On the Sum of Element Orders in Finite Abelian Groups
Mohsen Amiri
TL;DR
This paper resolves a conjecture of Tar̆năuceanu by showing that the sum-of-element-orders function ψ completely determines the order-type of finite abelian groups within the LCM-class: for finite LCM-groups G and H of the same order, ψ(G) = ψ(H) iff G and H have the same order type. It proves a stronger result for finite p-groups in 𝓛CM, establishing that equal ψ-values force identical order-type, and then extends the conclusion to all finite abelian groups, where ψ-values coincide precisely when the groups are isomorphic through their invariant factors. The work relies on a suite of reduction formulas, coset- and product-structure lemmas in LCM-groups and an inductive framework on group order, highlighting the role of Ω-structures and maximal subgroups. Overall, ψ serves as a practical invariant for classifying finite abelian groups and underscores the essential LCM condition for this equivalence.
Abstract
Let $ψ(G) = \sum_{g \in G} o(g)$ 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 $ψ$. Tărnăuceanu 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, $ψ(G) = ψ(H)$ if and only if $G$ and $H$ are the same order type.