About HAP: Introduction |
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HAP
can be used to make basic calculations in the cohomology of finite and
infinite groups.
For example, to calculate the integral homology Hn(S3,Z)
of the symmetric group S3 in dimensions 100 < n <105
we could proceed naively as follows. First choose a set of generators
for S3,
for instance the two permutations (1,2) and (1,2,3). Then, with HAP
loaded into GAP, perform the following commands. |
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gap>
gensG:=[(1,2), (1,2,3)]; [ (1,2), (1,2,3) ] gap> IntegralGroupHomology([gensG,101); [ 2 ] gap> IntegralGroupHomology([gensG,102); [ ] gap> IntegralGroupHomology(gensG,103); [ 6 ] gap> IntegralGroupHomology(gensG,104); [ ] |
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The
HAP command IntegralGroupHomology(gens,n) returns the abelian group
invariants of the n-dimensional homology of the group G generated by
gens with coefficients in the trivial G-module Z. We see that H101(S3,Z)
= Z2, H102(S3,Z) = 0, H103(S3,Z)
= Z6 and H104(S3,Z) = 0. This example has two features that dramatically help the computations. Firstly, S3 is an extremely small group. Secondly, S3 has periodic homology with period 4 (meaning that Hn(S3,Z)=Hn+4(S3,Z) for n>0) and so the homology groups themselves are also extremely small. Computations typically require a more careful approach. For example, to compute the low dimensional homology of the Sylow 2-subgroup P of the Mathieu simple group M23 we could try the same approach using using the set of six generators for P produced by GAP. However, we would run into a memory problem in dimension 4. The following commands show how to solve this problem by choosing a smaller set of generators. |
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>gap
P:=SylowSubgroup(MathieuGroup(23),2); <permutation group with 6 generators> gap> S:=GeneratorsOfGroup(P);; gensP:=[S[4],S[6],S[2]];; gap> Order(Group(gensP))=Order(P); true gap> IntegralGroupHomology(gensP,1); [2,2,2] gap> IntegralGroupHomology(gensP,2); [2,2,4] gap> IntegralGroupHomology(gensP,3); [2,2,2,4,4,8] gap> IntegralGroupHomology(gensP,4); [2,2,2,2,2,2,2,2] |
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Thus
H1(P,Z)=(Z2)3, H2(P,Z)=(Z2)2+Z4,
H3(P,Z)=(Z2)3+(Z4)2+Z8
and H4(P,Z)=(Z2)8. In order to compute
the higher homology of P, or the low dimensional homology of M23
itself, we need more sophisticated tricks. To explain
these, and also to explain the role of group generators in HAP's
homology computations, we should recall the definition of group
homology. |
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