Computing automatic coset systems and subgroup presentations

The concept of an automatic group can be generalized to a group that is aut omatic with respect to a specified subgroup. This means that there is a fin ite state automaton that recognizes a unique word in each coset of the subg roup, 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, an d to solve the generalized word problem for the subgroup efficiently. Algor ithms 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 comput e a finite presentation of the subgroup is also described. (C) 1999 Academi c Press.