


The
cohomology of a group G can be defined in terms of resolutions. Let Z
be the group of integers considered as a trivial ZGmodule. A free ZGresolution of Z is a
sequence of ZGmodule homomorphisms ... → M_{n} → M_{n1}
→ ... → M_{1 }→ M_{0}
satisfying:
... → TM_{n} → TM_{n1}
→ ... → TM_{1 }→ TM_{0} .
This sequence will generally not
satisfy the above exactness condition, and one defines the integral homology of
G to be
for all n>0. By changing the
definition of TM_{n}
one arrives at the definition of homology H_{n}(G,A) and
cohomology H^{n}(G,A) with coefficients in a ZGmodule A.
Needless to say, homology and cohomology are invariants of G, and do
not depend on the particular choice of free ZGresolution.
(See David Joyners intoduction to group cohomology for more details on this definition.) 

There
are two steps to computing group
homology:


gap>
F:=FreeGroup(2);;x:=F.1;;y:=F.2;; gap> G:=F/[x^2,y^512,(x*y)^2];; D_512:=Image(IsomorphismPermGroup(G));; gap> R:=ResolutionFiniteGroup(D_512,26); Resolution of length 26 in characteristic 0 for <permutation group of size 1024 with 2 generators> . gap> time; #The time, in milliseconds, for constructing R. 265262 gap> TR:=TensorWithIntegers(R);; Chain complex of length 26 in characteristic 0 . gap> Homology(TR,25); [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] 

We
see that H_{25}(D_{512},Z) = (Z_{2})^{14}.
(Quicker methods for computing this are given on subsequent pages!) Homology with other coefficients can also be calculated from the resolution R. For instance, the following additional command shows that the 25th homology of D_{512} with coefficients in the trivial module Z_{2} is the vector space H_{25}(D_{512},Z_{2}) = (Z_{2})^{26}. 

gap>
T2R:=TensorWithIntegersModP(R,2);; Chain complex of length 26 in characteristic 2 . gap> Homology(T2R,25); 26 

(At
this point we should note that there exist more efficient methods for
computing the homology of a finite group G over the finite field Z_{p}
with trivial action. One approach, which has been used very succesfully by Jon Carlson on small pgroups, is to regard the group ring Z_{p}G as a vector space of rank G. Each term in a Z_{p}Gresolution R is then also a vector space, and one can use linear algebra techniques to construct R. Details of Carlson's computations can be found here. The HAP function ResolutionPrimePowerGroup(G,n) uses this idea to compute a free Z_{p}Gresolution for a pgroup G. The resolution is minimal in the sense that the number of free generators in degree n is equal to the rank of the vector space H_{n}(G,Z_{p}) . For example, the following commands compute the ranks of the first 25 mod 2 homology groups of the Sylow 2subgroup of the Mathieu group M_{12} . 

gap>
G:=SmallGroup(64,134);; #This is the Sylow 2subgroup
of MathieuGroup(12). gap> R:=ResolutionPrimePowerGroup(G,25);; time; 436107 gap> #The dimensions of the first twentyfive mod 2 homology groups of G are: gap> List([0..25],n>Dimension(R)(n)); [ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351 ] 

The
linear algebra approach to mod p cohomology will not work so well on
large
groups G since one ends up having to compute nullspaces of large
matrices. Consider for instance the above computation for G=D_{512}.
Any resolution is going to have at least 26 Z_{2}Ggenerators
in dimension 25, and 25 generators in dimension 24. So to obtain the
26th term one would have
to compute the nullspace of a matrix with (1024)^{2}×26×25
= 681574400 entries. An efficient method for mod p cohomology of larger groups has been developed by David Green. It involves nonabelian Gröbner basis techniques and is described in the book [D.J. Green, Gröbner bases and the computation of group cohomology, Lecture Notes in Math., No. 1828 (Springer, 2003)] and on the corresponding web page.) 

For
integral (or mod p) computations to succeed the ZGrank of the
resolution R
should not
be too large in any given dimension. The construction of small
ZGresolutions is an interesting problem and at first glance might seem
to
be a purely algebraic one. However, we can benefit from the advice
of Sir Michael Atiyah.


