Volume 36 • Number 4 • 2013   • A Note on the Product of Element Orders of Finite Abelian Groups Marius Tarnauceanu Abstract. Given a finite group $G$, we denote by $\psi\,'(G)$ the product of element orders of $G$. Our main result proves that the restriction of $\psi\,'$ 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. 2010 Mathematics Subject Classification:Primary 20K01; Secondary 20D60, 20D15 Full text: PDF