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

TL;DR

This paper introduces a group-theoretical generalization of Euler's totient function, exploring divisibility properties of a new function defined on finite groups and characterizing groups with specific divisibility conditions.

Contribution

It determines finite groups where the generalized totient function respects subgroup inclusion, partially addressing a known open problem.

Findings

Characterization of groups satisfying the divisibility condition

Identification of structural properties of such groups

Partial solution to Problem 5.4 in the referenced work

Abstract

Let $G$ be a finite group and $\varphi(G)=|\{a\in G \mid o(a)=\exp(G)\}|$, 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 $\forall\, H,K\leq G$, $H\subseteq K$ implies $\varphi(H)\mid\varphi(K)$. This partially answers Problem 5.4 in \cite{6}.