A note on Frobenius quotient for prime-power divisor of the exponent of finite groups
Jiangtao Shi, Wenjing Liu
TL;DR
This work defines the maximum Frobenius quotient over prime-power divisors of the exponent, mf_pp(G), and classifies finite groups with mf_pp(G) <= q where q is the smallest prime dividing |G|. The main result identifies two structural families: (i) G = Q × R with R cyclic of order coprime to q and Q restricted to a short list of q-groups (cyclic, Z_{q^{m-1}} × Z_q, Q_8, or certain semidirect products), and (ii) a product involving a specific metacyclic group and a cyclic factor T with gcd(|T|,6) = 1. A central lemma provides a lower bound on |F_{|Q|}(G)| in terms of the Sylow q-subgroup structure, driving the necessity part of the classification. These results extend prior mf(G) classifications to prime-power Frobenius spectra, delivering explicit group-structure characterizations relevant to the Frobenius spectrum of finite groups.
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 $|G|$.