THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE
HIGMAN-SIMS GROUP
A. ADEM, J. F. CARLSON, D. B. KARAGUEUZIAN, AND R. JAMES MILGRAM
Abstract.In this paper we compute the mod 2 cohomology of the Sylow 2-su*
*bgroup
of the Higman-Sims group HS, one of the 26 sporadic simple groups. We o*
*btain its
Poincare series as well as an explicit description of it as a ring with *
*17 generators and
79 relations.
1.Introduction
Recently there has been substantial progress towards computing the mod 2 coho*
*mology
of low rank sporadic simple groups. In fact the mod 2 cohomology of every spor*
*adic
simple group not containing (Z=2)5 has been computed, with the notable exceptio*
*ns of
HS (the Higman-Sims group) and Co3 (one of the Conway groups). The reason for t*
*hese
exceptions is that these two groups have large and complicated Sylow 2-subgroup*
*s, with
many conjugacy classes of maximal elementary abelian subgroups. The Higman-Sims
group has order 44; 352; 000 = 29.32.53.7.11, and its largest elementary abelia*
*n 2-subgroup
is of rank equal to four. It is a subgroup of Co3 of index equal to 11178 = 2 .*
* 35. 23, hence
Syl2(HS) is an index 2 subgroup of Syl2(Co3).
In this paper we compute the mod 2 cohomology ring of S = Syl2(HS), obtaining
explicit generators and relations. In a sequel we will determine the necessary*
* stability
conditions for computing the cohomology of HS itself. This is a step towards ob*
*taining
a calculation of the mod 2 cohomology of Co3. This latter group is of particula*
*r interest
because of its relation to a homotopy-theoretic construction due to Dwyer and W*
*ilkerson
(see [2]) and it would seem that the most viable way of accessing this group is*
* via HS.
The calculation we present is long and highly technical, involving techniques*
* from
topology, representation theory and computer algebra. Our main result is the fo*
*llowing
Theorem 1.1. The mod 2 cohomology of S = Syl2(HS) has Poincare series
(1_+_x)2(1_-_x_+_x2)(1_+_2x_-_x5)_
:
(1 - x)2(1 - x4)(1 - x8)
___________
The first author was partially supported by the NSF, NSA and CRM-Barcelona.
The second author was partially supported by the NSF.
The third author was partially supported by an NSF postdoctoral fellowship an*
*d the NSA.
The fourth author was partially supported by the NSF and CRM-Barcelona.
1
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 2
As a ring, H*(S; F2) has seventeen generators, in degrees
1; 1; 1; 2; 2; 2; 2; 3; 3; 3; 3; 4; 4; 5; 6; 7; 8;
and a (minimal) set of seventy-nine relations.
Remark 1.2. Nine of the generators above can be given as Stiefel-Whitney classe*
*s associ-
ated to some of the irreducible representations of S (see x3), although the rem*
*aining eight
had to be determined by other means. The complete set of relations and the Ste*
*enrod
operations on a representative set of generators are described in two appendice*
*s at the
end of the paper.
We briefly outline our method of proof. The group S can be expressed as a sem*
*idirect
product (Z=4)3 : U3, where U3 ~=D8 is the group of upper triangular 3 x 3 matri*
*ces over
F2. Using a recent result in [7], we calculate the E2-term of the Lyndon-Hochsc*
*hild-Serre
spectral sequence associated to this extension. This involves using an explicit*
* decompo-
sition of the symmetric algebra of a module.
The second ingredient is a computer-assisted verification of the cohomology o*
*f S
through degree ten. Combined with the structure of the E2 term we obtain the fo*
*llowing
crucial result
Theorem 1.3. The mod 2 spectral sequence associated to S = 43 : U3 collapses *
*at E2,
and yields the Poincare series above.
A critical aspect of this is that the generators for the E2 term occur in low*
* degrees,
where the computer can provide enough information to ensure a collapse. This me*
*thod
seems to be quite effective for many cohomology computations and gives a very e*
*asy
calculation of the mod 2 cohomology of U4. Moreover, with somewhat more effort*
* it
yields the mod 2 cohomology of U5 as well. This last result is particularly in*
*teresting
since U5 is the Sylow 2-subgroup of the two sporadics M24 and He.
The next step consists of computing the restriction map from H*(S; F2) to the*
* cohomol-
ogy of centralizers of rank two elementary abelian subgroups, where we obtain a*
*n image
ring having the same Poincare series as the cohomology. From this we conclude
Theorem 1.4. The cohomology of S = Syl2(HS) is detected by the centralizers o*
*f rank
two elementary abelian subgroups, and its explicit image is the ring described *
*above, with
17 generators and 79 relations.
Remark 1.5. There are nine such subgroups up to conjugacy in S. Explicit gener*
*ators
are described in x3 via their restrictions to these nine centralizers.
As mentioned above, the results here will be used in a sequel to calculate th*
*e cohomology
of HS, and from there obtain information on the cohomology of Co3, thus bringin*
*g us
closer to a complete understanding of rank 4 sporadic simple groups.
Throughout this paper coefficients will be in F2, so they are suppressed. Occ*
*asionally
k will be used to denote this coefficient ring, especially if it appears in a r*
*epresentation-
theoretic context. We refer the reader to [1] for background on group cohomolog*
*y.
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 3
2. The subgroup structure of S = Syl2(HS)
The Sylow 2-subgroup of the Higman-Sims group (denoted S from here on) has a
description as a semi-direct product S = 43: D8 with v1, v2, v3 the generators *
*of the 43
while D8 = {t; s | t2 = s4 = (ts)2 = 1}, and
vt1= v-13; vt2= v-12; vs1= v2; vs2= v3; vs3= v-12v1v3:
In this section we will develop the required details about the subgroup structu*
*re of S to
enable us to understand its cohomology. In our notation, if x; y 2 G are eleme*
*nts in a
group, then xy = yxy-1.
To start we have
Lemma 2.1. < v1v3 >= Z=4 is a normal subgroup of S and S= < v1v3 >~=2 o 2 o 2.
2 -1 t -1
Proof. In the quotient we have vt1= v-13~ v1, while vts1= v1 . Also v2 = v2 w*
*hile
2 -1 -1
vts2= v2v3 v1 , which is equal to v2 in the quotient. Thus
< v1; ts2 >~=D8; < v2; t >~=D8
and these two copies of D8 commute with each other, giving a copy of D8 x D8 in*
* the
quotient. Next, note that ts exchanges t, ts2, and also exchanges v1, v-12, hen*
*ce the two_
copies of D8 above, and the extension (D8 x D8): < ts >~=2 o 2 o 2. *
* |__|
Definition 2.2. The group Kfiis the inverse image in S of the index two subgroup
D8 x D8 S= < v1v3 >.
Kfiis given explicitly as an extension
< v1; v2; v3 >: < t; s2 >= 43: 22:
Moreover, S is the split extension Kfi:2 = Kfi:< ts >.
Next we examine the maximal elementary abelian subgroups in S. We show that t*
*here
are precisely eight copies of 24 S. To begin note that there are exactly 5 co*
*njugacy
classes of maximal 2-elementaries in 2 o 2 o 2, first 3 conjugacy classes of 24*
*'s:
24I= < v21; ts2; v22; t >; 24II= < v21; v1ts2; v22; v-12t >
both normal, and 24I;II=< v21ts2; v22; v-12t > with Weyl group 22. Then there *
*are two
conjugacy classes of 23's, each with Weyl group D8,
23I= < (v1v2)2; s2; ts >; 23II= < (v1v2)2; v1v-12s2; ts >
which together generate a copy of D8 x 2,
(D8) x 2 = < v1v-12s2; ts > :
This is all standard and can be found in many references. Lifting these groups *
*to S we
find
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 4
Lemma 2.3. The following list describes the groups above and their lifts:
24I; D8 * D8 * 4
24II; D8 * D8 * 4
24I;II; (8: Aut(8)) * 4
(D8) x 2; D8 x D8
Proof. We note that the lift of 24Iis given as
< v1v3; v21; ts2; v22; t >=< v1v-13; ts2; t; v21; v1v-13v22>
On the other hand, note that v1v-13v22commutes with s2, t, consequently with ea*
*ch of
the remaining four generators, while the first two generators commute with the *
*third and
fourth, and < v1v-13; ts2 >= D8, < t; v21>= D8, and all three have the central *
*element
(v1v3)2 in common. The verification for the second group is similar. For the th*
*ird, note
that (v1s2)2 = v1v3 so that v1s2 has order 8. Also conjugation with v22takes th*
*e element
to its fifth power, while conjugation by t takes it to its inverse. This gives *
*the extension
8: Aut(8). The final generator can again be choosen as v1v-13v22.
It remains to check the lift of (D8) x 2. Note, first that < s2; v1v-12>= D8*
* as a
subgroup of S, while v1v3, ts both commute with the elements of this D8. On the*
* other __
hand < v1v3; ts >= D8 as well, and the final statement follows. *
* |__|
Now, to determine the structure of the set of 24's in S we check the inverse *
*images of
the conjugates of 23Iand 23II. The normalizer of each of these groups has order*
* 26 so there
are a total of 4 such groups in 2 o 2 o 2, with the remaining two given by conj*
*ugation with
v1,
(23I)v1 = < v1v2ts; v2v-13s2; (v1v2)2 >; (23II)v1 = < v1v2ts; v2v-13s2; (*
*v1v2)3 >
The lift of each to S is a copy of 22 x D8 which contains exactly two copies of*
* 24, so S
has at least eight copies of 24. In fact we have
Corollary 2.4. There are no 25's contained in S. There are exactly eight copi*
*es of 24
contained in S, breaking up into three conjugacy classes, two with two copies e*
*ach and
one with four.
