
follows for w4.
The result for a6 was done using a computer. *
* 
Corollary 3.5: We have that
(E*3+ 1)a6 + w2(fl4 + w41+ w4 + w1w3 ) = w23:

We now return to the proof of 3.3.
We can write F2[x1; : :;:x4]A4 = F2[w1; : :;:w4](1; b6) and
F2[w1; w2; w3; w4] = F2[w1; w2; w3; w4] = F2[w1; w3; fl4; fl8](1; w2; w4*
*; w2w4):
We set S = F2[w1; w3; fl4; fl8] and
S1 = S(1; w2; w4);
which is closed under the action of E3. There is an exact sequence of E3 modules
S! S1! S{w2} S{w4};
which leads to the long exact sequence
ffi
S! SE31!S{w2} S{w4}! H1(Z=2; S) = S! H1(Z=2; S1)! . .:.
Clearly ffi({w2}) = w21while ffi({w4}) = w1w3 so Ker(ffi) = S(5). In the H1 par*
*t of this
exact sequence we note for future use that the exact sequence becomes
0! S=(w21; w1w3 )! H1(Z=2; S1)! S5! 0:
Next, set S2 = S(1; w2; w4; w2w4) = F2[w1; w2; w3; w4]. We have an exact sequen*
*ce
S1! S2! S{w2w4}
which gives us the long exact sequence
ffi
SE31!SE32!S{w2w4}! H1(Z=2; S1)! . .:.
Moreover, an easy check gives that ffi({w2w4}) = w15, so ffi is an injection an*
*d SE31is the
entire invariant subring. This also gives us H1(Z=2; S2).
Finally, we add b6. We have the exact sequence
0! S2! S2(1; b6)! S2{b6}! 0;
and passing to cohomology we get the long exact sequence
ffi
SE32!S2(1; b6)E3! SE32{b6}! H1(Z=2; S2)! . .:.
12
Here the corollary above shows that ffi({b6}) = w23, and our calculation of H1(*
*S2; Z=2)
shows that Ker(ffi) = SE32w1{b6}. But the class w1b6 + w4w3 is an actual E3inv*
*ariant, so
it becomes b7 and we have proved the result. *
* 
The two A5's
There are two copies of A5 in A6. The first, given as , is *
*discussed
in [AM1] and its invariant subring is determined there. The other is studied as*
* follows.
First, from the Atlas, [C], it is direct to check that there are only two nonc*
*onjugate copies
of A5 A8, and consequently the second group may be realized from the usual act*
*ion
of A5 on F2[x1; : :;:x5] by projection onto the 5 invariant subring generated b*
*y the four
elements r1 = x1 + x2, r2 = x1 + x3, r3 = x1 + x4, and r4 = x1 + x5. The remai*
*ning
element in this basis for is x1 + x2 + x3 + x4 + x5 = w1, and rew*
*riting
F2[x1; : :;:x5]A5 = F2[w1] F2[r1; : :;:r4]A5:
Consequently, we can apply Hewett's result so
F2[r1; : :;:r4]A5 ~=F2[v2; v3; v4; v5](1; b10)
and arguing similarly,
F2[r1; : :;:r4]5 ~=F2[v2; v3; v4; v5]
where we can easily determine the explicit forms of the vi.
The A6 invariant subring
The Dickson element D8 is given as follows in terms of the above generators*
* for
F2[x1; : :;:x4]4 .
