#### Vol. 318, No. 1, 2022

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Generalized ideal classes in application to toroidal solenoids

### Maria Sabitova

Vol. 318 (2022), No. 1, 189–228
DOI: 10.2140/pjm.2022.318.189
##### Abstract

We study certain subgroups ${G}_{A}$ of ${ℚ}^{n}$ defined by nonsingular $n×n$-matrices $A$ with integer coefficients. In the first nontrivial 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.

##### Keywords
ideal classes, solenoids, odometers, matrices
Primary: 11Z05
##### Milestones
Received: 25 June 2021
Revised: 11 November 2021
Accepted: 2 April 2022
Published: 1 August 2022
##### Authors
 Maria Sabitova Department of Mathematics Queens College CUNY Flushing, NY United States