# Finite direct sums of cyclic embeddings

## TL;DR

This paper extends Kaplansky's classification of cyclic subgroup embeddings in finite abelian groups by introducing partial maps on Littlewood-Richardson tableaux, providing a new combinatorial characterization.

## Contribution

It introduces partial maps on Littlewood-Richardson tableaux to classify finite direct sums of cyclic embeddings, generalizing Kaplansky's earlier work.

## Key findings

- Partial maps on Littlewood-Richardson tableaux characterize isomorphism types.
- The classification applies to finite direct sums of cyclic embeddings.
- Generalizes Kaplansky's combinatorial approach.

## Abstract

In this paper we generalize Kaplansky's combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group given in his 1951 book ``Infinite Abelian Groups''. For this we introduce partial maps on Littlewood-Richardson tableaux and show that they characterize the isomorphism types of finite direct sums of such cyclic embeddings.