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 style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big><span
 style="font-weight: bold;">About HAP: Exact Cohomology Coefficient
Sequence<br>
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 href="aboutSuperperfect.html"><small style="color: rgb(0, 0, 102);">next</small></a><br>
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 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;">--> Hⁿ(G,A) 
--> Hⁿ(G,B) --> Hⁿ(G,C) --> Hⁿ⁺¹(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>
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 style="background-color: rgb(255, 255, 255); vertical-align: top;">Consider
the
symmetric
group
G=S₄ 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: H³(S₄,Z/4Z) 
----> 
H³(S₄,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>
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 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>
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 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;">H³(S₄,Z/8Z) 
----> 
H³(S₄,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>
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 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>
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 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;">H²(S₄,Z/2Z) 
----> 
H³(S₄,Z/4Z)<br>
      </div>
      <br>
and determine that the image of this homomorphism has order 2.<br>
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 style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
delta:=ConnectingCohomologyHomomorphism(psi,2);;<br>
gap> Size(ImageOfGOuterGroupHomomorphism(delta));<br>
2<br>
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      <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;">H²(S₄,Z/2Z) 
----> 
H³(S₄,Z/4Z) ----> 
H³(S₄,Z/8Z) ----> 
H³(S₄,Z/2Z) </div>
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