Proof. The only thing that needs to be pointed out is that if there were a 24 w*
*hich did
not contain the central element (v1v3)2, (and hence a 25 obtained by adjoining *
*(v1v3)2),
then it would project non-trivially to one of the four 24's in the quotient 2 o*
* 2 o 2. But
we have identified the lifts of these groups as copies of groups having rank th*
*ree! Hence,
every possible 24 contains (v1v3)2, and hence projects to one of the two conjug*
*acy classes
of extremal 23's. Consequently, it must lie in one or the other of the four co*
*pies_of
22 x D8 S that we have constructed. *
* |__|
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 5
We now list the eight 24's explicitly.
2A = < (v1v3)2; (v1v2)2; s2; ts >
2v1A= < (v1v3)2; (v1v2)2; v-11v3s2; v1v2ts >
2B = < (v1v3)2; (v1v2)2; v1v-12s2; ts >
2v1B= < (v1v3)2; (v1v2)2; v2v-13s2; v1v2ts >
2tB = < (v1v3)2; (v1v2)2; v3v-12s2; v3v-12ts >
2v1tB= < (v1v3)2; (v1v2)2; v1v-12s2; v1v3ts >
2C = < (v1v3)2; (v1v2)2; s2; v1v3ts >
2v1C= < (v1v3)2; (v1v2)2; v1v-13s2; v2v-13ts >
Lemma 2.5. There is an outer automorphism ff : S ! S which exchanges 2A and 2*
*C .
Proof. We use the description of S as 43: D8 where we write
D8 = 22: 2 =< ts; ts-1 >: < t > :
But we can replace this copy of D8 by
< v2v-13ts; v1v-12ts-1 >: < t >
*
* __
and ff is defined as the identity on 43 and the correspondence above on D8. *
* |__|
We show that there are precisely two conjugacy classes of D8 x D8 S.
Remark 2.6. The group D8 x D8 constructed in the proof above as the lift of (D8*
*) x 2
is
< s2; v1v-12> x < ts; v1v3 > :
We can also construct a second copy of
D8 x D8 S
as
< ts; v2v-13> x < ts-1; v1v-12> :
It is direct to check that the two (D8)2 above are not conjugate in S. In fact*
*, the
intersection of the second copy of (D8)2 and < v1v3 > is just < (v1v3)2 > and i*
*ts image
in 2 o 2 o 2 is easily seen to be D8 * D8. Thus there are at least two conjugac*
*y classes of
D8 x D8's contained in S. Shortly we will show that there are exactly two.
Lemma 2.7. The span < 2e1I; 2e2J> is always one of the three groups D8 x D8, *
*22 x D8,
or 22+4 = Syl2(L3(4)).
(a) It is 22+4 if and only if I = J and e2 = e1v1. Consequently, there are exa*
*ctly 3
conjugacy classes of 22+4's in S, two normal and one containing two elements.
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 6
(b) It is D8 x D8 in case
< 2A; 2v1C>=< 2B ; 2tB>
and their conjugates by v, and also in the cases
< 2A; 2v1B>=< 2C ; 2tB>
and their four conjugates by 22 =< t; v1 >. Consequently, there are exactly two*
* conjugacy
classes of D8 x D8 S, one containing four groups and the other containing 2.
(c) In each of the remaining cases it is 22xD8, and each 22xD8 is contained in *
*a D8xD8.
Proof. First we check the result for 2A; 2v1A. We have
(s2v-11v3s2)2 = (v1v3)2; (s2v1v2ts)2 = (v2v3)2;
(tsv-11v3s2)2 = (v2v3)2; (tsv1v2ts)2 = (v1v2)2;
and this is a presentation of 22+4. The same calculations result for 2C using *
*the auto-
morphism above. Moreover, ts commutes with v1v-12while s2 inverts it. So this c*
*hange
cancels out in the squares for the pair < 2B ; 2v1B>, and we have verified that*
* the groups
asserted to be 22+4's in fact are.
The remaining statements are now easily checked by comparing with the D8 x D8*
*'s
already constructed above. But this gives a complete list of possible pairs and*
* the_result
follows. *
* |__|
We show there there are exactly two conjugacy classes of 2 o 2 o 2 S. Note *
*that t
normalizes < 2A; 2v1C>, exchanging the two copies of D8, so
< 2A; 2v1C; t >~=2 o 2 o 2:
However, since s2, ve1ive2jts are not conjugate in S, it follows that no D8xD8 *
*in the second
conjugacy class is contained in a 2 o 2 o 2.
Corollary 2.8. There are exactly four copies of 2 o 2 o 2 contained in S formi*
*ng two
conjugacy classes with conjugation by v1 exchanging the groups in each class.
Proof. The normalizer of < 2B ; 2v1C> is obtained by adjoining t, v22. Thus the*
*re are three
degree two extensions of < 2B ; 2v1C> in S. The extension by t is 2 o 2 o 2. *
*Clearly, the
extension by v22does not give a 2 o 2 o 2.
Finally, consider the extension by tv22. Replace the second D8 by
< (v2v3)2ts-1; v1v-12> :
Then these two copies of D8 commute with each other and their span is D8xD8. Mo*
*reover, __
tv22exchanges them, giving an isomorphism of this group with a second copy of 2*
*o2o2. |__|
Remark 2.9. There is an automorphism of S fixing < v1; v2; v3 > and exchanging *
*the two
conjugacy classes of 2 o 2 o 2's constructed above. Indeed, such an automorphis*
*m can be
given by setting
t $ tv22; ts $ ts; ts-1 $ ts-1(v2v3)2
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 7
There are exactly six conjugacy classes of maximal 23 S.
A computer analysis of S using MAGMA or alternatively a direct analysis of th*
*e two
copies of D8 * D8 * 4 and the (8: Aut(8)) * 4 obtained as the lifts of the thre*
*e 24's in
2 o 2 o 2 shows that the maximal elementary abelian subgroups of S consist of t*
*he three
24's discussed above and 6 copies of 23, all contained in the subgroup Kfi: one*
* given as
I =< v21; v22; v23> with centralizer CI = 43, and the other five all with centr*
*alizer of the
form 22 x 4. These centralizers are
CII = < v1v-13v22; ts2; v23>
CIII = < v1v-13v22; t; s2 >
CIV = < v1v-13v22; s2; v1v3t >
CV = < v1v-13v22; v21; v1ts2 >
CV I = < v1v-13v22; v2t; v2v-13s2 >
with the 23 subgroups denoted II; : :;:V I respectively.
The group-theoretic information which we have described in this section will *
*be used
subsequently to establish a detection theorem for H*(S).
3. Explicit Detection and Stiefel-Whitney Classes
In this section we describe the seventeen generators of H*(S) mentioned in 1.*
*1 in terms
of their explicit restrictions to the cohomology of the nine detecting subgroup*
*s, which
completely determines them in view of 1.4. It turns out that nine of the genera*
*tors can
be given as Stiefel-Whitney classes though the remaining eight had to be determ*
*ined
by the computer. For the nine Stiefel-Whitney classes we will be very explicit*
*; for the
remaining generators we simply give convenient representatives.
We use the following notation for the cohomology of the subgroups:
H*(2m ) = F2[l1; : :;:lm ]
(where each li is 1-dimensional);
H*(43) = F2[b1; b2; b3] (e1; e2; e3)
(where the ei are 1-dimensional, and bi is the Bockstein of ei); and
H*(22 x 4) = (e) F2[l2; l3; b]
(where |e| = 1; b = fi(e) and |l2| = |l3| = 1).
One of the most powerful ways of constructing elements in cohomology is to us*
*e rep-
resentations, i.e., homomorphisms rI: S ! GLn(R ), which, in turn, induce maps*
* of
classifying spaces,
BrI:BS ! BGLn(R)
and pull back the Stiefel-Whitney classes. The restriction images of these coh*
*omology
classes in the detecting groups are then determined by taking the Stiefel-Whitn*
*ey classes
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 8
of the restrictions of the rI. Here the determinations of the Stiefel-Whitney *
*classes are
standard (see [8], [9]).
The real group-ring R(S) splits into 33 simple summands:
R(S) = 8 R12M2(R ) M2(C ) 8M4(R ) 3M8(R ) M8(C );
but we will only need a small number of these representations.
To begin consider the real representation r4. Here r4 is described by giving*
* matrix
images for the generators of the group:
0 1 0 1 0 1
-1 0 0 0 1 0 0 0 1 0 0 0
B 0 1 0 0 C B 0 1 0 0 C B 0 -1 0 0 C
v1 7! B@ 0 0 1 0 CA; v2 7! B@ 0 0 -1 0 CA; v3 7! B@ 0 0 -1 0 CA
0 0 0 -1 0 0 0 -1 0 0 0 1
0 1
0 0 0 1
B 1 0 0 0 C K 0
r4(s) = B@ 0 1 0 0 CA; r4(t) = 0 K
0 0 1 0
0 1
where K is the 2 x 2 matrix K = 1 0 . When we restrict to 2A we have that *
*the
0 I
first two generators map to I, while s2 7! I 0 . Consequently, when we dia*
*gonalize
-I 0
the image of s2 to 0 I , we see that ts can also be diagonalized to
0 1
-1 0 0 0
B 0 1 0 0 C
B C
@ 0 0 1 0 A
0 0 0 1
so the representation becomes the sum of four one-dimensional representations, *
*and the
total Stiefel-Whitney class becomes
(1 + l3 + l4)(1 + l3) = 1 + l4 + l3(l3 + l4):
Next we consider CI. Here we see at once that the total Stiefel-Whitney class is
(1 + e1)(1 + e3)(1 + e2 + e3)(1 + e1 + e2) = 1 + e2(e1 + e3):
As a final example of how to calculate these restrictions, consider CIII. Here,*
* v1v-13v227!
