On a bijection between a finite group and cyclic group (Q6117866)

Let \(G\) be a finite group of order \(n\), let \(o(g)\) be the order of an element \(g \in G\) and let \(\psi(G)=\sum_{g \in G}o(g)\). In [\textit{H. Amiri} et al., Commun. Algebra 37, No. 9, 2978--2980 (2009; Zbl 1183.20022)] it has been proven that if \(C_{n}\) is the cyclic group of order \(n\), then \(\psi(G) < \psi(C_{n})\) if \(G\) is non cyclic (see also the solution of Problem 10775 in [Am. Math. Monthly, Vol. 109, No. 3 (Mar., 2002), p. 299]). In relation to this result \textit{I. M. Isaacs} in [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.) The Kourovka notebook. Unsolved problems in group theory. 18th edition. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2014; Zbl 1372.20001)] proposed Problem 18.1: Does there necessarily exist a bijection \(f\) from \(G\) onto \(C_{n}\) such that for each element \(g \in G\), \(o(g)\) divides the order of \(f(g)\)? In the paper under review the author provides an affirmative answer to Isaacs' question.