A note on the product of element orders of finite abelian groups

Given a finite group $G$, we denote by $\psi {}^{\mathrm{\prime }}(G)$ the product of element orders of $G$. Our main result proves that the restriction of $\psi {}^{\mathrm{\prime }}$ to abelian $p$-groups of order $p^n$ is strictly increasing with respect to a natural order on the groups relating to the lexicographic order of the partitions of $n$. In particular, we infer that two finite abelian groups of the same order are isomorphic if and only if they have the same product of element orders.