-I while the images of s2 and t have already been discussed. Diagonalizing we h*
*ave that
the total Stiefel-Whitney class of the restriction is
(1 + e)(1 + e + l2)(1 + e + l3)(1 + e + l2 + l3) = 1 + d2(l2; l3) + (1 + e*
*)d3(l2; l3):
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 9
Here, d2(x; y) = x2 + xy + y2 while d3(x; y) = x2y + xy2 = xy(x + y) are the Di*
*ckson
elements.
The other representations that will be needed are first, an 8-dimensional rep*
*resentation,
r8, which is very similar to the 4-dimensional representation considered above:
0 1 0 1 0 1
J 0 0 0 I 0 0 0 I 0 0 0
B 0 I 0 0 C B 0 I 0 0 C B 0 J 0 0 C
v1 7! B@ 0 0 I 0 CA; v2 7! B@ 0 0 J 0 CA; v3 7! B@ 0 0 J 0 CA
0 0 0 J 0 0 0 J 0 0 0 I
0 1 0 1
0 0 0 I 0 K 0 0
B I 0 0 0 C B K 0 0 0 C
s 7! B@ 0 I 0 0 CA; t 7! B@ 0 0 0 K CA
0 0 I 0 0 0 K 0
0 -1
where I is the 2 x 2 identity matrix and J = 1 0 . The remaining represen*
*tations
we need are two-dimensional: the first of these, r2;1, has the form
vi7! J; 1 i 3; s 7! J; t 7! K;
the second, r2;2, is
vi7! J; 1 i 3; s 7! I; t 7! K;
and the third, r2;3, is
v1 7! J; v2 7! -J; v3 7! J; s 7! K; t 7! K:
Finally, we should mention the 1-dimensional representations which all factor i*
*n the form
S=S0= ~=23 -! Z =2 = {1}
The first Stiefel-Whitney classes of , ~~, respectively give the 1-dim*
*ensional gen-
erators for H*(S) while the second Stiefel-Whitney classes of the three 2-dimen*
*sional
representations above, together with the 4-dimensional representation, r4, give*
* the two-
dimensional generators.
The four 3-dimensional generators do not occur as Stiefel-Whitney classes, an*
*d were
obtained by a computer calculation using MAGMA as described previously.
The Stiefel-Whitney classes w4 and w8 for the eight-dimensional representatio*
*n are
generators.
The 5-dimensional generator is Sq2 of one of the computer generated three-dim*
*ensional
generators. The remaining 4-dimensional generator, n, as well as the seven-dime*
*nsional
generator, i, also had to be determined by MAGMA, and the 6-dimensional generat*
*or can
be given as Sq2(n).
The following tables give the restrictions of the generators discussed above.
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 10
Table 1: Restrictions of One Dimensional Stiefel-Whitney Classes
w1(v) w1(s) w1(t)
0 1
CI e1 + e2 + e3 0 0
CII B 0 0 l2 C
B C
CIII B 0 0 l2 C
B C
CIV B 0 0 l3 C
B C
CV B l3 0 l3 C
B C
CV I B l2 0 l2 C
B C
2A B 0 l4 l4 C
2B @ 0 l4 l4 A
2C 0 l4 l4
Table 2: Restrictions of Two-Dimensional Stiefel-Whitney Classes
w2(r2;1) w2(r2;2) w2(r2;3) w2(r4)
0 1
CI b1 + b2 + b3 b1 + b2 + b3 b1 + b2 + b3 e2(e1 + e3)
CII B el2 + l3(l2 + l3)el2 + l3(l2 + l3)el2 + l3(l2 + l3)l22 C
B C
CIII B el2 + l3(l2 + l3)el2 + l3(l2 + l3) el2 d2(2; 3) C
B C
CIV B el3 + l2(l2 + l3)el3 + l2(l2 + l3) el3 d2(2; 3) C
B 2 C
CV B el3 + l2(l2 + l3)el3 + l2(l2 + l3)el3 + l2(l2 + l3)l3 C
B C
CV I B el2 + l3(l2 + l3)el2 + l3(l2 + l3)el2 + l3(l2 + l3)d2(2; 3)C
B C
2A B l3(l3 + l4) 0 0 l3(l3 + l4)C
2B @ l3(l3 + l4) 0 l2 A
3 l3(l3 + l4)
2C l3(l3 + l4) 0 l24 l3(l3 + l4)
Table 3: Restrictions of w4 and w8 for the Eight-Dimensional Representation.
w4 w8
0 2 1
CI (b1 + b2 + b3) + b2(b1 + b3)b1b3(b1 + b2)(b2 + b3)
CII B d2(2; 3)2 L C
B 2 C
CIII B d2(2; 3) L C
B 2 C
CIV B d2(2; 3) L C
B 2 C
CV B d2(2; 3) L C
B 2 C
CV I B d2(2; 3) L C
B C
2A B d4(2; 3; 4) M C
2B @ d4(2; 3; 4) M A
2C d4(2; 3; 4) M
where d2(2; 3) = l22+ l2l3 + l23, d3(2; 3) = l2l3(l2 + l + 3), d4(2; 3; 4) is t*
*he fourth Dickson
invariant in the three classes l2, l3, and l4, while
L = b4 + b2d2(2; 3)2 + bd3(2; 3)2
M = l81+ l41d4(2; 3; 4) + l21Sq2(d4(2; 3; 4)) + l1Sq3(d4(2; 3; 4*
*)):
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 11
Next we give the restrictions of the computer generated indecomposables, s, r, *
*p, and q,
in dimension three.
s r
0 1
CI e1e2e3 e1e2e3
CII B 0 e1l22 C
B 2 2 2 C 2
CIII B 0 e1l2 + e1l2l3 + e1l3 + l2l3 +Cl2l3
B 2 2 C
CIV B 0 e1l2 + e1l2l3 + e1l3 C
B 2 C
CV B e1l3 0 C
B 2 2 2 3 C
CV I B e1l2 + e1l2l3 + e1l3 + l2l3 + l3 0 C
B C
2A B 0 0 C
2B @ l3 2 A
3 + l3l4 0
2C 0 l23l4 + l3l24
q p
0 1
CI e1e2e3 + e2b3 + e3b2 e1e2e3
CII B 0 e1l22 C
B 2 2 2 C
CIII B 0 e1l2 + l2l3 + l2l3 C
B 2 C
CIV B 0 e1l3 C
B 2 C
CV B e1l3 0 C
B 2 2 2 3 2 C
CV I B e1l2 + e1l2l3 + e1l3 + l2l3 + l3 e1l2l3 + e1l3 C
B 2 2 2 2 2 C *
* 2
2A B 0 l1l4 + l1l4 + l2l3 + l2l3 + l3l4C+ l3*
*l4
2B @ l3 2 2 2 2 2 2 A *
* 2
3 + l3l4 l1l4 + l1l4 + l2l3 + l2l3 + l3l4 + l3*
*l4
2C 0 l21l4 + l1l24+ l22l3 + l2l23
Note here that the computer has not always picked the simplest choices. For ex*
*ample
s + q restricts to e2b3 + e3b2 in CI and 0 in the remaining 8 centralizers.
Here is the expansion of the computer generated indecomposable, n in dimensio*
*n four:
n
0 1
CI e1e2b3 + e1e3b2 + e2e3b1 + e2e3b3
CII B e1l32+ e1l22l3 + e1l2l23 C
B 3 3 2 2 C
CIII B e1l2 + l2l3 + l2l3 C
B 2 2 3 C
CIV B e1l2l3 + e1l2l3 + e1l3 C
B 2 2 C
CV B e1l2l3 + e1l2l3 C
B 2 3 2 2 4 C
CV I B e1l2l3 + e1l3 + l2l3 + l3 C
B C
2A B 0 C
2B @ l2 2 2 2 3 4 3A 2 2
1l3l4 + l1l3l4 + l2l3 + l2l3 + l3 + l3l4 + l3l4
2C l21l24+ l1l34+ l22l3l4 + l2l23l4 + l44
This element, together with w4 for the eight-dimensional representation can be *
*taken as
the indecomposable generators in dimension 4. Finally, for the generator i in d*
*egree 7 the
restrictions are given as follows. For CI the restriction is given as
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 12
e1e2e3b23+ e1b22b3+ e1b2b23+ e2b21b3+ e2b1b23+ e2b22b3+ e2b2b23+ e3b21b2+*
* e3b1b22+ e3b32+
e3b22b3 + e3b33
while for CII through CV I we have
0 i 1
CII e1l62+ e1l52l3 + e1l42l23+ l42l33+ l32l43+ l22l53+ l2l63
CIIIB e l6 + e l4l2 + e l3l3 + e l l5 + e l6 + l6l + l4l3 +Cl2l5 + *
*l l6
B 1 2 1 2 3 1 2 3 1 2 3 1 3 2 3 2 3 C 2 3 *
* 2 3
CIV B e l6 + e l5l + e l3l3 + e l l5 + e l6 C
B 1 2 1 2 3 1 2 3 1 2 3 1 3 C
CV @ e1l22l43+ e1l2l53+ e1l63 A
CV I e1l62+ e1l22l43+ e1l2l53+ l22l53+ l2l63
The restrictions to 2A, 2B , and 2C are very long and complicated. For 2A we ob*
*tain
l21l42l4+l21l22l23l4+l21l22l3l24+l21l22l34+l21l2l23l24+l21l2l3l34+l1l42l2*
*4+l1l22l23l4+l1l22l3l34+l1l22l44+
l1l2l23l34+ l1l2l3l44+ l62l3 + l52l23+ l42l33+ l42l23l4 + l42l3l24+ l42l34+ l32*
*l43+ l22l43l4 + l22l33l24+ l22l23l34+
l22l3l44+ l22l54+ l2l23l44+ l2l3l54
For 2B the restriction is
l41l23l4+ l21l42l4+ l21l22l23l4+ l21l22l3l24+ l21l22l34+ l21l2l23l24+ l21*
*l2l3l34+ l21l43l4+ l21l23l34+ l1l42l24+
l1l22l23l24+ l1l22l3l34+ l1l22l44+ l1l2l23l34+ l1l2l3l44+ l1l43l24+ l62l3+ l52l*
*23+ l42l33+ l42l34+ l32l43+ l22l43l4+
l22l33l24+ l22l23l34+ l22l54+ l2l53l4 + l2l43l24+ l2l3l54+ l63l4 + l33l44
For 2C the restriction is
l41l34+ l21l42l4 + l21l22l23l4 + l21l22l3l24+ l21l22l34+ l21l2l23l24+ l21*
*l2l3l34+ l21l23l34+ l21l3l44+ l1l42l24+
l1l22l23l24+ l1l22l3l34+ l1l22l44+ l1l2l23l34+ l1l2l3l44+ l1l23l44+ l1l3l54+ l1*
*l64+ l62l3+ l52l23+ l42l33+ l42l3l24+
l32l43+ l22l23l34+ l22l3l44+ l2l43l24+ l2l33l34+ l2l23l44+ l63l4 + l53l24+ l43l*
*34+ l33l44
In the next section we begin the proof of our main theorem.
