Year: 2026
Type: preprint
Abstract: Let $G$ be a finite group and let $π(G)$ be the set of prime divisors of $|G|$. For each $p\in π(G)$, choose a Sylow $p$-subgroup $G_p$ of $G$. The subgroup $Ψ(G)= \prod_{p\in π(G)} Φ(G_p)_G$ is called the \textit{local Frattini subgroup} of $G$. Here, for $J \leqslant G$, the symbol $J_G$ denotes the core of $J$ in $G$. The subgroup $Ψ(G)$ is a characteristic subgroup of $G$ that is contained in $Φ(G)$, the Frattini subgroup of $G$. We prove that $Ψ(G)$ satisfies many properties similar to $Φ(G)$. Let $J$ be a normal $p$-subgroup of $G$ with $J \leqslant Φ(G)$, and let $S$ be a Sylow $p$-subgroup of $G$. John S. Rose proved that if $J \leqslant \mathbf{Z}(S)$, then $J \leqslant Φ(S)$. So, in fact, $J \leqslant Φ(S)_G \leqslant Ψ(G)$. Kaplan, Lev, Mann, and Solomon found various other conditions that would ensure $J$ is contained in $Φ(S)$. In this article, we add a few more conditions to this list.
Source: arXiv (Cornell University)
Author: Costanzo, David G.
Institution: Clemson University
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221811103