D8 = fl8 + fl24+ w81+ w21w23:
Also, the Dickson element D12 = Sq4(D8) has the following representation
r12 D12 = (D8 + (w1w3 + fl4 + w41)2)(fl4 + w1w3 + w41) + (w15 + w21fl4 + w23*
*)2:
Finally, there is the relation
r10 25= w21w35 + w21D8 + w23(w41+ fl4) + fl24w21+ w101:
Lemma 3.6: The class w3 satisfies E*1(w3 ) = w3 so w3 2 F2[x1; : :;:x4]A6. *
*Also, w3 ,
Sq2(w3 ) = 5, D8, and D12 are transcendentally independent so F2[x1; : :;:x4]A6*
* contains
the polynomial algebra
F2[w3 ; 5; D8; D12]:
Proof: w3 = S(x21x2) gives a representation of w3 as a symmetric sum. Now, E**
*1is
determined by the formula
x1 7! (x2 + x4)
x2 7! (x3 + x4)
x3 7! (x1 + x4)
x4 7! (x1 + x2 + x3)
13
We can write S(x21x2) = x21(x2 + x3 + x4) + x1(x2 + x3 + x4)2 + x22(x3 + x4) + *
*x2(x3 +
x4)2 + x23x4 + x3x24. Consequently
E*1(w3 ) =(x2 + x4)2x2 + (x2 + x4)x22
+ (x3 + x4)2(x2 + x3 + x4) + (x3 + x4)(x22+ x23+ x24)
+ (x1 + x4)2(x1 + x2 + x3) + (x1 + x4)(x21+ x22+ x23);
and one sees directly that this is again S(x21x2). The transcendental independe*
*nce of the
generators above is evident by inspection. *
* 
Corollary 3.7: The 9 dimensional class 9 = Sq4(5) is also invariant under A6.
The following result is our main technique in determining F2[x1; : :;:x4]A6*
* from
F2[x1; : :;:x4]4 .
Lemma 3.8: There exists an explicit projection operator
e: F2[x1; : :;:x4]4 ! F2[x1; : :;:x4]A6;
i.e., e2 = id restricted`to the A6 invariant subring, and im(e) F2[x1; : :;:x4*
*]A6.
Proof: We write A6 = 151vi4 for an explicit coset decomposition of A6. Then w*
*e set
15X
e = vi:
1
Clearly, if ff 2 F2[x1; : :;:x4]A6 we have e(ff) = 15ff = ff, while for ff 2 F2*
*[x1; : :;:x4]4
and g 2 A6 we have gvi = vg(i)si with si 2 4 so gvi(ff) = vg(i)(ff) and g(e(ff)*
*) = e(ff). 
Remark: The situation in 3.8 occurs for any subgroup H A6, as long as H contai*
*ns a
Sylow 2subgroup of A6.
Using 3.8 we obtain
Theorem 3.9: The ring of invariants F2[x1; : :;:x4]A6 is
F2[w3 ; 5; D8; D12](1; 9; b15; 9b15)
where b15 = e(fl24b7).
Proof: The major step in the proof is to reduce the determination of the A6 inv*
*ariants
to a finite calculation. This is a direct consequence of
Lemma 3.10: F2[x1; : :;:x4]4 is freely generated over the A6invariant polynom*
*ial sub
ring,
B = F2[w3 ; 5; D8; D12]
on the 60 generators 8
>><1; w1; w21; w31; : :;:w91
2fl ; : :;:w9fl
(1; b7) > fl4;2w1fl4;2w1 4 5 2 1 4
>:fl4; w1fl4; : :;:w1fl4
w21fl34; w31fl34; w41fl34; w51fl34:
Proof of 3.10: We have that the Poincare series for the free Bmodule on 60 gen*
*erators
having the dimensions of the generators above is
____(1_+_x7)(1_+_x4_+_x8)(1__x10)___= _______(1_+_x5)(1_+_x7)_____
(1  x)(1  x3)(1  x5)(1  x8)(1  x12)(1  x)(1  x3)(1  x4)(1  x8)
14
which is the Poincare series for F2[x1; : :;:x4]4 . Hence, if we can show that*
* the B
submodule of F2[x1; : :;:x4]4 generated by the 60 elements above is the entire*
* invariant
ring the lemma will follow.
Let A = F2[w1; w3; fl4; D8](1; 5) F2[x1; : :;:x4]4 . We now wish to deter*
*mine
T orB0(A; F2) which determines a generating set for A over B.
Let R = F2[w1; w3; fl4; 5; D8; D12] be the polynomial algebra on (formal) g*
*enerators.
