Character codegrees, kernels, and Fitting heights of solvable groups

2025/02/04 by Guohua Qian, Yu Zeng, Qian, Guohua +1 · 1 citation
Engineering · Mathematics · #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2502.01950

openalex publication_date 2025/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an irreducible character χ of a finite group G, let cod(χ):=|G: ker(χ)|/χ(1) denote the codegree of χ, and let cod(G) be the set of irreducible character codegrees of G. In this note, we prove that if ker(χ) is not nilpotent, then there exists an irreducible character ξ of G such that ker(ξ)<ker(χ) and cod(ξ)> cod(χ). This provides a character codegree analogue of a classical theorem of Broline and Garrison. As a consequence, we obtain that for a nonidentity solvable group G, its Fitting height ℓ_F(G) does not exceed |cod(G)|-1. Additionally, we provide two other upper bounds for the Fitting height of a solvable group G as follows: ℓ_F(G)≤ (1)/(2)(|cod(G)|+2), and ℓ_F(G)≤ 8log₂(|cod(G)|)+80.

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