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| A
short
exact
sequence
of
ZG-modules
A >--> B -->> C
induces a long exact sequence of
cohomology groups
--> Hn(G,A)
--> Hn(G,B) --> Hn(G,C) --> Hn+1(G,A)
-->
.
The implementation of this sequence is joint work with Daher Al-Baydli. |
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| Consider
the
symmetric
group
G=S4 and the sequence Z/4Z
>-----> Z/8Z ---> Z/2Z
of trivial ZG-modules. We can represent a ZG-module as a GOuterGroup.
The following commands use this representation to compute the induced
cohomology homomorphismf: H3(S4,Z/4Z)
---->
H3(S4,Z/8Z)
and determine that the image of this induced homomorphism has order 8
and that its kernel has order 2. |
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| 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 |
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| The
following
commands
then
compute
the
homomorphism H3(S4,Z/8Z)
---->
H3(S4,Z/2Z)
induced by Z/4Z >----->
Z/8Z ---->> Z/2Z .
and determine that the kernel of
this homomorphsim has order 8.
|
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| 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 |
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| The
following commands then compute the connecting homomorphism H2(S4,Z/2Z)
---->
H3(S4,Z/4Z)
and determine that the image of this homomorphism has order 2. |
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| gap>
delta:=ConnectingCohomologyHomomorphism(psi,2);; gap> Size(ImageOfGOuterGroupHomomorphism(delta)); 2 |
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| Note
that
the
various
orders are consistent with exactness of the sequence H2(S4,Z/2Z)
---->
H3(S4,Z/4Z) ---->
H3(S4,Z/8Z) ---->
H3(S4,Z/2Z)
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