Year: 2026
Type: preprint
Abstract: For a finite group $H$, let $ν(H)$ denote the maximum order of a nilpotent subgroup of $H$. We prove that every finite solvable transitive permutation group $P$ on a finite set $Ω$ has a subset $Δ\subseteqΩ$ such that $|P:P_Δ|\geν(P_Δ)$. We also prove that if a finite solvable group $H$ acts faithfully and completely reducibly on a finite module $V$, then some $x\in V$ satisfies $|H:H_x|\geν(H_x)$. Consequently, we settle Gluck's well-known conjecture, open since 1985: every finite solvable group $G$ satisfies $|G:\mathbf{F}(G)|\le b(G)^2$, where $\mathbf{F}(G)$ is the largest normal nilpotent subgroup of $G$ and $b(G)$ is the largest degree of an irreducible complex character of $G$.
Source: arXiv (Cornell University)
Author: Zhang, Baoyu
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: University of Warwick
OpenAlex ID: W7202355133
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