Computing with group homomorphisms

✍ Scribed by Charles R. Leedham-Green; Cheryl E. Praeger; Leonard H. Soicher

• Publisher: Elsevier Science
• Year: 1991
• Tongue: English
• Weight: 388 KB
• Volume: 12
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1016/s0747-7171(08)80102-2

✦ Synopsis

Let G = (X) and H be finite groups and let r : X ~ H be a map from the generating set X of G into H. We describe a simple approach for deciding whether or not r determines a group homomorphism from G to H, and if it does, for computing the kernel of r If O and H are permutation groups the algorithm is a simple application of standard algorithms for bases and strong generating sets. If G and H are soluble groups given in the usual way by PAG-systems and corresponding power conjugate presentations then the algorithm is a simple application of the non-commutative Gauss algorithm for constructing a subgroup of a soluble group. Further, a probabilistic algorithm is given for finding the kernel and image of 4~ when each of G and H is given as a permutation group or a soluble group, IGI is known, and r is known to determine a homomorphism.