There is an obvious surjective map R! A and the kernel is the ideal generated *
*by the two
elements
R12 = D12+ (D8 + (w1w3 + fl4 + w41)2)(fl4 + w3w1 + w41) + (w15 + w21fl4 + w*
*23)2;
and
R10 = 25+ w21w35 + w101+ (w41+ fl4)w23+ (D8 + fl24)w21:
Consequently we obtain a resolution of A over R,
@ @
3:11 0! Rs22! R(r10; r12)! R! A! 0
where @(r10) = R10, @(r12) = R12, and @(s22) = R12r10+ R10r12.
Since R is free over B the resolution 3.11 is also a resolution of A over B*
*. Moreover
R=B = F2[w1; fl4]
so the complex for computing T orB*(A; F2) becomes
@ @
F2[w1; fl4]s22! F2[w1; fl4](r10; r12)! F2[w1; fl4]! T orB0(A; F2)*
*! 0:
Here @(r10) = w101+ fl24w21and @(r12) = (fl4 + w41)3 + w41fl24. Now a direct ca*
*lculation gives
that T orB0(A; F2) has 30 generators,
8
>><1; w1; w21; w31; : :;:w91
2fl ; : :;:w9fl
T orB0(A; F2) = > fl4;2w1fl4;2w1 4 5 2 1 4
>:fl4; w1fl4; : :;:w1fl4
w21fl34; w31fl34; w41fl34; w51fl34
and it follows that F2[x1; : :;:x4]4 = A(1; b6) is free over B on the sixty ge*
*nerators asserted
in 3.10. *
* 
The proof of 3.9 is now a direct computer computation. The projection opera*
*tor e is
evaluated, in turn, on each of the sixty generators above. Note that B and thes*
*e images
certainly generate F2[x1; : :;:x4]A6, so the final determination of the ring be*
*came simply
a matter of identifying the independent generators among these images. Again th*
*is was
done using the computer. *
* 
The A7 invariant subring
The argument for A7 is similar to that for A6. We start with the Dickson a*
*lgebra
D = F2[D8; D12; D14; D15] = F2[x1; : :;:x4]GL2(4)and show to begin that F2[x1; *
*: :;:x4]A6
15
is freely and finitely generated over it on 56 generators. Then we use a projec*
*tor similar
to the e of the previous section,
f: F2[x1; x2; x3; x4]A6! F2[x1; x2; x3; x4]A7
on each of the 56 generators above to obtain a generating set for F4[x1; : :;:x*
*4]A7.
Lemma 3.12: There exists an explicit projection operator
f: F2[x1; : :;:x4]A6! F2[x1; : :;:x4]A7;
i.e,. f2 = id when restricted to the A7 invariant subring and im(f) F2[x1; : :*
*;:x4]A7.P
(The proof is exactly like that of 3.8. Once more A7 : A6 = 7 is odd and so *
* 71wi = f
where the wi are a set of coset generators for A7 over A6.)
Theorem 3.13: F2[x1; : :;:x4]A7 is freely and finitely generated over the Dicks*
*on algebra
D on eight generators, {1}, f(w339) in dimension 18, f(w535) in dimension 20, f*
*(w73) in
dimension 21, f(w539) in dimension 24, f(w5325) in dimension 25, f(w639) in dim*
*ension
27, and f(w73915) in dimension 45. In particular it has Poincare series
1_+_x18+_x20+_x21+_x24+_x25+_x27+_x45__ :
(1  x8)(1  x12)(1  x14)(1  x15)
Proof: The proof follows that of the previous theorem on the A6 invariant subri*
*ng quite
closely. We begin with
Lemma 3.14: The subring of F2[x1; : :;:x4]A6 given as B = F2[w3 ; 5; D8; D12](*
*1; 9)
contains the Dickson algebra D and is freely and finitely generated over D on t*
*he 28
generators 8
>><1; w3; w23; w33; w43; w53; w63; w73;
9; 9w3 ; 9w23; 9w33; 9w43; 9w53; 9w63; 9w73;
>>:5; 5w3 ; 5w23; 5w33; 5w43; 5w53;
25; 25w3; 25w23; 25w33; 25w43; 25w53:
Consequently F2[x1; : :;:x4]A6 is free and finitely generated over D on the 56 *
*generators
consisting of the 28 generators above and their products with 15.
