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In this paper, a formula for the calculation of the number of right cosetscontained in a double coset with respect to a unimodular group over a Dedekinddomain is developed, and applications of this formula in the theory ofcongruence subgroups -- an index formula -- and the theory of abstract Heckealgebras -- a reduction theorem and an algorithm for the explicit calculationof products -- are given.

Counting cosets of unimodular groups over Dedekind domains