4. Preliminaries on Modules
We now turn to the proof of 1.1. As our intial step we have to determine H*(4*
*3) as a
module over F2(U3).
Let U3 ~=D8 denote the Sylow 2-subgroup of L3(2). In this section we describ*
*e the
symmetric algebra of the natural U3 module M by giving a "factorization" of thi*
*s algebra.
This information will be used to describe the E2 term of a spectral sequence co*
*nverging
to H*(S). In order to describe this factorization we must make a number of defi*
*nitions
and recall a few results. We will be using ideas and methods from [7]. We wri*
*te the
symmetric algebra as k[x; y; z], where the action of U3 on the homogeneous poly*
*nomials
of degree 1 preserves the flag of subspaces < x >< x; y >< x; y; z >. Let's den*
*ote by
b; c; d the elements of U3 which send y 7! y + x, z 7! z + x, and z 7! z + y, r*
*espectively.
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 13
(b is supposed to fix z, c to fix y and d to fix x.) We write H1 for the subgro*
*up < b; c >
and H2 for the subgroup < d >.
Remark 4.1. It is worth noting that the module M is not isomorphic to M*, but t*
*hat the
two differ by an outer automorphism of the group U3. This means that the symme*
*tric
algebras S*(M) and S*(M*) also differ by an outer automorphism of the group; th*
*us if
we are only interested in Poincare series it doesn't matter which one we study.*
* However,
it is not true that S*(M)* S*(M*).
Definition 4.2. d1 := x; d2 := y2 + xy; d4 := z(z + y)(z + x)(z + y + x)
The following result is well-known:
Proposition 4.3. k[x; y; z]U3 = k[d1; d2; d4]
Definition 4.4. OE := z2 + yz; := z2 + xz. We denote by N, K, the U3-submo*
*dule
of the symmetric algebra generated by OE, , respectively.
Lemma 4.5. N is isomorphic to the permutation module k[U3=H2] and has socle d*
*21. K
is isomorphic to the permutation module k[U3=H1] and has socle d2.
We will now exhibit three U3-submodules of k[x; y; z].
Definition 4.6.
A := k[d4]; B := k K d2K d22K . . .
C := k M N d1N d21N d31N . . .
Note that A is a k[d4][U3]-submodule of k[x; y; z], while B is a k[d2][U3]-su*
*bmodule and
C a k[d1][U3]-submodule. (The only point to check is that d1M N, but this can*
* be
done by working directly with the generating polynomials.)
Now we can describe the "factorization" mentioned above:
Proposition 4.7. There is an isomorphism of U3-modules
A B C ! k[x; y; z]
induced by multiplication in the symmetric algebra.
Proof. For any triple of indecomposable modules Ai, Bj, Cl appearing in the dir*
*ect sum
decompositions of A, B, and C above, we have soc(Ai Bj Cl) = soc(Ai) soc(Bj)
soc(Cl). Thus soc(ABC) = soc(A)soc(B)soc(C) = k[d4]k[d2]k[d1]. This shows
that the map above is an isomorphism on the socle, hence it is injective. The P*
*oincare __
series of the two sides are equal, by a calculation, so we have an isomorphism.*
* |__|
Remark 4.8. No analogous factorization exists for larger fields or for more var*
*iables.
We can rewrite this factorization as a direct sum decomposition
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 14
Proposition 4.9. There is an isomorphism of U3-modules:
k[x; y; z]= k[d4] (k M)
k[d1; d4] N
k[d2; d4] (K W )
k[d1; d2; d4] F:
Here F = N K is a free U3-module, appearing in degree 4, while k, M, N, W = M K,
and K appear in degrees 0,1,2,3, and 2 respectively.
5.The Sylow 2-subgroup of the Higman-Sims group: Modules and
Cohomology
As we mentioned, the Sylow 2-subgroup can be written as the semidirect produc*
*t (Z4)3o
U3, where the action of U3 on the vector space H1((Z4)3) gives this cohomology *
*group
the structure of the natural U3-module M. It follows that as a U3-module, H*((Z*
*4)3) =
*(M) S*(M).
In this section we compute the Poincare series of H*(U3; *(M) S*(M)). We wil*
*l also
try to find a bound on the degrees of generating elements. To do this we must c*
*ompute the
Poincare series of the cohomologies of all the U3-modules appearing in the deco*
*mposition
of *(M) S*(M). Since *(M) = k M M* k, it is enough to compute the
cohomologies of the modules appearing in the decomposition (4.9) of S*(M), plus*
* the
cohomologies of the tensor products of these modules with M and M*.
The first thing to understand is how the tensor products of the modules k, M,*
* N, K,
W , and F with the modules M and M* break up into direct sums of indecomposable*
*s.
The answers are displayed in the table below. A couple of new modules appear;*
* they
are described below the table.
__||___M____||__M*______
k || M || M*
M || Y9 ||F k
N || F N || F N
K || W || W *
W ||F X10 ||F 2 K
F || F 3 || F 3
The module X10 fits in an exact sequence X10 ,! F F i W . The module Y9 is by
definition M M, and there is an exact sequence M ,! T F i Y9. This second exa*
*ct
sequence is constructed by taking the exact sequence k ,! T i M and applying - *
*kM.
Here we note that T is the permutation module U3= < b >.
The next step is to compute cohomologies for each of the modules listed in th*
*e table;
we will need Poincare series and degrees of generators for the cohomologies H*(*
*U3; X) as
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 15
modules over the cohomology H*(U3; k), for each indecomposable module X. We dis*
*play
the results in the form of a table and give the methods of proof afterwards.
_Module_||Poincare_SeriesB||ound_on_Generator_Degrees_
k || __1__(1-x)2|| 0
M || __1__(1-x)2|| 1
M* || __1__(1-x)2|| 1
N || _1__(1-x)|| 0
K || __1__(1-x)2|| 1
W || 1 + __x__(1-x)2|| 2
W * || _2-x_(1-x)2|| 1
F || 1 || 0
2
Y9 || 2-2x-x_(1-x)2|| 2
2
X10 ||2 + x + _x___(1-x)2|| 3
Now we present brief arguments for these cohomology computations.
In the case of a permutation module, i.e. the cases k, N, K, and F , the coho*
*mology
is just the cohomology of the appropriate subgroup, regarded as a module over t*
*he co-
homology of U3 via the restriction map. The degrees of the module generators c*
*an be
worked out from a knowledge of the restriction image.
Now we turn to the more subtle cases. First let us note that there is an exac*
*t sequence
k ,! T i M. In fact, more is true, there is also an exact sequence M ,! N i k, *
*and
by dualizing we can get exact sequences for M*. We'll just work with the first*
* exact
sequence.
Lemma 5.1. The cohomology of M, i.e. H*(U3; M), has Poincare series (1 - x)-2*
* and
is generated by elements in degrees 0 and 1.
Proof. Consider the long exact sequence in cohomology arising from the short ex*
*act se-
quence k ,! T i M. The maps Hi(U3; k) ! Hi(U3; T ) are just the restriction h*
*o-
momorphisms for < b >= Z2 U3, which can be shown to be surjective in all de-
grees i. This implies that Hi(U3; M) is isomorphic to the kernel of the restri*
*ction map
Hi+1(U3; k) ! Hi+1(U3; T ) for all i 0. This means that we can write the Poinc*
*are series
of the cohomology of M as
1 1 1
x-1 ________- _______ = ________:
(1 - x)2 (1 - x) (1 - x)2
*
* __
The module generators may be taken to be in degrees 0, 1. *
* |__|
Lemma 5.2. The cohomology of M*, i.e. H*(U3; M*), has Poincare series (1-x)-2*
* and
is generated by elements in degrees 0 and 1.
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 16
*
* __
Proof. M and M* are interchanged by an outer automorphism of U3. *
* |__|
Lemma 5.3. H*(U3; W ), has Poincare series 1+x(1-x)-2 and is generated by ele*
*ments
in degrees 2.
*
* __
Proof. There is an exact sequence W ,! F i K. *
*|__|
Lemma 5.4. H*(U3; W *), has Poincare series (2 - x)(1 - x)-2 and is generated*
* by ele-
ments in degrees 1.
*
* __
Proof. There is an exact sequence K ,! F i W *. *
* |__|
Lemma 5.5. H*(U3; Y9), has Poincare series (2 - 2x + x2)(1 - x)-2 and is gene*
*rated by
elements in degrees 2.
Proof. There is an exact sequence M ,! T M* i Y9, and we consider the associat*
*ed
long exact sequence in cohomology. Since T M* T F , we see that Hi(U3; T F *
*) =
Hi(U3; T ) is one-dimensional if i > 0. Furthermore, the induced maps Hi(U3; M*
*) !
