Automorphism Classes of Elements in Finitely Generated Abelian Groups

TL;DR

This paper characterizes automorphism classes of elements in finitely generated abelian groups, showing each element is automorphically equivalent to a representative in a repeat-free subgroup, and provides counts for finite cases.

Contribution

It introduces a new classification method for elements in finitely generated abelian groups and offers a way to count automorphism classes in finite groups.

Findings

Every element is automorphically equivalent to a representative in a repeat-free subgroup.

Provides a method to count automorphism classes in finite abelian groups.

Establishes a structural understanding of automorphism classes in these groups.

Abstract

We will show that every element of a finitely generated abelian group is automorphically equivalent what we will define to be a {\em representative element} in a {\em repeat-free subgroup}, and for finite abelian groups we can count the number of automorphism classes of elements.