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The mechanical computation of first and second cohomology groups. (English) Zbl 0587.20035
The author continues his work of ”A computer program for the calculation of the Schur multiplier of a permutation group” [Computational group theory, Proc. Symp., Durham 1982, 307-319 (1984; Zbl 0544.20004); J. Pure Appl. Algebra 35, 287-295 (1985; Zbl 0552.20006)].
This paper describes the theory and implementation of algorithms to compute the dimension of the first and second cohomology groups. The input is (1) a finite group G given by generating permutations, and (2) matrices for the generators’ action on a finite module M over $$K=GF(p)$$, p prime.
A combination of hand and mechanical calculation produces (3) a Sylow p- subgroup P of G, (4) a chain of subgroups $$P=H_ 0\leq H_ 1=N(P)\leq...\leq H_ n=G$$, (5) suitable double coset representatives $$D_ i$$ of $$H_{i-1}$$ in $$H_ i$$, $$i=1,2,...,n$$, (6) $$P\cap P^ q$$, $$P\cap P^{g^{-1}}$$, for each g in the $$D_ i$$ as further requirements of the algorithm.
The algorithm uses variants of the nilpotent quotient algorithm to compute $$H^ x(P,M)$$, and then computes $$H^ x(G,M)$$ as the subgroup of stable elements of $$H^ x(P,M)$$, $$x=1,2$$. The author has a stand alone implementation written in C for the Unix operating system. Two examples - the natural 5-dimensional module of $$L_ 5(2)$$, and a 6-dimensional module over GF(3) for $$L_ 3(9)$$- illustrate the performance of the program.
Reviewer: G.Butler

MSC:
 20J06 Cohomology of groups 20C25 Projective representations and multipliers 20-04 Software, source code, etc. for problems pertaining to group theory 20F05 Generators, relations, and presentations of groups
Cayley
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References:
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