Hi(U3; T ) are surjective for all i > 0. This, plus the fact that the socle of*
* Y9 is two-
dimensional, determines the Poincare series. For the information on the degree*
*s of the
generators, we must study the kernel of the map H*(U3; M) ! H*(U3; T ). This ke*
*rnel_is
generated by elements in degrees 2. *
* |__|
Lemma 5.6. H*(U3; X10), has Poincare series 2 + x + x2(1 - x)-2 and is genera*
*ted by
elements in degrees 3.
*
*__
Proof. Use the exact sequence X10,! F F i W . |*
*__|
6. The E2-term for the Sylow 2-subgroup of the Higman-Sims Group
The algebraic computations in the preceding section will now be assembled to *
*describe
the E2-term of the Lyndon-Hochschild-Serre spectral sequence associated to the *
*group
extension S = 43 : D8. It can of course be described precisely as H*(U3; *(M) *
*S*(M)).
To start we make a table of the modules appearing in the E2-term, which we re*
*gard as
*(M) S*(M), and the Poincare series of their cohomologies.
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 17
_Module__||Decomposition_||Poincare_SeriesD||egree_Bound_
k || k || __1__(1-x)2|| 0
M || M || __1__(1-x)2|| 1
M* || M* || __1__(1-x)2|| 1
2
M M || Y9 || 2-2x+x_(1-x)2|| 2
M M* || F k || 1 + __1__(1-x)2|| 0
N || N || _1__(1-x)|| 0
N M || F N || 1 + __1_(1-x)|| 0
N M* || F N || 1 + __1_(1-x)|| 0
K || K || __1__(1-x)2|| 1
K M || W || 1 + __x__(1-x)2|| 2
K M* || W * || _2-x_(1-x)2|| 1
W || W || 1 + __x__(1-x)2|| 2
2
W M || F X10 ||3 + x + _x___(1-x)2||3
W M* || F F K || 2 + __1__(1-x)2|| 1
F || F || 1 || 0
F M || F 3 || 3 || 0
F M* || F 3 || 3 || 0
Now we must describe the propagation of these modules in the cohomology of (Z*
*4)3.
This follows from the description of S*(M) in section 4. It is important to not*
*e, however,
that we have doubled all degrees in the symmetric algebra. For this reason, we *
*will refer
to the invariants as d2, d4, and d8, by abuse of notation. We want to compute t*
*he Poincare
series of the E2-term, and what we shall do is compute the Poincare series of t*
*he "tensored
with a trivial" part first, then compute the Poincare series of the "tensored w*
*ith M" part,
and then compute the Poincare series of the "tensored with M*" part. Finally we*
* combine
these pieces.
We make a table for the "tensored with a trivial part". Since there are two *
*trivial
modules in *(M), we just consider the one in degree 0 and then multiply our Poi*
*ncare
series by 1 + x3. In the table below, "Degree" means the degree in which the pr*
*opagated
module first appears in the symmetric algebra, and "C-Deg." means the degree i*
*n the
E2-term after which we know no further generators in cohomology appear.
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 18
_Module_||Propagators_||Degree_||_Poincare_Series_____||C-Deg._
k || d8 || 0 || _1__1-x8. __1__(1-x)2||0 + 0 = 0
2 1
M || d8 || 2 || _x__1-x8. _____(1-x)2||2 + 1 = 3
4 1
N || d8, d2 ||4 || ____x_____(1-x8)(1-x2).4___1-x+||0 =*
* 4
4 1
K || d8, d4 ||4 || ____x_____(1-x8)(1-x4).4_____(1-x)2||*
*+ 1 = 5
6 x
W || d8, d4 ||6 ||__x_____(1-x8)(1-x4). (16++_____(1-x)2*
*)2||= 8
8
F ||d8, d4, d2 ||8 || ______x_______(1-x8)(1-x4)(1-x2)||8 *
*+ 0 = 8
Thus the total Poincare series for the "tensored with a trivial" part is:
1 1 x2 1 x4 1
(1 + x3) . ______. ________+ ______. ________+ _______________. _____+
1 - x8 (1 - x)2 1 - x8 (1 - x)2 (1 - x8)(1 - x2) 1 - x
_______x4______ 1 x6 x x8
. ________+ _______________. 1+ ________+ ______________________
(1 - x8)(1 - x4) (1 - x)2 (1 - x8)(1 - x4) (1 - x)2 (1 - x8)(1 - x4)(1 - x*
*2)
Note that the maximum degree of an algebra generator for the E2-term is 8. Al*
*though
we are tensoring with a trivial module in degree 3, this is really just multipl*
*ying by an
element of the E2-term and so no new algebra generators in degrees 11 or greate*
*r are
produced.
Now let's move on to the "tensored with M" part.
We produce the desired information in the form of a table and a Poincare seri*
*es, as
above.
_Module_||Propagators_||Degree_||____Poincare_Series_______||C-Deg._
M || d8 || 1 || _x__1-x8. __1__(1-x)21 ||+ 1 = 2
3 2-2x+x2
M M || d8 || 3 || _x__1-x8. _______(1-x)23||+ 2 = 5
5 1
N M || d8, d2 ||5 || ___x______(1-x8)(1-x2). (15++___1-x)0 =*
* ||5
5 x
K M || d8, d4 ||5 || ___x______(1-x8)(1-x4). (15++_____(1-x)2*
*)2 =||7
7 x2
W M || d8, d4 ||7 ||___x_____(1-x8)(1-x4). (37++x3+=_____(1-x*
*)2)1||0
9
F M || d8, d4, d2 ||9 || ______3x______(1-x8)(1-x4)(1-x2)||9 +*
* 0 = 9
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 19
Thus the total Poincare series for the "tensored with M" part is:
__x___ 1 x3 2 - 2x + x2 x5 1
. ________+ ______. ___________+ _______________. (1 + _____) +
1 - x8 (1 - x)2 1 - x8 (1 - x)2 (1 - x8)(1 - x2) 1 - x
______x5_______ x
. (1 + ________) +
(1 - x8)(1 - x4) (1 - x)2
______x7_______ x2 3x9
. (3 + x + ________) + ____________________*
*__
(1 - x8)(1 - x4) (1 - x)2 (1 - x8)(1 - x4)(1 -*
* x2)
Now we do the "tensored with M*" part:
__Module__||Propagators_||Degree_||Poincare_Series_____||C-Deg.___
2 1
k M* || d8 || 2 || _x__1-x8. _____(1-x)2||2 + 1 = 3
4 1
M M* || d8 || 4 || _x__1-x8. (1 + _____(1-x)2)4 + ||0 = 5
6 1
N M* || d8, d2 ||6 ||___x______(1-x8)(1-x2).6(1++0___1-x)= |*
*|6
6 2-x
K M* || d8, d4 ||6 || ____x_____(1-x8)(1-x4).6_____(1-x)2||+*
* 1 = 7
8 1
W M* || d8, d4 ||8 ||__x______(1-x8)(1-x4). (28++_____(1-x)2*
*)1||= 9
10
F M* || d8, d4, d2 ||10 || ______3x______(1-x8)(1-x4)(1-x2)||10 *
*+ 0 = 10
Thus the total Poincare series for the "tensored with M*" part is:
__x2__ 1 x4 1 x6 1
. ________+ ______. (1 + ________) + _______________. (1 + _____) +
1 - x8 (1 - x)2 1 - x8 (1 - x)2 (1 - x8)(1 - x2) 1 - x
_______x6______ 2 - x x8 1 3x10
. ________+ _______________.(2+ ________)+ ______________________
(1 - x8)(1 - x4) (1 - x)2 (1 - x8)(1 - x4) (1 - x)2 (1 - x8)(1 - x4)(1 - *
*x2)
Adding these series up, we obtain the Poincare series for the E2-term. We sum*
*marize
our computation in the following
Theorem 6.1. The Poincare Series for the E2 term of the Lyndon-Hochschild-Ser*
*re spec-
tral sequence for the cohomology of the group extension S = Syl2(HS) = 43 : D8 *
*is given
by the rational function
(1 + x)2(1 - x + x2)(1 + 2x2 - x5)
p(x) = _______________________________:
(1 - x)2(1 - x4)(1 - x8)
Moreover a complete set of algebra generators for the E2 term of the spectral s*
*equence
occur in degrees eight and below.
Using Maple, this series can be expanded to yield
1 + 3x + 7x2 + 14x3 + 23x4 + 34x5 + 48x6 + 65x7 + 84x8 + 105x9 + 131x10+
163x11+ 198x12+ 236x13+ 280x14+ 330x15+ 383x16+ 439x17+ 503x18+ . . .
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 20
Our goal is to analyze the behaviour of this spectral sequence. Our descripti*
*on provides
us with very good control of the generators, especially given that they arise i*
*n low degree.
To make effective use of this we require an explicit computer-assisted calculat*
*ion of H*(S)
in low degrees. This can indeed be implemented; details are provided in a subs*
*equent
section. For now we use this information to prove the main result in this paper.
Theorem 6.2. Let S = (Z=4)3 o U3 denote the Sylow 2-subgroup of HS, the Higma*
*n-
Sims group. The Lyndon-Hochschild-Serre spectral sequence for this semidirect p*
*roduct
collapses at E2 and hence the polynomial p(x) is the Poincare series for H*(S; *
*F2).
Proof. A computer calculation using MAGMA for H*(S; F2) shows that through degr*
*ee
10 the coefficients of the polynomial p(x) agree with the ranks of the cohomolo*
*gy. Now
according to the preceding tables, all algebra generators occur by this degree.*
* Hence using
the multiplicative structure of the spectral sequence we infer that it must col*
*lapse_at E2,
i.e. E2 = E1 . *
* |__|
7. Computer Calculations of the Cohomology of S = Syl2(HS)
The cohomology H*(S; F2) was calculated directly through degree 10 using a co*
*mputer.
