In order to compute the cohomology of a group with respect to a G-module M, it is first necessary to construct a data structure known as a cohomology module.
Given a group G and a G-module M with acting group G this function returns a cohomology module for the action of G. The group G may be a finite permutation group, a finite matrix group, a PC-group, or any finitely presented group. For the PC-group case, however, the PC-presentation of G must be conditioned. This can be achieved by first executing the statement G := ConditionedGroup(G);
Let G be a group which acts on a finitely-generated abelian group with invariants given by the sequence Q, and action described by T. The action T is given in the form of a sequence of d x d matrices over the integers, where d is the length of T, and T[i] defines the action of the i-th generator of G on the abelian group. The function returns a cohomology module for the action of G. The group G may be a finite permutation group, a finite matrix group, a PC-group or any finitely presented group. For the PC-group case, however, the PC-presentation of G must be conditioned. This can be achieved by first executing the statement G := ConditionedGroup(G);
> G := PSL(3, 2); > Irrs := AbsolutelyIrreducibleModules(G, GF(2)); > Irrs; [ GModule of dimension 1 over GF(2), GModule of dimension 3 over GF(2), GModule of dimension 3 over GF(2), GModule of dimension 8 over GF(2) ] > M := Irrs[2]; > CM := CohomologyModule(G, M); > CM; Cohomology Module
For a permutation group G acting on some abelian group A through M, compute the cohomology module. M has to be either a map from G into the endomorphisms of A, or a sequence of endomorphisms of A, one for each of the generators of G.[Next][Prev] [Right] [Left] [Up] [Index] [Root]