Characteristic Subgroups of Finite Abelian Groups

TL;DR

This paper investigates when two finite abelian groups have isomorphic lattices of characteristic subgroups, providing a complete classification for groups with at least one of odd order, and highlighting a notable exceptional isomorphism.

Contribution

It offers an explicit description of characteristic subgroups and fully characterizes isomorphisms of their lattices in the odd order case, revealing a unique exceptional case.

Findings

Complete classification for groups with odd order

Explicit description of characteristic subgroups

Identification of a unique exceptional isomorphism

Abstract

We consider the question: When do two finite abelian groups have isomorphic lattices of characteristic subgroups? An explicit description of the characteristic subgroups of such groups enables us to give a complete answer to this question in the case where at least one of the groups has odd order. An "exceptional" isomorphism, which occurs between the lattice of characteristic subgroups of $Z_p\times Z_{p^2}\times Z_{p^4}$ and $Z_{p^2} \times Z_{p^5}$, for any prime $p$, is noteworthy.