Proof of 3.14: One checks directly that we have
D14 = 59 + w23D8 + w335;
D15 = 35+ w239 + w53:
This shows that D B.
Let C = F2[w3 ; 5; 9; D8; D12; D14; D15] be the polynomial algebra on 7 gen*
*erators in
the stated dimensions. There is a surjection C! B taking the generators to the*
*ir images
with the same names. To determine a resolution of B over C note the three relat*
*ions
R18: 29+ D825+ w63+ 35w3+ 9w33+ D12w23
R15: D15+ w239 + 35+ w53
R14: D14+ 59 + w23D8 + w335:
16
We clearly obtain a resolution of B over C as
0! C(s47)! C(s29; s32; s33)! C(s14; s15; s18)! C! B! 0
where @(s14) = R14, @(s15) = R15, @(s18) = R18, and so on. Consequently, since *
*C is free
over the Dickson algebra D we obtain a resolution of B over D as follows. Set
E = F2[w3 ; 5; 9] C;
then a chain complex for determining T orD*(B; F2) is given as
E(s47)! E(s29; s32; s33)! E(s14; s15; s18)! E! T orD0(B; F2)*
*! 0
where @(s14) = 59+w335, @(s15) = w239+35+w53, and @(s18) = 29+35w3+w339+w63,
and we have
ffi 3 2 3 5 2 3 3 6
T orD0(B; F2) = F2[w3 ; 5; 9] (59 + w35; w3 9 + 5 + w3; 9 + 5w3 + w39 + w3):
Within the ideal which is being factored out note the following. First the rela*
*tion for 29
has the form 29+ w3r15 where r15 is the second relation above. Thus 292 the rel*
*ation
set. Similarly, 5r15 = 45+w23r14 so 45is in the relation set. Also, 9r14 = w335*
*9+529
so w3359 is in the relation set, and expanding out 5r18 gives that w635 = 0. Al*
*so, note
that r14 implies that w3335= 359 and r15 implies that w53+ w239 + 35. Consequen*
*tly
w33r15 = w539 + 359 + w83= (w239 + 35)9 + 359 + w83= w83
and w83is in the relation set. Thus, T orD0(B; F2) can be no larger than the se*
*t asserted in
3.14.
We now show that it also cannot be smaller. Note that the Poincare series *
*for the
free module over D on generators in the stated dimensions above is
1x24_ 9 1x18_ 5 10
__1x3(1_+_x_)_+__1x3(x__+_x__)_
(1  x8)(1  x12)(1  x14)(1  x15)
but this is
_____(1_+_x9)(1__x14)(1_+_x5_+_x10)___= ____________1_+_x9____________
(1  x3)(1  x8)(1  x12)(1  x14)(1  x15)(1  x3)(1  x5)(1  x8)(1  x1*
*2)
and 3.14 follows. *
* 
The next step in the proof is contained in
Lemma 3.15: F2[x1; : :;:x4]A7 is freely and finitely generated over D on eight *
*generators.
