Characterizing finite groups using the sum of the orders of the elements. (Q471798)

Let \(G\) be a finite group and \(X\) be a non-empty subset of \(G\). Then \(s(X)\) denotes the sum of the orders of all elements in \(X\). Recently some researchers have paid atention to classifying finite groups using the function \(s\). For example [in Commun. Algebra 37, No. 9, 2978-2980 (2009; Zbl 1183.20022)] \textit{H. Amiri} et al. showed that if \(C\) is a finite cyclic group and \(G\) is a group of the same order as \(C\), then \(s(G)<s(C)\). In the paper under review the authors prove some theorems concerning the function \(s\), such as if \(G\) is a finite abelian group, then \(s(H)\) divides \(s(G)\) for every subgroup \(H\) of \(G\) if and only if \(G\) is cyclic of square free order.