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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
Software:
Cayley
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References:
[1] Butler, G., An imnplementation and proof of Holt’s algorithm, ()
[2] Cannon, J.J., An introduction to the group theory language, Cayley, (), 145-183
[3] Cartan, H.; Eilenberg, S., ()
[4] Gaschütz, W., Über modulare darstellungen endlicher gruppen, die von freien gruppen induziert werden, Math. zeit, 60, 274-286, (1954) · Zbl 0056.02401
[5] Glazunov, N.M., Algorithms for calculations of cohomology groups of finite groups, (), 208-218, (Math. Reviews 82f 20080)
[6] Havas, G.; Newman, M.F., Applications of computers to questions like those of Burnside, (), 211-230 · Zbl 0432.20033
[7] Holt, D.F., A computer program for the calculation of the Schur multiplier of a permutation group, (), 307-319 · Zbl 0699.20040
[8] Holt, D.F., A computer program for the calculation of a covering group of a finite group, J. pure applied algebra, 35, 287-295, (1985) · Zbl 0552.20006
[9] Leon, J.S., On an algorithm for finding a base and strong generating set for a group given by generating permutations, Math. comp, 35, 941-974, (1980) · Zbl 0444.20001
[10] MacLane, S., Cohomology theory in abstract groups III, Ann. math, 50, 736-761, (1949) · Zbl 0039.25703
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