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

TL;DR

This paper classifies finite groups based on the maximum Frobenius quotient for prime-power divisors of their exponent, linking group structure to prime divisors of group order.

Contribution

It provides a complete classification of finite groups with maximum Frobenius quotient bounded by the smallest prime divisor of their order.

Findings

Characterization of groups with bounded Frobenius quotient

Relation between Frobenius quotient and prime divisors of group order

Complete classification for the case when maximum Frobenius quotient is small

Abstract

Let $G$ be a finite group and $n$ be any prime-power divisor of ${\rm exp}(G)$, 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 ${\rm exp}(G)$$\}$ and ${\rm mf}_{pp}(G)$ be the maximum Frobenius quotient in $\mathcal{F}_{pp}(G)$. In this paper, we provide a complete classification of finite group $G$ with ${\rm mf}_{pp}(G)\leq q$, where $q$ is the smallest prime divisor of $|G|$.