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The cohomology of a group G with coefficients in a ZG-module A is defined as:
Hn(G,A) = | Ker( HomZG(Rn,A) → HomZG(Rn+1,A) ) Image( HomZG(Rn-1,A) → HomZG(Rn,A)
When the abelian group underlying A is free of rank n we can encode A as a group homomorphism A:G → GLn(Z). When G is a permutation group of degree n the free abelian group Zn admits a canonical G-action defined by g·(x1, x2, ... , xn) = (xg'(1) , xg'(2) , ... , xg'(n)) where g'=g-1, for g in G and xi in Z. This canonical permutation module A can be constructed for any permutation group G using the HAP command PermToMatrixGroup(). For example:
gap> G:=AlternatingGroup(5);; gap> A:=PermToMatrixGroup(G,5); [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ], [ 1, 0, 0, 0, 0 ] ], [ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ] ] ]
The following commands show that: the 6th cohomology of the alternating group G=A5 with coefficients in its 5-dimensional canonical permutation module A is H6(G,A) = Z2+Z6. The 3rd cohomology of the even subgroup B+ of the 5-string Braid group, again with coefficients in the permutation module A (considered as a B+-module via the quotient homomorphism B+ → A5) is H3(B+,A) = Z2+Z6+Z3.
gap> Alt5:=AlternatingGroup(5);; gap> A:=PermToMatrixGroup(SymmetricGroup(5),5);; gap> R:=ResolutionFiniteGroup(Alt5,7);; gap> TR:=HomToIntegralModule(R,A);; gap> Cohomology(TR,6); [ 2, 6 ] gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]]];; gap> R:=ResolutionArtinGroup(D,10);; gap> Brd5:=R!.group;; Brd5Gens:=GeneratorsOfGroup(Brd5);; gap> ImGens:=[Image(A,(1,2)),Image(A,(2,3)),Image(A,(3,4)),Image(A,(4,5))];; gap> B:=GroupHomomorphismByImages(Brd5,Image(A),Brd5Gens,ImGens);; gap> EvBrd5:=EvenSubgroup(Brd5);; gap> S:=ResolutionSubgroup(R,EvBrd5);; gap> TS:=HomToIntegralModule(S,B);; gap> Cohomology(TS,3); [ 2, 6, 0, 0, 0 ]
A group G can act non-trivially on the integers Z. For example, a permutation group G can act on Z according to the formula g.n= -n if g is an odd permutation, g.n= n if g is an even permutation. The following commands show that, with this twisted action of S6 on Z, we have third twisted integral homology H3(S6,Z)=Z2+Z10 .
gap> G:=SymmetricGroup(6);; gap> R:=ResolutionFiniteGroup(G,4);; gap> C:=TensorWithTwistedIntegers(R,SignPerm);; gap> Homology(C,3); [ 2, 10 ]
With the analogous twisted action of S6 on Z5, the following commands show that the twelvth homology is H12(S6,Z5)=Z5 . (The calculation relies on the fact that Hn(G,Zp) is equal to its p-primary part Hn(G,Zp)(p) .)
gap> G:=SymmetricGroup(6);; gap> P:=SylowSubgroup(G,5);; gap> R:=ResolutionFiniteGroup(P,15);; gap> F:=function(R);return TensorWithTwistedIntegersModP(R,5,SignPerm);end;; gap> PrimePartDerivedFunctor(G,R,F,12); [ 5 ]
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