Counting cosets of unimodular groups over Dedekind domains

In this paper, a formula for the calculation of the number of right cosets contained in a double coset with respect to a unimodular group over a Dedekind domain is developed, and applications of this formula in the theory of congruence subgroups – an index formula – and the theory of abstract Hecke algebras – a reduction theorem and an algorithm for the explicit calculation of products – are given.