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

Abstract

Let $G$ be a finite group and $\phi (G)=|\{a\in G\mid o(a)=\exp (G)\}|$, where $o(a)$ denotes the order of $a$ in $G$ and $\exp (G)$ denotes the exponent of $G$. Under a natural hypothesis, in this note we determine the groups $G$ such that $\forall H,K\le G$, $H\subseteq K$ implies $\phi (H)\mid \phi (K)$. This partially answers Problem 5.4 in \cite{6}.