On a conjecture of Gluck

2014/02/03 by James P. Cossey, Zoltán Halasi, Cossey, James P. +5
Computer Science · Engineering · Mathematics · #20C30 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary 20C15 #Secondary 20D06 #graph theory and CDMA systems

paper · doi:10.48550/arxiv.1402.0366

openalex publication_date 2014/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F(G) and b(G) respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group G. A well-known conjecture of D. Gluck claims that if G is solvable then |G:F(G)|≤ b(G)². We confirm this conjecture in the case where |F(G)| is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure.

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