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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Counting extensions of groups 
(& computing extensions using HAPcocyclic) 
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 <div style="text-align: center;"></div>
 <div style="text-align: center;">
 <h3>Some classical theory</h3>
 </div>
 The second <a href="aboutTwistedCoefficients.html">cohomology</a>
H2(G,A) of G with coefficients in a G-module A, together
with the corresponding 2-<a href="aboutCocycles.html">cocycles</a>,
have
a
useful group-theoretic interpretation. 
 
 Any group extension N >---> E --->> G gives rise to 
 
 <ol>
 <li>an outer action alpha:G ---> Out(N) of G on N. 
 </li>
 <li>an action G ---> Aut(Z(N)) of G on the centre of N,
uniquely induced
by the outer action alpha and the canonical action of Out(N) on Z(N). 
 </li>
 <li>a "2-cocycle" f:G×G ---> N with values in
the
group N.</li>
 </ol>
 Any outer homomorphism alpha:G--->Out(N) gives rise to a
cohomology class [c] in H3(G,Z(N)). 
 
 It was shown
by Eilenberg and Mac Lane that the cohomology class [c] is
trivial if and only if the outer action alpha arises from some group
extension N >---> E --->> G. 
 
 If [c] is
trivial then there is a bijection between the second cohomology group H2(G,Z(N))
and
Yoneda
equivalence classes of extensions of G by N that are
compatible with alpha. 
 
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style="vertical-align: top; background-color: rgb(255, 255, 255);">
 <h3>An example</h3>
 <div style="text-align: left;">Consider the group
H=SmallGroup(64,134). This is the Sylow 2-subgroup
of the Mathieu group M12. Consider the normal subgroup
N:=NormalSubgroups(G)[16] and quotient group G=H/N. We have N=C2×D4
, A=Z(N)=C2×C2 and G=C2×C2
. 
 
Suppose that we wish to classify all extensions C2×D4
 >---> E ---> C2×C2
which induce the given outer action of G on N. The following commands
show
that, up to Yoneda equivalence, there are two such extensions. 
 </div>
 
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style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
H:=SmallGroup(64,134);; 
gap> N:=NormalSubgroups(H)[15];; 
 
gap> A:=Centre(GOuterGroup(H,N));; 
gap> G:=ActingGroup(A);; 
 
gap> R:=ResolutionFiniteGroup(G,3);; 
gap> C:=HomToGModule(R,A);; 
gap> Cohomology(C,2); 
[ 2 ] 
 </td>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following additional commands return a standard 2-cocycle
f:G×G-->A =Z2×Z2 corresponding
to
the
non-trivial
element in H2(G,A). The value f(g,h) of the
2-cocycle is calculated for all 16 pairs g,h in G. </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
CH:=CohomologyModule(C,2);; 
 
gap> Elts:=Elements(ActedGroup(CH)); 
[ <identity> of ..., f1 ] 
gap> x:=Elts[2];; 
 
gap> c:=CH!.representativeCocycle(x); 
Standard 2-cocycle 
gap> f:=Mapping(c);; 
 
gap> for g in G do for h in G do 
> Print(f(g,h),"\n"); 
> od; 
> od; 
<identity> of ... 
<identity> of ... 
<identity> of ... 
<identity> of ... 
<identity> of ... 
f6 
<identity> of ... 
f6 
<identity> of ... 
<identity> of ... 
<identity> of ... 
<identity> of ... 
<identity> of ... 
f6 
<identity> of ... 
f6 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following extended example illustrates how to construct a cohomology
class C in H2(G, A) from a cocycle c:G x G --> A. 
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style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
#We'll construct G=SL(2,Z_4) as a permutation group.

gap> G:=SL(2,ZmodnZ(4));; 
gap> G:=Image(IsomorphismPermGroup(G));; 
 
gap> #We'll construct Z_8=Z/8Z as a G-outer group

gap> z_8:=Group((1,2,3,4,5,6,7,8));; 
gap> Z_8:=TrivialGModuleAsGOuterGroup(G,z_8);; 
 
gap> #We'll compute the group h=H^2(G,Z_8)

gap> R:=ResolutionFiniteGroup(G,3);; #R is a free resolution 
gap> C:=HomToGModule(R,Z_8);; # C is a chain complex 
gap> H:=CohomologyModule(C,2);; #H is the second cohomology
H^2(G,Z_8) 
gap> h:=ActedGroup(H);; #h is the underlying group of H 
 
gap> #We'll compute cocycles c2, c5 for the second and fifth
cohomology classs 
gap> c2:=H!.representativeCocycle(Elements(h)[2]); 
Standard 2-cocycle 
 
gap> c5:=H!.representativeCocycle(Elements(h)[5]); 
Standard 2-cocycle 
 
gap> #Now we'll construct the cohomology classes C2, C5 in the group
h corresponding to the cocycles c2, c5. 
gap> C2:=CohomologyClass(H,c2);; 
gap> C5:=CohomologyClass(H,c5);; 
 
gap> #Finally, we'll show that C2, C5 are distinct cohomology
classes, both of order 4. 
gap> C2=C5; 
false 
gap> Order(C2); 
4 
gap> Order(C5); 
4 
 </td>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
 HAPcocyclic package
developed by Robert Morse can be used to construct an extension of G by
N for each cohomology class in H2(G,A). An initial
developmental version of this package can be <a
href="../../../hapcocyclic.dev.1.tar.gz">downloaded</a> and untarred
in
directory ../gap/pkg . 
 
The following commands will then construct and identify all extensions
of N by G corresponding to the given outer action of G on N. 
 </td>
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style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
#The
following
commands only work if the HAPcocyclic package by Robert
Morse is loaded. 
gap> LoadPackage("HAPcocyclic");; 
 
gap> H := SmallGroup(64,134);; 
gap> N := NormalSubgroups(H)[16];; 
gap> ON := GOuterGroup(H,N);; 
gap> A := Centre(ON);; 
gap> G:=ActingGroup(A);; 
gap> R:=ResolutionFiniteGroup(G,3);; 
gap> C:=HomToGModule(R,A);; 
gap> CH:=CohomologyModule(C,2);; 
 
gap> Elts:=Elements(ActedGroup(CH));; 
gap> lst :=
List(Elts{[1..Length(Elts)]},x->CH!.representativeCocycle(x));; 
gap> ccgrps := List(lst, x->CcGroup(ON, x));; 
gap> #So ccgrps is a list of groups, each being an extension of G by
N, corresponding 
gap> #to the two elements in H^2(G,A). 
 
gap> #The following command produces the GAP identification number
for each group. 
gap> L:=List(ccgrps,IdGroup); 
[ [ 64, 134 ], [ 64, 135 ] ] 
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