A note on Frobenius quotient for prime-power divisor of the exponent of finite groups

Abstract

Let $G$ be a finite group and $n$ be any prime-power divisor of $\exp (G)$, the exponent of $G$. Frobenius' theorem indicates that $|\{g\in G\mid g^n=1\}|=f_n\cdot n$ for some positive integer $f_n$. We call $f_n$ a Frobenius quotient of $G$ for $n$. Let ${\mathcal{F}}_{pp}(G)=\{f_n\mid n$ is any prime-power divisor of $\exp (G)$$\}$ and ${\mathrm{mf}}_{pp}(G)$ be the maximum Frobenius quotient in ${\mathcal{F}}_{pp}(G)$. In this paper, we provide a complete classification of finite group $G$ with ${\mathrm{mf}}_{pp}(G)\le q$, where $q$ is the smallest prime divisor of $|G|$.