The calculation included a determination of the Betti numbers, a minimal set of*
* ring
generators, and a complete collection of relations among the generators in the *
*first ten
degrees. We also obtained the images of the generators under various restrictio*
*ns. Using
the latter information and the analysis of the last section we were able to con*
*struct the
cohomology ring with all generators and relations using computer technology. We*
* give a
summary of the calculations in this section.
The calculation was begun by first obtaining a minimal projective resolution
: :-:! P2 -! P1 -! P0 -! k
of the trivial module k = F2. The process is mostly linear algebra. The free mo*
*dule over
S is generated as a collection of matrices representing the actions of the gene*
*rators of
the group. At each stage in the construction we get a minimal set of generators*
* for the
kernel of the previous boundary map, create the free module with exactly that n*
*umber
of generators, make the matrix for the boundary map from the free module to the*
* kernel
of the previous boundary map - this is the new boundary map - and find its kern*
*el. The
programs are conservative with both time and memory in that they save only the *
*minimal
amount of information necessary to reconstruct the boundary homomorphisms and c*
*reate
module structure only when it is necessary. The method is described in detail i*
*n [5], [4].
Because the resolution is minimal, any nonzero homomorphism fl0 : Pn - ! k is*
* a
cocycle representing a nonzero cohomology class fl. Any such cocycle can be lif*
*ted to a
chain map ^fl: P* -! P* of degree n which also represents fl. Again this is a *
*exercise
in linear algebra on the computer and details of the implementation can be foun*
*d in
the references given above. The point of this operation is that the cup produc*
*t of two
cohomology elements is the class of the composition of the representing chain m*
*aps. Thus
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 21
we can compute the product structure of the cohomology ring. Note that the pro*
*gram
computes the chain maps for only a minimal set of generators for the cohomology*
*. That
is, at each stage it constructs the subspace of Hom kG (Pn; k) that can be obta*
*ined by
compositions with chain maps of lower degrees.
The same implementations have been used to compute the cohomology rings of al*
*l but
a few of the groups of order dividing 64.1 The programs are written in the MAG*
*MA
language and run in the MAGMA computer algebra system [3]. The computations of
the cohomology of S were all run on an SUN ULTRA2200 (named the sloth) that has
approximately 1Gb. of RAM and 14Gb. of hard disk. The computation of the projec*
*tive
resolution out to degree 10 for the trivial module k of S took slightly under 3*
*1 hours. The
computation of the chain maps of the minimal generators of the cohomology ring *
*took
more than 55 hours. The attempt to compute an 11th step in the projective reso*
*lution
failed for lack of memory. Some of the results of this calculation are given in*
* the following.
Note that this is precisely the information that is needed to complete the proo*
*f of Theorem
6.2.
Proposition 7.1. The Betti numbers for the cohomology H*(S; k) through degree *
*10 are
1; 3; 7; 14; 23; 34; 48; 65; 84; 105; 131;
and the degrees of the minimal generators of the cohomology ring through degree*
* 10 are
1; 1; 1; 2; 2; 2; 2; 3; 3; 3; 3; 4; 4; 5; 6; 7; 8:
We also computed the restrictions of the cohomology to the elementary abelian*
* sub-
groups and to the centralizers of the elementary abelian subgroups. The restric*
*tion maps
are obtained by first converting the projective resolution of k for the group t*
*o a projective
resolution of the subgroup and then constructing the chain map, lifing the iden*
*tity on k,
from a minimal projective resolution of k for the subgroup to the converted pro*
*jective
resolution for the group. The cocycles for the cohomology elements of the group*
* are then
pulled back along the chain map. The process is described in more detail in [6].
Of particular interest was the restrictions to the centralizers of the elemen*
*tary abelian
groups of order 4. This calculation was made on the assumption (hope) that the *
*depth
of the cohomology ring H*(S; k) is at least two and hence that the cohomology o*
*f S is
detected on these centralizers. An easy calculation shows the following.
Lemma 7.2. If E is an elementary abelian subgroup of order four in S then the*
* central-
izer of E is contained in the subgroup Kfi(defined in Section 2) or in one of t*
*he three
subgroups
M = ;
D1 = <(v1v2)2; v1v3; t; s>;
___________
1See the second author's web page, http://www.math.uga.edu/"jfc/groups2/cohom*
*ology2.html for the
results and some further discussion of the methods.
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 22
D2 = <(v1v2)2; v1v3; v-12t; ts; v1v-12s2>:
Here M is the centralizer of (v1v2)2, D1 is the centralizer of s2 and D2 is the*
* centalizer
of v1v-12s2. The orders of M is 256 while the orders of D1 and D2 are 64.
The computer was able to calculate the cohomology ring of all four of the gro*
*ups
in the Lemma. In the case of Kfiit was known that the cohomology is detected on*
* the
centralizers of the maximal elementary abelian subgroups and this aided the com*
*putation.
In all three of the other cases the minimal set of generators for the cohomolog*
*y ring
contained no element of degree greater than 4 and the computation was possible *
*even
though all three of the groups have 2-rank 4 and the Betti numbers of the cohom*
*ology
grow rapidly. In fact, the cohomology rings of D1 and D2 are included in the ca*
*lculations
of the second author (see the web page). These groups have Hall-Senior numbers *
*170 and
110 respectively.
We were able to compute the restriction maps and then get the kernels of the *
*restriction
to each of the subgroups. Then the intersection of the kernel of the restrictio*
*n maps was
computed. Actually the problem of getting the intersection of the kernels seeme*
*d to be too
difficult for the computer to attack directly using Gr"obner basis machinery. I*
*nstead we
employed an indirect method of turning the restriction maps into linear transfo*
*rmations
on the spaces of monomials in each degree and computing the intersection of the*
* kernels
out to degree 17. Thus we had a complete set of generators for the intersectio*
*n of the
kernels out through degree 17.
There is one uncertainty that should be noted. The cohomology of M was only c*
*alcu-
lated out to degree 10 and that is not far enough to pass our test for complete*
*ness of the
calculation [4]. Nonetheless, the generators of the cohomology of S are in degr*
*ees at most
8 and so their restrictions to M are expressed as polynomials in the calculated*
* generators
of the cohomology of M. Any polynomial in those generators which is computed to*
* be in
the kernel of restriction to M is, in fact, in that kernel. Thus we get the fol*
*lowing result,
which is used in the next section.
Proposition 7.3. The ideal I of Theorem 8.1 is generated by the elements of de*
*gree at
most 17 that are in the intersection of the kernels of the restriction maps to *
*Kfi, M, D1
and D2. Hence the ideal is in the intersection of the kernels of the restricti*
*ons to the
centralizers of the elementary abelian subgroups of order four.
In the next section we argue that this information together with some extra v*
*erification
is sufficient.
8. Generators and Relations for the Cohomology of S
Our purpose in this section is to give a proof of the following.
Theorem 8.1. The cohomology of S has the form
H*(S; F2) ~=F2[z; y; x; w; v; u; t; s; r; q; p; n; m; k; j; i; h*
*]=I
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 23
where the degree of z; y; x is 1, the degree of w; v; u; t is 2, the degree of *
*s; r; q; p is 3, the
degree of n; m is 4 and the degrees of k; j; i; h are 5; 6; 7 and 8 respectivel*
*y. The ideal I is
generated by the relations in the appendix.
Let R = F2[z; y; x; w; v; u; t; s; r; q; p; n; m; k; j; i; h] be the polynomi*
*al ring in 17 vari-
ables in weighted degrees as in the theorem. For n 0 let Rn be the space of ho*
*mogenous
polynomials of degree n. From Proposition 7.1 we know that there is a natural h*
*omomor-
phism OE : R -! H*(S; F2) taking each variable to the computed cohomology eleme*
*nt.
Let J be the kernel of OE. Our task is to establish that I J .
Recall that the following is a complete list of representatives for the conju*
*gacy classes
of centralizers of maximal elementary abelian subgroups in S:
1. CI = 43, a characteristic subgroup of S
2. CII = 22 x 4,
3. CIII = 22 x 4,
4. CIV = 22 x 4,
5. CV = 22 x 4,
6. CV I = 22 x 4,
7. 24A,
8. 24B
9. 24C
The following basic detection result involving these groups, these groups was*
* proved by
computer calculation.
Lemma 8.2. Any homogeneous polynomial of Rn in degree n 20 is in I if and on*
*ly if
it is in the kernel of restriction to all of the centralizers listed above.
Proof. First it was shown that the elements of I are in the kernels of restrict*
*ion by com-
puting the restriction maps on the elements. Then for each degree n 20 the res*
*triction
maps Rn=In -! H*(C; F2) for C in the list was written as a linear transformatio*
*n and __
the intersection of the null spaces was computed to be zero *
* |__|
Proof of Theorem 8.1.From the lemma we see that J n In for n 20. But by the
Poincare series (Theorem 6.2) we have that Rn=In has the same dimension as Rn=J*
* n~=
H*(S; F2). On the other hand we know that I is generated by elements of degree*
* at
most 14. So I J , since all generators are in J . So OE induces a surjective h*
*omomor-
phism R=I -! H*(S; F2). But again because the Poincare series are the same this*
* is_an
isomorphism. *
*|__|
9. Appendix I: Relations in the cohomology of S
The following is a list of the relations in the cohomology ring H*(S; F2). Th*
*ese elements
generate the ideal I in Theorem 8.1. Note that there are 79 relations generatin*
*g I and
that the relations are minimal in the sense that no collection of fewer than 79*
* elements will
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 24
generate the ideal. This set is not a Gr"obner basis for the ideal. The compute*
*d Gr"obner
basis using the grevlex ordering on the variables consisted of 884 elements in *
*degrees up
to 73. The elements in the list are ordered by degree. The largest degree of an*
*y element
in the list is 14.
