Enumeration of Double Cosets

Let H and K be subgroups of a group G. The double cosets of H and K in G are the sets HgK, g E G. In this paper we describe a procedure, P, for determining the cardinality of the set H\G/K of double cosets of H and K in G given a finite presentation for G and finite sets of generators for H and K. It is well known that the problem of determining whether or not a group G defined by a finite presentation is finite is unsolvable. As this problem is the same as enumerating H\G/K when H= K= 1, every double coset enumeration procedure must fail for some inputs. P fails by running forever and never terminating. Enumeration of H\G/K when H= 1 is coset enumeration, and the CoxeterTodd procedure [2, Chapter 21 solves this problem exactly when the number of cosets is finite. By enumerating G/H and G/K in parallel one can use the Coxeter-Todd method to count H\G/K whenever G/H or G/K is finite. Thus our procedure, P, is of interest when G/H and G/K are both infinite. Some examples are given in Section 4. Unfortunately Example 3 shows that P need not terminate when H\G/K is finite. A version of P for the case H= K = 1 appears in [3].