Coset enumeration problem

Given data
A group $$G$$ is specified by a presentation, viz a generating set $$X$$ subject to a set of relations $$R$$. A subgroup $$H$$ of $$G$$ is specified (again by means of generators).

Goal
We need to count the number of cosets of $$H$$ in $$G$$, and also to explicitly compute the permutation representation of $$G$$ on the coset space of $$H$$.

Two popular algorithms
The following are the typical algorithms for this problem:


 * The Todd-Coxeter algorithm
 * The Knuth-Bendix algorithm

Difficulty with algorithms
The problem is that the algorithms may take very long. This is essentially because even for the trivial groups, the presentations could be arbitrarily complicated.

For finite groups, we have the guarantee that the Todd-Coxeter algorithm for coset enumeration terminates eventually in finitely many steps. However, this finite number could be arbitrarily large even if we restrcti attention to the trivial group.