On a question of Jaikin-Zapirain about the average order elements of finite groups (Q6645105)

Let \(G\) be a finite group. We denote by \(o(G)\) the average order of the elements of \(G\). In 2011 \textit{A. Jaikin-Zapirain} [Adv. Math. 227, No. 3, 1129--1143 (2011; Zbl 1227.20014)] asked whether is true that \(o(G) \geq o(N)^{1/2}\) for every normal subgroup \(N\) of \(G.\) Khukhro, Moretó and Zarrin gave a negative answer. However in this paper the authors prove that the answer is affirmative for abelian groups and in this case, an even stronger statement holds: if \(G\) is a finite abelian group, then \(o(H) \leq o(G)\) for all subgroups \(H\) of \(G.\)