Proof of 3.15: An easy induction using 3.14 and the projector f shows that the *
*ring
of invariants F2[x1; : :;:x4]A7 is freely and finitely generated over D. It rem*
*ains to show
that the number of generators is eight. To see this we pass to quotient fields*
*. We have
that the degree of F2(x1; : :;:x4)A7 over F2(x1; : :;:x4)GL4(2)is eight by the *
*fundamental
17
theorem of Galois theory. On the other hand F2[x1; : :;:x4]GL4(2)= D and thus *
*QD =
F2(x1; : :;:x4)GL4(2)so
Q(F2[x1; : :;:x4]A7) = QD (w1; : :;:w8)
where w1; : :;:w8 are the eight generators of F2[x1; : :;:x4]A7 over D. *
* 
For the final step the operator f was evaluated on the 28 generators in the*
* first lemma
above using a computer, and image classes in dimensions 18, 20, 21, 24, 25, and*
* 27 were
found. The resulting table is given as follows where Sxa1xb2xc3xd4denotes the *
*symmetric
sum
The class f(w33fl9):Sx81x62x23x24+ Sx91x52x23x24+ Sx91x62x33+ Sx81x42x43x*
*24
+ Sx91x52x43+ Sx101x62x13x14+ Sx101x42x23x24+ Sx101x62x*
*23
+ Sx101x42x33x14+ Sx91x62x23x14+ Sx111x42x23x14+ Sx81x4*
*2x33x34
+ Sx101x52x33+ Sx121x42x23+ Sx91x42x33x24+ Sx81x72x23x14
+ Sx121x42x13x14+ Sx121x22x23x24+ Sx91x42x43x14
The class f(w53fl5):Sx121x42x33x14+ Sx81x62x43x24+ Sx101x82x23+ Sx121x42x*
*23x24
+ Sx91x62x43x14+ Sx81x82x33x14+ Sx91x82x33+ Sx101x42x43*
*x24
+ Sx121x62x23+ Sx81x52x53x24+ Sx121x52x33+ Sx91x52x43x24
+ Sx101x82x13x14+ Sx131x42x23x14+ Sx121x62x13x14+ Sx81x*
*72x43x14
+ Sx91x62x53+ Sx91x42x43x34
The class f(w73):Sx121x62x33+ Sx101x62x53+ Sx141x42x23x14+ Sx121x42x33x24
+ Sx81x62x53x24+ Sx101x52x43x24+ Sx81x52x43x44+ Sx121x52*
*x43
+ Sx81x62x63x14+ Sx91x82x23x24+ Sx81x72x43x24+ Sx101x82x*
*23x14
+ Sx91x42x43x44+ Sx101x42x43x34+ Sx121x52x23x24+ Sx81x82*
*x33x24
+ Sx101x62x43x14+ Sx121x42x43x14+ Sx91x82x43+ Sx81x82x43*
*x14
+ Sx121x62x23x14+ Sx101x82x33
The class f(w53fl9):Sx161x42x23x24+ Sx101x82x43x24+ Sx101x82x53x14+ Sx161*
*x52x33
+ Sx121x82x33x14+ Sx161x62x23+ Sx101x92x53+ Sx91x82x53x*
*24
+ Sx81x82x63x24+ Sx111x82x43x14+ Sx121x120x23+ Sx121x82*
*x23x24
+ Sx161x62x13x14+ Sx131x82x23x14+ Sx121x92x33+ Sx161x42*
*x33x14
+ Sx171x42x23x14+ Sx91x92x43x24+ Sx81x82x53x34+ Sx121x1*
*20x13x14
The class f(w53fl25):Sx81x82x53x44+ Sx181x42x23x14+ Sx161x42x43x14+ Sx91x8*
*2x63x24
+ Sx121x120x33+ Sx161x52x43+ Sx101x92x63+ Sx161x52x23x24
+ Sx121x120x23x14+ Sx121x92x43+ Sx101x92x43x24+ Sx141x82*
*x23x14
+ Sx81x82x63x34+ Sx101x82x63x14+ Sx101x120x43x14+ Sx161x*
*62x23x14
+ Sx161x42x33x24+ Sx121x82x33x24+ Sx91x82x43x44+ Sx121x9*
*2x23x24
+ Sx161x62x33+ Sx111x82x43x24
18
The class f(w63fl9):Sx131x82x43x24+ Sx161x42x43x34+ Sx161x62x53+ Sx141x82x*
*43x14
+ Sx161x62x43x14+ Sx101x82x83x14+ Sx101x92x43x44+ Sx91x8*
*2x63x44
+ Sx121x82x63x14+ Sx161x52x43x24+ Sx81x82x83x34+ Sx121x1*
*22x23x14
+ Sx91x82x83x24+ Sx121x120x53+ Sx81x82x63x54+ Sx101x92x83
+ Sx101x82x53x44+ Sx161x82x33+ Sx121x82x53x24+ Sx121x92x*
*63
+ Sx201x42x23x14
These classes are easily checked and seen to be independent over D. Thus these*
* seven
classes, together with {1} and one other class freely generate the A7 invariant*
* subring
over D. To find the last class note that the numerator in the Poincare series f*
*or a Cohen
Macaulay ring satisfies a symmetry condition of the form, the coefficient of xn*
* is equal to
the coefficient of xln for some fixed l. Here, the only possibility for l is 4*
*5, and the result
follows. *
* 
19
x4 The cohomology of some sporadic simple groups
In this section we will provide explicit applications of our invariant calc*
*ulations to
determine the mod (2)cohomology of certain important simple groups. We begin wi*
*th a
"small" group.
