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

Let $G$ be a finite group and $n$ be any prime-power divisor of $\mathrm{e}\mathrm{x}\mathrm{p}(G)$, the exponent of $G$. Frobenius’ theorem indicates that $\mathrm{\mid }\{g\in G\mid g^n=1\}\mathrm{\mid }=f_n\cdot n$ 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 $\mathrm{e}\mathrm{x}\mathrm{p}(G)$$\}$ and ${\mathrm{m}\mathrm{f}}_{pp}(G)$ be the maximum Frobenius quotient in ${\mathcal{F}}_{pp}(G)$. In this paper, we provide a complete classification of finite group $G$ with ${\mathrm{m}\mathrm{f}}_{pp}(G)\le q$, where $q$ is the smallest prime divisor of $\mathrm{\mid }G\mathrm{\mid }$.