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). For modules defined over the ring of integers only, n may also be equal to 3. (In this case, H3(G, M) is computed as the second cohomology group of M regarded as a module over Q/Z.) 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) must be a module returned by invoking CohomologyModule.
> G:=CyclicGroup(4); > mats := [ Matrix(Integers(),2,2,[1,2,1,3]) ]; > invar := [2,4]; > CM := CohomologyModule(G,invar,mats); > CohomologyGroup(CM,0); Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 2 ] > CohomologyGroup(CM,1); Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 2 ] > CohomologyGroup(CM,2); Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 2 ]
> G := Group<x,y | x^2,y^3,(x*y)^7 >; > L := LowIndexSubgroups(G, <7,7>); > Q := CosetImage(G,L[1]); > PM := PermutationModule(Q, Integers()); > cons := Constituents(PM); > mats := ActionGenerators(cons[2]); > M := GModule(G,mats); > CM := CohomologyModule(G,M); > CohomologyGroup(CM,0); Full Quotient RSpace of degree 0 over Integer Ring Column moduli: [ ] > CohomologyGroup(CM,1); Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 7 ] > CohomologyGroup(CM,2); Runtime error: Second cohomology groups are not implemented for GrpFP
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.Note that there are some alternative functions for performing these calculations described in other manual chapters.
For K[G]-module M (with K a finite field and G a finite group), compute and return the K-dimension of the cohomology group Hn(G, M) for n ≥0. For n=0 and 1, this is carried out by using the function CohomologicalDimension(CM,n) just described. For n ≥2, it is done recursively using projective covers and dimension shifting to reduce to the case n=1. See Section Projective Indecomposable Modules in Chapter MODULES OVER AN ALGEBRA AND GROUP REPRESENTATIONS for further details and examples.
For K[G]-module M (with K a finite field and G a finite group), compute and return the sequence of K-dimensions of the cohomology groups Hk(G, M) for 1 ≤k ≤n. On account of the recursive method used, this is quicker than computing them all individually.
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 data structure.
This function returns the dimension of the first cohomology group of the F-module MF, where F is a finitely presented group, f is an epimorphism from F onto a finite group G and M is a G-module that has been lifted to MF. Again, this function is generally much faster, since it uses sparse matrices.
This function is equivalent to CohomologicalDimension(G, M, 1) where M := PermutationModule(P, K) and P is defined by the images of generators of G under the map f, but is generally much faster, since it uses sparse matrices and avoids the explicit construction of the permutation module.
These functions are equivalent to CohomologicalDimension(G, M, 1) where M is the symmetric square or exterior square, respectively, of PermutationModule(P, R), and P is defined by the images of generators of G under the map f. But the functions are generally much faster, since they use sparse matrices and avoid the explicit construction of the modules.
> 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)
> G := ASL(3,5);
> N := pCore(G,5);
> M := GModule(G,N);
> CM := CohomologyModule(G,M);
> time [CohomologyGroup(CM,i) : i in [0..2]];
[
Full Vector space of degree 0 over GF(5),
Full Vector space of degree 1 over GF(5),
Full Vector space of degree 1 over GF(5)
]
Time: 46.940
> CM := CohomologyModule(G,M);
> time [CohomologicalDimension(CM,i) : i in [0..2]];
[ 0, 1, 1 ]
Time: 0.320
> 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