A Proof of Gluck's Conjecture

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 $G$.

Disclosure

“, University of Warwick. I also thank Prof. Derek Holt for a helpful discussion concerning possible corrections to the classification in [12] during the conference Modern Flavours of Finite Groups. During the preparation of this paper, the GPT-5.4 model was used to assist with debugging and improving some verification programs and translating GAP code into Magma. I checked and revised the resulting text and code, verified the computations, and take full responsibility for the conten”

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Classification

Code generation, completion, or debugging

Multiplier

2

Verified

2026-08-23T05:10:22.898159Z

Structural counts