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

Given a finite group G, we denote by ? ʹ(G) the product of element orders of G. Our main result proves that the restriction of ? ʹ to abelian p-groups of order pn 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.