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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

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
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