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Given a cohomology module CM for the group G acting on the module
M and a non-negative integer n taking one of the values 0, 1 or 2,
this function returns the cohomology group Hn(G, M).
must be a module returned by invoking CohomologyModule.
If the group used to define CM was a finitely presented group, then n may
only be equal to 0 or 1. Note that CM
Given a cohomology module CM for the group G acting on the module
M defined over a finite field K and
a non-negative integer n taking one of the values 0, 1 or 2,
this function returns the dimension of Hn(G, M) over K. Note
that this function may only be applied to the module returned
by a call to CohomologyModule(G, M), where M
is a module over a finite field K. When n = 2, this function
is faster and may be applied to much larger examples than
CohomologyGroup(CM, n) but, unlike that function, it does
not enable the user to compute with explicit extensions and
two-cocycles.
Given the permutation group G, the K[G]-module M and an integer
n (equal to 1 or 2), return the dimension of the n-th cohomology
group of G acting on M. Note that K must be a finite field of
prime order. This function invokes Derek Holt's original C cohomology
code (see [Hol85b]). In some cases it will be faster than the
function that uses the cohomology module datastructure.
We examine the first and second cohomology groups of the
group A8.
> G := Alt(8);
> M := PermutationModule(G, GF(3));
We first calculate the dimensions of H1(G, M) and H2(G, M)
using the old functions.
> time CohomologicalDimension(G, M, 1);
0
Time: 0.020
> time CohomologicalDimension(G, M, 2);
1
Time: 0.020
We now recalculate the dimensions of H1(G, M) and H2(G, M)
using the new functions.
> X := CohomologyModule(G, M);
> time CohomologicalDimension(X, 1);
0
Time: 0.020
> time CohomologicalDimension(X, 2);
1
Time: 0.920
> X := CohomologyModule(G, M);
> time C:=CohomologyGroup(X, 2);
Time: 4.070
> C;
Full Vector space of degree 1 over GF(3)
In the case of Ω^ - (8, 3) acting on its natural module,
the new function succeeds, but the old function does not.
> G := OmegaMinus(8, 3);
> M := GModule(G);
> X := CohomologyModule(G, M);
> time CohomologicalDimension(X, 2);
2
Time: 290.280
> phi, P := PermutationRepresentation(G);
> MM := GModule(P, [ActionGenerator(M, i): i in [1..Ngens(G)]] );
> time CohomologicalDimension(P, MM, 2);
Out of space.
>> time CohomologicalDimension(P, MM, 2);
^
Runtime error in 'CohomologicalDimension': Cohomology failed
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