On a group-theoretical generalization of the Euler's totient function (Q6669294)

Let \(G\) be a finite group and let \(\varphi(G)=\big | \big \{ g \in G \; \big | \; o(g)=\exp(G) \big \} \big | \). Since \(\varphi(\mathbb{Z}_{n})=\varphi(n)\) for all \(n\geq 1\), it generalizes the well-known Euler's totient function.\N\NIn the paper under review the author proves that, if \(\varphi(S) \not = 0\) for any subgroup \(S\) of \(G\) and \(\varphi(H)\, \big |\, \varphi(K)\) whenever \(H \leq K \leq G\), then \(G\) is nilpotent and its Sylow subgroups are cyclic, \(Q_{8}\) or \(\mathbb{Z}_{p} \times \mathbb{Z}_{p}\) (\(p\) a prime).