Algorithms for polycyclic-by-finite groups

Abstract

A set of fundamental algorithms for computing with polycyclic-by-finite groups is presented.Polycyclic-by-finite groups arise naturally in a number of contexts; for example, as automorphism groups of large finite soluble groups, as quotients of finitely presented groups, and as extensions of modules by groups. No existing mode of representation is suitable for these groups, since they will typically not have a convenient faithful permutation representation.A mixed mode is used to represent elements of such a group, utilising either a power-conjugate presentation or a polycyclic presentation for the elements of the normal subgroup, and a permutation representation for the elements of the quotient.