Leedham-Green, Charles R.; Praeger, Cheryl E.; Soicher, Leonard H. Computing with group homomorphisms. (English) Zbl 0789.20001 J. Symb. Comput. 12, No. 4-5, 527-532 (1991). Based on the elementary remark that a mapping \(\phi\) from a group \(G\) into a group \(H\) is a homomorphism if and only if the set \(\{(g,\phi(g))\mid g\in G\}\) is a subgroup of \(G\times H\), the authors describe simple but highly efficient algorithmic methods for deciding if a mapping \(\phi\) from a generating set \(X\) of \(G\) into \(H\) determines a homomorphism, and in case it does to determine the kernel of that homomorphism. For these methods \(G\) and \(H\) are supposed to be permutation groups or polycyclicly presented groups and standard methods for these classes of groups, like the “Schreier-Sims method” and “base change” for permutation groups or the “non-commutative Gauß Algorithm” for polycyclic groups, are employed. Reviewer: J.Neubüser (Aachen) Cited in 1 ReviewCited in 4 Documents MSC: 20-04 Software, source code, etc. for problems pertaining to group theory 20F05 Generators, relations, and presentations of groups 20F16 Solvable groups, supersolvable groups 20E36 Automorphisms of infinite groups 20B40 Computational methods (permutation groups) (MSC2010) Keywords:Schreier-Sims method; base change; homomorphism; efficient algorithmic methods; generating set; permutation groups; polycyclicly presented groups; polycyclic groups Software:Cayley PDFBibTeX XMLCite \textit{C. R. Leedham-Green} et al., J. Symb. Comput. 12, No. 4--5, 527--532 (1991; Zbl 0789.20001) Full Text: DOI References: [1] Cannon, J. J., An introduction to the group theory language Cayley, (Atkinson, M. D., Computational Group Theory (1984), Academic Press: Academic Press London), 145-183 · Zbl 0544.20002 [2] Laue, R.; Neubüser, J.; Schoenwaelder, U., (Atkinson, M. D., Algorithms for finite soluble groups and the SOGOS system (1984), Academic Press: Academic Press London), 105-135 · Zbl 0547.20012 [3] Leedham-Green, C. R.; Soicher, L. H., Collection from the left and other strategies, Symbolic Computation., 9, 665-675 (1990) · Zbl 0726.20001 [4] 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 [5] Praeger, C. E., Computing the kernel of a group homomorphism with soluble domain, (Research Report (1987), University of Western Australia) · Zbl 0675.05013 [6] Sims, C. C., Determining the conjugacy classes of a permutation group, (Birkhoff, G.; Hall, M., Computers in Algebra and Number Theory, 4 (1971), Amer. Math. Soc.), 191-195, SIAM-AMS Proc. · Zbl 0253.20001 [7] Thiemann, P., SOGOS III, (Diplomarbeit (1987), RWTH Aachen) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.