yx; zx; zy; xv; xw; yw + yv + yu; zt; zw; vt; xs;
yq + xq + w2 + wu; yq + wv; yr + xr + xq + xp + ut; yr + wt; ys + yq;
y2u + yq + xq + wu; zp + yq + xq + v2 + vu; zr; zs + zq + xq;
tq; ts; vr; vs + vq; ws + wq; x2r + xn + wp + vp + up;
x2r + xn + wr + ur; yu2 + yn + wp; yvu + ws;
y2r + wr; z2p + zvu + zn + ws + vs + us;
z3u + z2p + zvu + zu2 + zn + zm + x2r + xu2 + xn + xm;
rq; sp + qp; sq + q2; sr; s2 + sq; xuq + vu2 + vn + sp;
ytr + x3p + x2n + xup + xtp + r2; ytr + x3p + xuq + xtr + xk + r2;
yur + x3p + xuq + xk + tn + rp; y2n + yvp + yuq + wn;
z3p + z2n + zvq + zuq + y2m + ytr + ytp + yk + x3p + xk + vn + vm + tn + *
*tm + p2;
z3p + zvq + zup + yuq + xuq + vu2 + sq; z3q + z3p + zuq + zk; sn + qn;
ytn + xtm + xp2 + rn; yvn + yqp + vuq + vup + qn;
y5v + y4q + y4p + y3n + y3m + y2vq + y2tp + y2k + yum + ytn + yr2+ yrp + *
*yj + wk;
zvm + zq2 + zj + yvm + yum + yr2 + yrp + x3u2 + x3m + vup + vk + u2q + u2*
*p +
uk + sm + rn + rm + pn;
z2vq + zvm + zq2 + zqp + zj + yvn + yqp + vuq + u2s;
z3u2 + zu3 + zum + yvn + yqp + x3u2 + x3n + xu3 + xum + xtn + xj + vuq + *
*u2s;
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 25
z5u+z3m+zu3+zum+x3n+x3m+xu3+xum+xtn+xj +wk+vk+u2s+u2q+uk;
y4m + y3k + y2vn + y2vm + y2p2+ yt2p + ytk + yrm + x4u2+ x3k + x2un + x2t*
*n +
x2j + xu2p + xtk + xrm + xpn + tr2 + tj + rk;
y5p + y3k + y2vn + y2vm + yrm + ypm + yi + x4u2 + x3k + x2un + x2tn + x2j*
* +
xt2p + xqm + wun + wum + wj + tr2 + sk + qk;
zpn + ypn + xpn + wun + vun + vp2 + uqp + n2;
z2q2 + zuk + zqn + zqm + y4m + y3vq + y3tp + y3k + y2vm + y2tm + y2p2+ yq*
*m +
ypn + x2un + xuk + xrm + wun + wj + vum + vp2 + vj + up2 + qk;
z3k + z2vm + z2q2 + zi + y3tp + y2tm + yuk + yt2p + ytk + yqm + x3k + x2u*
*3 +
x2un + x2tn + xt2p + xi + wum + vun + vum + urp + up2 + tr2 + tj + sk + rk + qk;
z3k + z2vm + zqn + zpn + zi + y5q + y4n + y2q2+ yqm + x2u3+ xu2p + xuk + *
*xrm +
wj + vq2 + vp2 + u2n + uq2 + uj + sk;
z4u2+ z2u3+ z2un + z2um + z2q2 + zqn + y5q + y4n + y2q2 + yqm + wun + vun*
* +
vq2 + vp2 + vj + uq2 + sk;
z5p + z3k + z2un + z2j + zqn + zi + y3vq + y2vn + yuk + yrm + ypn + x2u3+*
* xuk +
xqm + wum + vp2 + u2n + urp + uj + sk;
y5n + y3q2 + y2vk + y2qm + yvqp + yup2+ yuj + yt2n + yrk + yqk + ynm + wu*
*k +
wqn + wpn + wi + trm + sj + r2p + rp2 + qj;
z2pn + zuq2 + zuj + zn2+ znm + y6p + y5m + y3vm + y3tm + y2qn + y2qm + y2*
*i +
yvqp + yvj + yt2n + yt2m + ytj + yrk + ypk + ynm + wqn + wpm + wi + vpm + u3q +
u2k + urn + trn + q3 + pj + nk;
z2qn + zuj + zn2 + znm + yup2 + yt2n + yrk + ynm + xpk + wpn + wpm + vuk +
vqm + u3q + u2k + upn + trm + sj + rp2 + rj + qj;
z3un + z2qn + z2pn + zvj + zuq2 + zuj + znm + sj + qj;
y2tk + y2qm + yup2+ yt2m + ytp2+ ytj + yqk + ynm + x3tn + xrk + xn2+ wuk +
wqn + wpm + t3r + trn + tpm + ti + sj + r2p + rp2 + qj;
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 26
z3vn + z3un + z2vk + z2pn + zvj + y5n + y4vq + y3vm + y2qn + y2pn + yvq2+*
* yvqp +
yup2 + yn2 + xnm + wuk + vqm + vpm + vi + u3q + u3p + u2k + upn + q3;
z3vn+z2qn+z2pn+zvj+y5n+y2vk+y2tk+y2qn+yvqp+yup2+yt2n+yt2m+ytp2+
ytj +yrk+yn2+x3tn+wpn+wpm+wi+vuk+vqm+urn+t3r+trm+tpm+ti+sj +r2p;
y3t2p + y2t2m + yt3p + yt2k + yr2p + yrp2 + x2u2n + x2t2n + xrj + t3n + t*
*2rp +
t2j + trk + r2n + r2m + rpm + ri;
z2uq2+ z2qk + zpj + y5vq + y4j + y3vk + y3t2p + y3qn + y3qm + y3pm + y3i *
*+ y2vq2+
y2vj + y2t2m + y2tp2 + y2tj + y2pk + y2h + yvpn + yupn + yt3p + yt2k + yr2p + y*
*qp2 +
yqj + x2t2n + xu3p + xtpn + xrj + xpj + wqk + wpk + wnm + vup2 + upk + t3n + t2*
*p2 +
t2j + trk + r2n + r2m + rpn + rpm + q2n + p2n + pi + k2;
zuqn + y7q + y4q2+ y3vk + y3qm + y2vq2+ y2vj + yvqm + yvpn + yvi + yupn +*
* yqj +
x2u2n+x2nm+xu3p+xpj +wqk+wpk+vuj +upk+un2+tn2+r2n+q2n+qpn+qpm+nj;
z3uk + zuqm + zupm + zmk + y3vk + y3qn + y3pm + y2tp2 + y2qk + yvpn + yvi*
* +
yupn + yupm + yr2p + yq2p + yqj + ypj + x2u2n + xpj + wpk + vup2+ upk + t2rp + *
*rpn +
qpm + p2n + p2m + pi;
z3uk + z2u2n + z2uj + z2qk + zupm + zui + zpj + y3qn + y3qm + y3pn + y2vq*
*2 +
y2nm + yq2p + x3uk + x2u4+ x2nm + xu3p + xui + xrj + wpk + wnm + vuj + usk + up*
*k +
un2 + tn2 + si + r2n + qpn + nj;
z4vm+z4j+z3vk+zuqn+zuqm+zupm+zpj+y7q+y4q2+y3vk+y3qn+y3pn+y2vj+
y2nm+yvpn+yvi+yq2p+yqj +x2u2n+x2nm+vup2+u3n+usk +un2+unm+qpn+qi;
z2u2k + z2uqm + z2q3 + z2qj + zvqk + zu3n + zu2j + zupk + zun2+ y6vq + y4*
*qm +
y4pn + y3tp2+ y2qj + yvpk + yvn2+ yvnm + yupk + yq2n + yqpm + yqi + x2u2k + x2n*
*k +
xunm + xtpk + xtnm + xri + xnj + wupm + wpj + wnk + vq2p + vpj + urp2 + unk +
t2rn + t2pm + trp2 + tp3 + tpj + rpk + qpk + qnm + pnm + ni;
z2vqm+z2u2k+z2uqm+z2q3+zvqk+zu3n+zu2j+zun2+y9v+y6vq+y6k+y5vm+
y5tm+y5q2+y4qn+y4qm+y3tp2+y3tj +y3pk +y3nm+y3m2+y3h+y2vi+y2qj +y2nk +
yvpk+yvnm+yvm2+yvh+yupk+yuh+ytm2+x2mk+xunm+xtnm+xqi+wupm+wpj+
wnk+wmk+vmk+urj+uqp2+upj+unk+t2pm+tp3+r2k+rn2+qpk+qn2+qnm+ni+kj;
z2q2n + zvqj + zvmk + zupj + zqn2 + zni + y8m + y7vq + y7tp + y7k + y6tm +
y5vk + y5qn + y4pk + y4nm + y3p3+ y2vnm + y2vh + yvq3+ yvqj + yvnk + yvmk + yup*
*3+
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 27
yumk + ytrj + ytnk + ytmk + yrpk + yqm2+ yp2k + ypnm + yni + ykj + x2mj + xqm2+
xkj + wum2+ wq2m + wpi + vqpn + vp2m + u2qk + u2pk + u2n2+ urpm + uri + uq2m +
up2n + t2nm + t2m2 + tr2n + tp2m + rpj + q4 + q3p + q2j + qp3 + p2j + pnk + j2;
z5pm+z2q2n+z2qi+zvqj +zupj +zpm2+zkj +y8m+y7vq +y7k +y6vm+y5tk +
y5pm + y4pk + y3vqn + y3t3p + y3ti + y2vnm + y2vh + y2th + y2q2m + y2k2+ yvqj +*
* yvnk +
yumk +yt4p+yt3k +ytnk +ytmk +yrpk +yrn2+yq2k +yqn2+yp2k +ypnm+yni+ykj +
x8u2+x8m+x7k +x6u3+x5tk +x5i+x4tj +x2t2j +x2mj +xt2pn+xni+wun2+wum2+
wq2m+wmj +vum2+vqpn+vnj +vmj +u2qk +uri+uqi+up2n+upi+uk2+t4n+t3rp+
t3p2+t3j+t2nm+t2m2+trpm+tk2+r3p+r2p2+rpj+q4+q3p+q2j+qp3+p2j+pmk+ki+j2;
z11u + z9u2+ z9m + z7vn + z6vk + z6pm + z3vnm + z3un2+ z3nj + z2unk + z2p*
*m2+
z2mi+z2kj+zumj+y7p2+y6vk+y4pj+y4nk+y3qi+y3pi+y2vnk+y2tmk+y2q2k+y2qnm+
y2qh+y2p2k+yvq2m+yvqpm+yvqi+yvk2+yup2n+yup2m+yumj +yuk2+ytri+yr2p2+
yr2j+yq4+yq2j+yqpj+yn2m+x6uk+x5u2m+x4ui+x4mk+x3nj+xu4n+xu2nm+xupi+
xumj+xtnj+xpnk+wunk+wq2k+wmi+vupj+vumk+vqnm+u5q+u4k+u2p3+u2nk+
usm2+uq2k+uqn2+t3rn+trnm+tpnm+tpm2+q3n+qp2n+qnj +p3m+pnj +nmk+ji;
z11p + z5vi + z5u2k + z4pi + z3umk + z2qmk + z2j2 + zvqn2 + zuqnm + zumi +
zukj + zq3n + zm2k + y10n + y9tp + y8q2+ y8j + y7tk + y7qn + y7i + y6vj + y6tj *
*+ y6nm +
y6h + y5vi + y5ti + y4vn2+ y4vm2+ y4mj + y4k2+ y3vqj + y3tmk + y3p2k + y3mi + y*
*2vqi +
y2vnj + y2t2m2+ y2q2j + y2n2m + y2mh + y2ki + yvq2k + yvqm2+ yvqh + yvpnm + yvm*
*i +
yupn2+yuph+yuni+yt3pm+yt2pj +ytp2k +ytmi+yr2pm+yq3n+yp2i+yn2k +x8u3+
x7uk + x6u2m + x6uj + x4um2+ x3mi + x3kj + x2u4n + x2umj + x2j2 + xu2mk + xrk2 +
xji + wunj + wqnk + wnm2+ wnh + vumj + vq2p2+ vqmk + vn2m + vj2+ u2qi + u2p2m +
u2mj + urpj + urmk + uqnk + upmk + un3+ unm2+ uki + t3nm + t2r2m + t2rpm + t2ri*
* +
t2p2n + tr2j + trp3 + trnk + tp4 + skj + r3k + r2pk + rpn2 + rni + qmi + pni + *
*mk2 + i2
10. Appendix II: Steenrod Operations
In this appendix we describe the Steenrod operations on the cohomology genera*
*tors
listed previously. As before this was done on a computer using MAGMA. Note that*
* the
program used puts all of the polynomials in "`normal form" relative to the Groe*
*bner basis
of the ideal of relations.
