


A
short
exact
sequence
of
ZGmodules
A >> B >> C
induces a long exact sequence of
cohomology groups
> H^{n}(G,A)
> H^{n}(G,B) > H^{n}(G,C) > H^{n+1}(G,A)
>
.
The implementation of this sequence is joint work with Daher AlBaydli. 

Consider
the
symmetric
group
G=S_{4} and the sequence Z/4Z
>> Z/8Z > Z/2Z
of trivial ZGmodules. We can represent a ZGmodule as a GOuterGroup.
The following commands use this representation to compute the induced
cohomology homomorphismf: H^{3}(S_{4},Z/4Z)
>
H^{3}(S_{4},Z/8Z)
and determine that the image of this induced homomorphism has order 8
and that its kernel has order 2. 

gap>
G:=SymmetricGroup(4);; gap> x:=(1,2,3,4,5,6,7,8);; gap> a:=Group(x^2);; gap> b:=Group(x);; gap> ahomb:=GroupHomomorphismByFunction(a,b,y>y);; gap> A:=TrivialGModuleAsGOuterGroup(G,a);; gap> B:=TrivialGModuleAsGOuterGroup(G,b);; gap> phi:=GOuterGroupHomomorphism();; gap> phi!.Source:=A;; gap> phi!.Target:=B;; gap> phi!.Mapping:=ahomb;; gap> Hphi:=CohomologyHomomorphism(phi,3);; gap> Size(ImageOfGOuterGroupHomomorphism(Hphi)); 8 gap> Size(KernelOfGOuterGroupHomomorphism(Hphi)); 2 

The
following
commands
then
compute
the
homomorphism H^{3}(S_{4},Z/8Z)
>
H^{3}(S_{4},Z/2Z)
induced by Z/4Z >>
Z/8Z >> Z/2Z .
and determine that the kernel of
this homomorphsim has order 8.


gap>
bhomc:=NaturalHomomorphismByNormalSubgroup(b,a); gap> B:=TrivialGModuleAsGOuterGroup(G,b); gap> C:=TrivialGModuleAsGOuterGroup(G,Image(bhomc)); gap> psi:=GOuterGroupHomomorphism(); gap> psi!.Source:=B; gap> psi!.Target:=C; gap> psi!.Mapping:=bhomc; gap> Hpsi:=CohomologyHomomorphism(psi,3); gap> Size(KernelOfGOuterGroupHomomorphism(Hpsi)); 8 

The
following commands then compute the connecting homomorphism H^{2}(S_{4},Z/2Z)
>
H^{3}(S_{4},Z/4Z)
and determine that the image of this homomorphism has order 2. 

gap>
delta:=ConnectingCohomologyHomomorphism(psi,2);; gap> Size(ImageOfGOuterGroupHomomorphism(delta)); 2 

Note
that
the
various
orders are consistent with exactness of the sequence H^{2}(S_{4},Z/2Z)
>
H^{3}(S_{4},Z/4Z) >
H^{3}(S_{4},Z/8Z) >
H^{3}(S_{4},Z/2Z)