Example 4.1: Set G = M11, the first Mathieu group having order 11 x 10 x 9 x 8*
* =
7920 = 24 32 5 11. M11 has 2rank two with one conjugacy class of groups (Z=2)2*
* and one
conjugacy class of involutions. From the Atlas, [C], N(2A) = 2 . 4 = GL2(F3), a*
*nd we
can also check that N((Z=2)2) = 4. Thus the quotient A2(M11=M11 has the form
4 .................................................*
*.........................................................................@
*
* @
Apply 2.1 to obtain the formula
H*(M11) H*(D8) = H*(GL2(3)) H*(4)
and substitute for H*(4) using the formula in 2.5 to get
H*(M11) H*(V2)Z=2 ~=H*(GL2(3)) H*(V2)GL2(F2):
From Quillen's results, [Q2], the Poincare series for H*(GL2(F3)) is
_(1_+_t)(1_+_t3)_= 1_+_t_+_t2_+_t3_+_t4_+ t5
(1  t2)(1  t4) (1  t3)(1  t4)
and so the Poincare series for M11 is (as first computed in [We1], compare [BC])
(1_+_t_+_t2_+_t3_+_t4_+_t5)_+_(1_+_t2)__(1_+_t2)(1_+=t_+_t3)1_+_t5__:
(1  t3)(1  t4) (1  t3)(1  t4)
The group GL2(F3) contains a Sylow 2subgroup of M11 and has its mod (2)cohomol*
*ogy
detected on its elementary 2subgroups. Consequently the same is true for M11. *
*It follows
that
H*(M11; F2) F2[x1; x2]GL2(F2)= F2[D2; D3];
and since there is only one element in each of the dimensions 3, 4, and 5 in th*
*is ring we have
an independent proof of the result that H*(M11) ~=F2[D3; D22](1; D2D3). In par*
*ticular
note that H*(M11) is CohenMacaulay.
Example 4.2: We now consider the second Mathieu group M12of order 95040 = 26 33*
* 5 11.
Its poset space is much more complex since it has two distinct conjugacy classe*
*s of involu
tions as well as three distinct conjugacy classes of (Z=2)3's and four conjugac*
*y classes of
(Z=2)2. For the details of its structure see [AMM2]. In particular, from [AMM2]*
* we have
that the Poincare series for H*(M12; F2) is
2 + 3t3 + t4 + 3t5 + 4t6 + 2t7 + 4t8 + 3t9 + t10+ 3t11+ t1*
*2+ t14
PM12(t) = 1_+_t_________________________________________________________(*
*1  t4)(1  t6)(1  t7)
20
and, in fact, H*(M12; Z=2) is CohenMacaulay over the polynomial subring F2[D4;*
* D6; D7].
