is much interest in the ring structure of the mod p cohomology H*(G,Zp)
of p-groups G.
At present these HAP functions work differently to those for integral cohomology in that they rely heavily on matrix algebra and minimal resolutions. More work needs to be done on improving the effeciency of these functions.
G be the group G=SmallGroup(64,135) in the small groups
library. The following HAP commands compute the ring H*(G,Zp)
modulo all elements of degree greater than 10. The ring is returned as
a structure constant algebra A over the field of two elements.
Resolution of length 10 in characteristic 2 for <pc group of size 64 with
6 generators> .
No contracting homotopy available.
A partial contracting homotopy is available.
<algebra of dimension 187 over GF(2)>
|The following additional command shows that the ring H*(G,Zp) is generated by three elements in degree 1, two in degree 2, one in degree 3, one in degree 5, one in degree 8 and possibly some generators of degree greater than 10.|
[ 0, 1, 1, 1, 2, 2, 3, 5, 8 ]
|The Poincare series
cohomology ring H*(G,Zp) is the infinite series
a0 + a1x + a2x2 + a3x3 + ...
where ak is by definition the dimension of the vector space Hk(G,Zk) . The Poincare function is a rational function P(x)/Q(x) equal to the Poincare series.
The following commands compute the Poincare function for the Sylow 2-subgroup of the Mathieu group M12. They rely on an algorithm which seems unlikely to produce a wrong answer but for which we have no proof that the answer is always correct.
|Poincare series for groups of order 32 and most of the groups of order 64 are listed here.|