Year: 2026
Type: preprint
Abstract: For a finite group $G$, let $ρ^{*}(G)$ be the set of primes dividing some sub-class size of $G$, and let $σ^{*}(G)$ be the maximum number of distinct prime divisors of a sub-class size. Motivated by the conjugacy-class version of Huppert's $ρ$--$σ$ problem, we prove that \[ |ρ^{*}(G)|<2σ^{*}(G) \] for every finite nonnilpotent group $G$. This answers \cite[Remark~5.3]{QY} and extends to arbitrary finite nonnilpotent groups the coefficient-$2$ bound proved in \cite[Theorem~1.7]{QY} for solvable nonnilpotent groups. The coefficient $2$ is best possible. The proof does not use the classification of finite simple groups.
Source: arXiv (Cornell University)
Author: Yang, Yong
Institution: Texas State University
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214288727