Since H*(BG2; F2) ~=F2[D4; D6; D7] as well, this gives the indication of some c*
*onnections
between M12 and the exceptional Lie group G2. Of course M12 6 G2 so the connect*
*ion is
not nearly as simple as group inclusion. However, in M12 there are two maximal *
*subgroups
of order 192. The first is the holomorph of the quaternion group Q8 which we wr*
*ite W ,
and the second is a split extension
(Z=4)2 o (Z=2 x 3)
where the Z=2 acts to invert elements in (Z=4)2. But W 0is also seen to be the *
*extension
of the elements of order 4 in the usual torus of G2(q) for q ~=3; 5 mod (8) by *
*the Weyl
group of G2. These groups both contain Syl2(M12) so there is a configuration
W o..............................................*
*.........................................................................@
Syl2(M12)
contained in M12. In [AMM2] it is shown that this configuration completely dete*
*rmines
H*(M12; F2) as the intersection in H*(Syl2(M12)) of the restriction images of H*
**(W ) and
H*(W 0). On the other hand, recalling the group E discussed in 2.7 and the fac*
*t that
W ~= ss1 (P1), W 0~=ss1 (P2), we have a very similar configuration in E. Howe*
*ver, the
two configurations are not isomorphic and here the deviation between the cohomo*
*logy of
E and that of M12 is explained by 2.2. Indeed, applying 2.2 we have
4:3 H*(E) ~=H*(M12) (H*(V3) St(GL3(F2))) GL3(F2):
On the other hand, using [M] we can give an independent evaluation of H*(E) usi*
*ng
the fact that E appears as the normalizer of one of the two (Z=2)3 tori in G2(F*
*q) for
q ~=5 mod (8), [FM]. In particular the result is
PE (t) = ________q(t)_________(1  t4)(1  t6)(1  t*
*7)
where q(t) = 1 + x2 + 3x3 + 2x4 + 4x5 + 5x6 + 4x7 + 5x8 + 4x9 + 2x10+ 3x11+ x12*
*+ x14.
Note that E is also a subgroup of the compact 14 dimensional Lie group G2 and q*
*(t) is
the Poincare series of the compact 14 dimensional manifold G=E, [M]. It follows*
* that the
error term (H*(V3) St(GL3(F2))) GL3(F2)has Poincare series
4 + t5 + t6 + 2t7 + t8 + t9 + t10
e3(t) = t____________________________(1; t4)(1  t6)(1 *
* t7)
a result which is useful for understanding the group O0N.
Example 4.4: The O'Nan group O0N has order 460; 815; 505; 920 = 29 34 5 73 11 1*
*9 31,
and in [AM2] we determine its poset space, obtaining the following picture.
2 4 . L3(4) : 21 *
* @
(4 x 2 )D8.................................................*
*.........................................................................@
...............................................*
*.........................................................................@
2 ...................................................*
*.........................................................................@
(3 .........................................................*
*.........................................................................@
4 ..............................................................*
*.........................................................................@
(3 :4 x A4) ..................................................................*
*.........................................................................@
o..................................................................*
*.........................................................................@
...............................................................*
*.........................................................................@
.............................................................*
*.........................................................................@
(4x22)S3..........................................................*
*.........................................................................@
.......................................................*
*.........................................................................@
.....................................................*
*.........................................................................@
(4 x 2 x 2) . S4 (4x22)D8 43 . S4
21
From this picture some easy cancellations give
H*(O0N) H*((Z=4)3 . 4) ~=H*((Z=4)3 . GL3(F2)) H*(Z=4 . SL3(F4) o Z=2);
and using 2.2 this can be reduced to
H*(O0N) H*(S)Z=2 ~=H*(S)3 (H*(V3) St))GL3(F2) H*(Z=4 . SL3(F4) o Z=2)
where S ~=(Z=4)3 . (Z=2)2. The terms involving S are reasonably direct to evalu*
*ate while
the term involving the Steinberg module is discussed in 4.2. The final term is *
*essentially
determined in [AM1]. The A5 invariants discussed in x3 play a key part in the w*
*ork there.
Example 4.5: For the third Mathieu group M22 we use a sporadic geometry describ*
*ed in
[RSY] which also satisfies the hypotheses necessary for 2.1 to remain valid. Th*
*e associated
complex has the form
V4 o A6 o.....................o...........................*
*.........................................................................@
................................................*
*.....
...............................................*
*...............V o D
V3o4 .............................................*
*.................V3o438
...........................................*
*...................
.o..............................
V3 o GL3(F2)
We apply 2.1, 2.2, to this picture and, after some cancellation we have
H*(M22) H*(Syl2(M22)) ~= H*(V4 o 4) H*(V4 o 5)
(H*(V3) St) GL3(F2) (H*(V4) St)A6 :
From this, the results in [AM3], and the discussion of invariants in x3 we obta*
*in the
Poincare series for M22. The formula is very messy and not too illuminating, so*
* we defer
details to a further paper.
22
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* (in
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23