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# Given a map f:H->G, this function produces
# the induced map H*(G)->H*(H) to degree n.
InstallGlobalFunction(InducedHomomorphismOnCohomology,
function(C,D,f,n)
local B,P,Q,M,L,F,c,i,j,k,I,J;
if not C!.K=D!.K then
Error("Fields not the same\n");fi;
if not Source(f)=C!.G then
Error("Source(f) not the correct group\n");fi;
if not Range(f)=D!.G then
Error("Range(f) not the correct group\n");fi;
if not C!.p=D!.p then
Error("Primes not the same\n");fi;
CohomologyRing(C,n);
CohomologyRing(D,n);
B:=Basis(C!.A);
# kG is a kH module via f. The list L below
# gives the action of H on kG.
P:=Pcgs(C!.G);
Q:=Pcgs(D!.G);
M:=IdentityMat(Size(D!.G),D!.K);
L:=List(P,x->
LinearCombinationPcgs(
D!.L[1]+M,
ExponentsOfPcElement(Q,x^f),
IdentityMat(Size(D!.G),C!.K)
)-M
);
# Lift 1:K->K to a chain map.
F:=[[ListWithIdenticalEntries(Size(D!.G),Zero(C!.K))]];
F[1][1][1]:=One(C!.K);
for j in [1..n] do
c:=C!.R[j]*LiftHom(L,F[j],C!.p);
F[j+1]:=List(c,x->SolutionMat(D!.d[j],x));
od;
# Bookeeping. Produce the list J of
# images of the map we're constructing.
I:=[];
for j in [1..n+1] do
for k in [1..D!.B[j]] do
Append(I,
[Concatenation(
ListWithIdenticalEntries(Sum(C!.B{[1..j-1]}),Zero(C!.K)),
ExtractColumn(F[j],(k-1)*Size(D!.G)+1),
ListWithIdenticalEntries(Sum(C!.B{[j+1..n+1]}),Zero(C!.K))
)]
);
od;
od;
J:=List(I,x->LinearCombination(B,x));
return AlgebraHomomorphismByImages(D!.A,C!.A,Basis(D!.A),J);
end
);
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