Geodesic
A Note on the Product of Element Orders of Finite Abelian Groups
Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 4 Cet article a éte moissonné depuis la source Bulletin of the Malaysian Mathematical Society website

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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.
Classification : Primary 20K01; Secondary 20D60, 20D15 
@article{BMMS_2013_36_4_a20,
     author = {Marius Tarnauceanu},
     title = {A {Note} on the {Product} of {Element} {Orders} of {Finite} {Abelian} {Groups}},
     journal = {Bulletin of the Malaysian Mathematical Society},
     year = {2013},
     volume = {36},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/BMMS_2013_36_4_a20/}
}
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Marius Tarnauceanu. A Note on the Product of Element Orders of Finite Abelian Groups. Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 4. http://geodesic.mathdoc.fr/item/BMMS_2013_36_4_a20/