On the Sum of Element Orders in Finite Abelian Groups

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 $\psi$. 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, $\psi (G)=\psi (H)$ if and only if $G$ and $H$ have the same order type.