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big><
span
style=
"font-weight: bold;">About HAP: Exact Cohomology Coefficient
Sequence<
br>
</
span></
big></
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<
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<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">A
short
exact
sequence
of
ZG-modules
<
br>
<
div style=
"text-align: center;">A >--> B -->> C<
br>
<
div style=
"text-align: left;">induces a long exact sequence of
cohomology groups<
br>
<
div style=
"text-align: center;">--> H<
sup>n</
sup>(G,A)
--> H<
sup>n</
sup>(G,B) --> H<
sup>n</
sup>(G,C) --> H<
sup>n+1</
sup>(G,A)
-->
.<
br>
<
br>
<
div style=
"text-align: left;"><
br>
<
br>
The implementation of this sequence is joint work with <
span
style=
"font-weight: bold;">Daher Al-Baydli</
span>. </
div>
</
div>
</
div>
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<
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<
td
style=
"background-color: rgb(255, 255, 255); vertical-align: top;">Consider
the
symmetric
group
G=S<
sub>4</
sub> and the sequence<
br>
<
div style=
"text-align: center;"> Z/4Z
>-----> Z/8Z ---> Z/2Z<
br>
<
br>
</
div>
of trivial ZG-modules. We can represent a ZG-module as a GOuterGroup.
The following commands use this representation to compute the induced
cohomology homomorphism<
br>
<
br>
<
div style=
"text-align: center;">f: H<
sup>3</
sup>(S<
sub>4</
sub>,Z/4Z)
---->
H<
sup>3</
sup>(S<
sub>4</
sub>,Z/8Z)<
br>
<
br>
</
div>
and determine that the image of this induced homomorphism has order 8
and that its kernel has order 2. <
br>
</
td>
</
tr>
<
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<
td
style=
"background-color: rgb(255, 255, 204); vertical-align: top;">gap>
G:=SymmetricGroup(4);;<
br>
gap> x:=(1,2,3,4,5,6,7,8);;<
br>
gap> a:=Group(x^2);;<
br>
gap> b:=Group(x);;<
br>
gap> ahomb:=GroupHomomorphismByFunction(a,b,y->y);;<
br>
gap> A:=TrivialGModuleAsGOuterGroup(G,a);;<
br>
gap> B:=TrivialGModuleAsGOuterGroup(G,b);;<
br>
gap> phi:=GOuterGroupHomomorphism();;<
br>
gap> phi!.
Source:=A;;<
br>
gap> phi!.Target:=B;;<
br>
gap> phi!.Mapping:=ahomb;;<
br>
<
br>
gap> Hphi:=CohomologyHomomorphism(phi,3);;<
br>
<
br>
gap> Size(ImageOfGOuterGroupHomomorphism(Hphi));<
br>
8<
br>
<
br>
gap> Size(KernelOfGOuterGroupHomomorphism(Hphi));<
br>
2<
br>
</
td>
</
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<
tr>
<
td
style=
"text-align: left; background-color: rgb(255, 255, 255); vertical-align: top;">The
following
commands
then
compute
the
homomorphism<
br>
<
br>
<
div style=
"text-align: center;">H<
sup>3</
sup>(S<
sub>4</
sub>,Z/8Z)
---->
H<
sup>3</
sup>(S<
sub>4</
sub>,Z/2Z)<
br>
</
div>
<
br>
induced by <
br>
<
br>
<
div style=
"text-align: center;">Z/4Z >----->
Z/8Z ---->> Z/2Z .<
br>
<
br>
<
div style=
"text-align: left;">and determine that the kernel of
this homomorphsim has order 8. <
br>
</
div>
</
div>
</
td>
</
tr>
<
tr>
<
td
style=
"background-color: rgb(255, 255, 204); vertical-align: top;">gap>
bhomc:=NaturalHomomorphismByNormalSubgroup(b,a);<
br>
gap> B:=TrivialGModuleAsGOuterGroup(G,b);<
br>
gap> C:=TrivialGModuleAsGOuterGroup(G,Image(bhomc));<
br>
gap> psi:=GOuterGroupHomomorphism();<
br>
gap> psi!.
Source:=B;<
br>
gap> psi!.Target:=C;<
br>
gap> psi!.Mapping:=bhomc;<
br>
<
br>
gap> Hpsi:=CohomologyHomomorphism(psi,3);<
br>
<
br>
gap> Size(KernelOfGOuterGroupHomomorphism(Hpsi));<
br>
8<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands then compute the connecting homomorphism<
br>
<
br>
<
div style=
"text-align: center;">H<
sup>2</
sup>(S<
sub>4</
sub>,Z/2Z)
---->
H<
sup>3</
sup>(S<
sub>4</
sub>,Z/4Z)<
br>
</
div>
<
br>
and determine that the image of this homomorphism has order 2.<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
delta:=ConnectingCohomologyHomomorphism(psi,2);;<
br>
gap> Size(ImageOfGOuterGroupHomomorphism(delta));<
br>
2<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">Note
that
the
various
orders are consistent with exactness of the sequence<
br>
<
br>
<
div style=
"text-align: center;">H<
sup>2</
sup>(S<
sub>4</
sub>,Z/2Z)
---->
H<
sup>3</
sup>(S<
sub>4</
sub>,Z/4Z) ---->
H<
sup>3</
sup>(S<
sub>4</
sub>,Z/8Z) ---->
H<
sup>3</
sup>(S<
sub>4</
sub>,Z/2Z) </
div>
<
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