Finite soluble groups G are represented in a computer by a (consistent) power-commutator-presentation passing through a normal series \(G=N_ 0>N_ 1>...>N_ r=1\) with elementary abelian factors; subgroups have a canonical generating series; many computations proceed inductively from \(G/N_ i\) to \(G/N_{i+1}\). The present paper contains an algorithm for the computation of \(H^ 1(G,M)\) (i.e. representatives for the conjugacy classes of complements to the normal subgroup M in the semidirect product GM) for a soluble group G acting on an elementary abelian p-group M and explanations of its use in the computation of representatives for the conjugacy classes of complements to normal subgroups and of its use in the computation of the normalizer of a subgroup. In these algorithms the authors seek to avoid general ‘orbit-stabilizer algorithms’ which would involve time consuming collection processes; they are replaced by affine group actions on vector-spaces which can be processed by solving systems of linear equations. While there are no comparable previous implementations of complement algorithms, the effectiveness of the present refinement of the S. Glasby and M. Slattery normalizer algorithm [J. Symb. Comput. 9, 637-651 (1990; Zbl 0707.20001)] is demonstrated by relevant examples run under the GAP-system at Aachen.
\(\{\) Several references have now appeared in J. Symb. Comput. 9, No.5/6 (1990).\(\}\) This article is reprinted in: G. M. Piacentini Cattaneo, E. Strickland (eds.), Topics in Computational Algebra (Computational Algebra Seminar, Rome ‘Tor Vergata’, 9-11 May 1990), Kluwer Acad. Publ. (1990; Zbl 0723.00021).