Computing Automatic Coset Systems and Subgroup Presentations

Abstract

The concept of an automatic group can be generalized to a group that is automatic with respect to a specified subgroup. This means that there is a finite state automaton that recognizes a unique word in each coset of the subgroup, and others that essentially recognize the permutation action on these cosets induced by multiplying by a group generator. These automata make it possible to enumerate coset representatives as words in the generators, and to solve the generalized word problem for the subgroup efficiently. Algorithms to construct these automata have been described previously by Redfern. Here we describe improved versions, together with implementation details and some examples of successful calculations. A related algorithm to compute a finite presentation of the subgroup is also described.