Orbit sizes, character degrees and Sylow subgroups.
type: Item
instance of: scholarly article
title: Orbit sizes, character degrees and Sylow subgroups.
zbMATH Open document ID: 1068.20008
DOI: 10.1016/S0001-8708(03)00093-8
author: Alexander Moretó
author: Thomas Wolf
published in: Advances in Mathematics
publication date: 2004-08-06T00:00:00Z
review text: Let \(G\) be a finite group. The \(i\)-th Fitting subgroup is inductively defined by \(F_{i+1}(G)/F_i(G)=\) Fitting subgroup of \(G/F_i(G)\), where \(F_1(G)\) stands for the Fitting subgroup of \(G\). The authors show some significant connections between lengths of conjugacy classes of \(G\), degrees of irreducible characters of \(G\), and the ascending Fitting series \(G\geq\cdots\geq F_i(G)\geq\cdots\geq F_1(G)\). We give an overview of their results, without being complete in respect to all the results obtained in the paper. Part of Theorem A (paraphrased): If \(|F_{10}(G)|\) is odd, then there exists a product of three distinct irreducible characters \(\chi_1\), \(\chi_2\), \(\chi_3\) of \(G\) such that \(\chi_1\chi_2\chi_3(1)\) is a multiple of \(|G/F_1(G)|\). Theorem C: If \(G\) is solvable, then there exists \(\mu\in\text{Irr}(F_{10}(G))\) such that \(\mu^G=\chi\in\text{Irr}(G)\). So \(|G:F_{10}(G)|\) divides \(\chi(1)\). Theorem E: If \(V\) is a finite completely reducible faithful \(G\)-module for a solvable group \(G\), then (1) there exists \(v\in V\) such that \(C_G(v)\leq F_9(G)\), (2) if \(|GV|\) is odd, there exists \(v\in V\) in a regular orbit of \(F_1(G)\) such that \(C_G(v)\leq F_2(G)\). As applications the authors provide results on zeroes of characters, the height-zero-conjecture, Huppert's \(\rho\)-\(\sigma\) conjecture, large Abelian subgroups and large orbit sizes, large orbits in actions of nilpotent groups. All together, a wealth of nice new results is presented.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 2081954
zbMATH Keywords: transitive solvable permutation groups
zbMATH Keywords: nilpotent groups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: character degrees
zbMATH Keywords: Fitting subgroup
zbMATH Keywords: derived lengths
zbMATH Keywords: zeroes of characters
zbMATH Keywords: height-zero conjecture
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2081954