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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 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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