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Let FG be the group algebra of a finite group over the field F of p elements, and let  M be an FG-module.  The abelian groups

TornFG(M,F)
and
ExtnFG(M,F)

can be calculated from a free resolution of M.

We illustrate this for the module M arising from the canonical action of the group G=GL3(2) on the 3-dimensional column vector space over GF(2). The module M can be entered as a meat-axe module using the following standard GAP commands.
gap> G:=GL(3,2);;
gap> M:=GModuleByMats(GeneratorsOfGroup(G),GF(2));;
The module can be converted to an FpG-module DM using the following command. The "desuspended module" DM is mathematically related to M via a short exact sequence

0 → DM → PM → M → 0

where PM is a free module. Thus

TornFG(DM,F)  =  Torn+1FG(M,F)
and
ExtnFG(DM,F)  =  Extn+1FG(M,F)
gap> DM:=DesuspensionMtxModule(M);;
The following commands now compute the 2-dimensional vector spaces

Tor5FG(M,F)  =  Tor4FG(DM,F) = F2
and
Ext5FG(M,F)  =  Ext4FG(DM,F)  = F2 .

gap> R:=ResolutionFpGModule(DM,5);;
gap> p:=2;;
gap> Homology(TensorWithIntegersModP(R,p),4);
2
gap> Cohomology(HomToIntegersModP(R,p),4);
2
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