On the Sum of Element Orders in Finite Abelian Groups

TL;DR

This paper confirms a conjecture that two finite abelian groups are isomorphic if and only if they have the same sum of element orders, extending known results from p-groups to all finite abelian groups.

Contribution

It proves that for finite LCM-groups, equal sum of element orders implies the groups have the same order type, confirming a conjecture for all finite abelian groups.

Findings

Finite abelian groups with the same sum of element orders are isomorphic.

The result extends the known characterization from p-groups to all finite abelian groups.

The paper establishes a stronger criterion based on the order type of the groups.

Abstract

Let $\psi(G) = \sum_{g \in G} o(g)$ denote the sum of element orders of a finite group $G$. It is known that among groups of order $n$, the cyclic group $C_n$ maximizes $\psi$. T\u{a}rn\u{a}uceanu proved that two finite abelian $p$-groups of the same order are isomorphic if and only if they have the same sum of element orders, and conjectured this for arbitrary finite abelian groups. In this paper, we confirm the conjecture by proving a stronger result: for finite $LCM$-groups $G$ and $H$ of the same order, $\psi(G) = \psi(H)$ if and only if $G$ and $H$ have the same order type.