Sq1w = yv; Sq1v = zv + yv; Sq1u = zu + xu; Sq1t = yt + xt
Sq1s = vu + yq + xq; Sq2s = z2q + us + zn
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 28
Sq1r = xr + xp; Sq2r = ur + tr + xn
Sq1q = vu + yq + xq; Sq2q = xu2 + z2q + uq + wp + vp + up + zn + xm
Sq1p = zp; Sq2p = z2q + z2p + y2p + tr + wq + uq + vp + ym + k
Sq1n = x2p + wp + vp; Sq2n = y4v + y3q + y3p + zvq + yvq + xuq + yvp + zu*
*p +
yup + ytp + z2n + y2n + x2n + y2m + x2m + q2 + rp + qp + tn + vm + yk + xk + j
Sq1m = zu2 + xu2 + y2p + us + vq + up + zm + xm
Sq2m = y4v + y3q + y3p + u3 + zvq + xuq + yvp + xup + xtp + z2m + y2m + r*
*2 +
q2 + rp + vn + un + wm + vm + um + tm + xk + j
Sq1k = y3p + ytr + zvq + yvq + yvp + yup + xtp + z2n + y2m + qp + p2 + vn
Sq2k = y4p + xu3 + x3m + u2s + u2p + yrp + yp2 + zun + yun + zvm + yvm +
xum + xtm + y2k + x2k + rn + qn + pn + sm + rm + vk + xj
Sq4k = xu4 + y4k + yvq2 + xt2n + z2pn + xu2m + yt2m + y2vk + y3j + r2p +
qp2 + p3 + urn + uqn + tpn + zn2 + uqm + wpm + upm + znm + xnm + ym2 + wuk +
vuk+u2k+t2k+yrk+xrk+zqk+yqk+xpk+xuj +ytj +xtj +z2i+x2i+mk+pj +vi+yh
Sq1j = y4q + y2vq + y3n + y3m + x3m + zq2 + vup + zqp + yvn + ytn + zvm +
ytm + y2k + qn + pn + vk + zj + yj + xj
Sq2j = y2p2 + x2tn + z2vm + ur2 + vq2 + tp2 + t2n + zqn + vum + t2m + zqm*
* +
xqm + zpm + zvk + yvk + zuk + xtk + z2j + n2 + sk + vj
Sq4j = y2t2m + y3pm + z4j + x4j + yq3+ t2rp + yqp2+ yp3+ yvqn + yupn + y2*
*n2+
yvqm + yupm + z2nm + xu2k + z2qk + z2vj + y2tj + x2tj + y3i + qpn + un2+ q2m + *
*qpm +
wnm + tnm + wm2 + vm2 + tm2 + urk + trk + wqk + uqk + wpk + upk + znk + xnk +
ymk + vuj + yrj + zqj + zpj + xpj + xti + y2h + nj + mj + qi + pi + wh
Sq1i = y2q2+ y2p2+ z2vm + y2tm + ur2+ tr2+ wq2+ urp + trp + vqp + vp2+ tp*
*2+
vun + xrn + yqn + zpn + xpn + vum + t2m + yqm + xqm + zpm + ypm + zuk + xuk +
z2j + n2 + sk + rk + wj + vj + tj + yi
Sq2i = y3tm + yvq2 + xt2n + y2pn + y2tk + q2p + rp2 + qp2 + p3 + vqn + uq*
*n +
zn2+ xn2+ urm + trm + wqm + uqm + wpm + vpm + upm + tpm + ynm + xnm + ym2+
THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 29
xm2 + vuk + xrk + zqk + xpk + zvj + xuj + z2i + x2i + nk + mk + sj + rj + qj + *
*vi + ti
Sq4i = y4tk + xu3m + yt3m + y3vj + y3tj + z4i + y4i + x4i + vq3 + vupn + *
*yqpn +
yvn2 + yun2 + t2pm + yqpm + xunm + yum2 + xum2 + ytm2 + u3k + xupk + z2nk +
y2nk + z2mk + y2mk + yt2j + xt2j + z2qj + y2qj + z2pj + y2vi + y2ti + rn2 + pn2*
* +
rnm + rm2 + qm2 + r2k + vnk + tnk + vmk + umk + tmk + yk2 + urj + wqj + vqj +
uqj +wpj +ynj +xnj +zmj +ymj +xmj +t2i+yqi+ypi+xpi+yvh+kj +ni+mi+sh+qh
Sq1h = z7v + x7u + y2qn + z2pn + y2pn + yt2m + y2pm + y2vk + y2tk + x2tk *
*+ y3j +
rp2 + p3 + vpn + upn + tpn + xn2 + usm + trm + wqm + vqm + wpm + vpm + xnm +
wuk + vuk + yrk + zpk + ypk + xpk + zvj + yvj + zuj + yuj + xuj + x2i + nk + pj
Sq2h = z8v+x8u+u5+y3pm+y3tk+t2rp+yrp2+yqp2+yupn+y2n2+u3m+yvqm+
ytpm + z2nm + y2nm + x2nm + z2m2+ y2m2+ x2m2+ xu2k + xt2k + z2qk + y2vj + x2uj +
y2tj+x2tj+x3i+r2n+q2n+p2n+wn2+vn2+un2+q2m+rpm+qpm+wnm+vnm+trk+
wqk +uqk +wpk +upk +ynk +xnk +zmk +vuj +zqj +ypj +yvi+xui+yti+xti+y2h+pi
Sq4h = z6j +x6j +u6+y2t3m+y3t2k+x5i+t3p2+t4n+t4m+z2um2+xt3k+y3nk+
y3pj+y3vi+x3ti+z4h+y4h+x4h+q4+p4+vq2n+tp2n+u2n2+yrn2+yqn2+ypn2+tr2m+
vq2m+vqpm+vp2m+tp2m+vunm+u2nm+t2nm+yrnm+ypnm+wum2+vum2+u2m2+
yqm2+xqm2+ypm2+zq2k+yrpk+yqpk+zvmk+zumk+yumk+u3j+t3j+yvpj+xupj+
ytpj+xtpj+y2nj+x2nj+z2mj+xu2i+xt2i+z2qi+y2qi+y2pi+x2pi+y2vh+z2uh+nm2+
m3+pmk +wk2+vk2+r2j +q2j +rpj +vnj +unj +tnj +wmj +tmj +zkj +ykj +xkj +
usi+uri+vqi+uqi+wpi+upi+zni+yni+ymi+vuh+t2h+yrh+xrh+zph+ki+nh+mh
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THE COHOMOLOGY OF THE SYLOW 2-SUBGROUP OF THE HIGMAN-SIMS GROUP 30
Mathematics Department, University of Wisconsin, Madison WI 53706
E-mail address: adem@math.wisc.edu
Mathematics Department, University of Georgia, Athens GA 30602
E-mail address: jfc@math.uga.edu
Mathematics Department, University of Wisconsin, Madison WI 53706
E-mail address: dikran@math.wisc.edu
Mathematics Department, Stanford University, Stanford CA 94305
E-mail address: milgram@math.stanford.edu
~~