A graphical approach to coset enumeration

In 1936, J.A. Todd and H.S.M. Coxeter published a paper in which they described a technique for enumerating the cosets of a finite group given only the defining relations for the group and the generators of the subgroup written in terms of the generators of the group. This process has an interesting interpretation in terms of directed graphs.

The technique seemed well adapted for use with a stored program computer and was among the first programs written for such machines in the field of finite group theory. Although the first programs enjoyed some success, their performance was not entirely satisfactory. Consequently, the algorithm has been evolving and the programs as well as the machines have become several orders of magnitude more powerful. The major evolutionary trends are described as are the primary problems involved in the design of such a program, in the hope that some enterprising reader will make some headway with these problems or perhaps provide another evolutionary advance.