Generalized ideal classes in application to toroidal solenoids

Abstract

. We study certain subgroups G A of Q n defined by non-singular n × n -matrices A with integer coefficients. In the first non-trivial case when n = 2, we give necessary and sufficient conditions for two such groups to be isomorphic. Namely, in the generic case when the characteristic polynomial of A is irreducible, we attach a generalized ideal class to A and essentially, two groups are isomorphic if and only if the corresponding ideal classes are equivalent. The obtained results can be applied to studying associated toroidal solenoids.