# On separable abelian $p$-groups

## TL;DR

This paper classifies abelian p-groups that are separable with respect to the class of all abelian groups, based on the properties of Schur rings and algebraic isomorphisms.

## Contribution

It provides a complete classification of abelian p-groups that are separable with respect to the class of abelian groups, advancing understanding of Schur rings.

## Key findings

- Identifies which abelian p-groups are separable
- Characterizes algebraic isomorphisms in Schur rings
- Provides criteria for separability of abelian p-groups

## Abstract

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial isomorphism. A finite group is said to be \emph{separable} with respect to $\mathcal{K}$ if every $S$-ring over this group is separable with respect to $\mathcal{K}$. We provide a complete classification of abelian $p$-groups separable with respect to the class of abelian groups.