On the modular version of Huppert's \(\rho\)-\(\sigma\) conjecture
type: Item
instance of: scholarly article
title: On the modular version of Huppert's \(\rho\)-\(\sigma\) conjecture
zbMATH Open document ID: 0949.20007
author: Ziqun Lu
author: Jiping Zhang
published in: Journal of Algebra
publication date: 2000-07-24T00:00:00Z
review text: Let \(p\) be a prime divisor of a finite solvable group \(G\), \(\text{IBr}_p(G)\) the set of all irreducible \(p\)-Brauer characters of \(G\). Let \(\sigma_p(G)=\max\{\sigma(\chi(1))\mid\chi\in\text{IBr}_p(G)\}\), where \(\sigma(n)\) is the number of distinct prime divisors of \(\chi(1)\). Next, \(\rho_p(G)\) denotes the set of prime divisors of irreducible \(p\)-Brauer characters of \(G\). There is the conjecture that \(|\rho_p(G)|\leq f(\sigma_p(G))\) for some function \(f\) (this is the \(\rho_p\)-\(\sigma_p\) conjecture). That conjecture for solvable groups is reduced to the case of nilpotent-by-metabelian groups. For the last groups, the authors find the function \(f\) in the following two cases: (1) the Sylow \(p\)-subgroups of \(G\) are cyclic. (2) \(\rho=2\) or \(p\) is a Fermat number.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1441804
zbMATH Keywords: irreducible \(p\)-Brauer characters
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: finite solvable groups
reviewed by: Yakov Berkovich
Wikidata QID: Q123121909
MaRDI profile type: Publication
full work available at URL: jabr.1999.8164
OpenAlex ID: W2032940393
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DOI: 10.1006/JABR.1999.8164
description: scientific article; zbMATH DE number 1441804