Ordinary and p-modular character degrees of solvable groups
type: Item
instance of: scholarly article
title: Ordinary and p-modular character degrees of solvable groups
zbMATH Open document ID: 0714.20005
DOI: 10.1016/0021-8693(91)90053-B
author: Udo Leisering
published in: Journal of Algebra
publication date: 1991-01-01T00:00:00Z
review text: Let G be a finite solvable group, let p be a prime and let \(cd(G)=\{\chi (1)|\chi \in Irr_{{\mathbb{C}}}(G)\}\) resp. \(cd_ p(G)=\{\beta (1)|\beta \in IBr_ p(G)\}\) denote the set of irreducible complex resp. p-modular character degrees. Note that \(cd_ p(G)\) is a subset of cd(G) by the Fong-Swan theorem. The paper under review is concerned with the following question: Given arithmetical conditions on \(cd_ p(G)\). What can we say about cd(G). Most of the paper is spent for the proof of the following main result: If \(O_ p(G)=1\) and \(cd_ p(G)\) is squarefree (i.e. \(q^ 2\nmid \beta (1)\) for all primes q and all \(\beta \in IBr_ p(G))\) then cd(G) is cubefree. In this case the group structure is described to some extend very precise. Easy examples show that the bound is best possible. In a final section the author assumes more generally that \(q^ n\nmid \beta (1)\) for all \(\beta (1)\in cd_ p(G)\), all primes q and some \(n\in {\mathbb{N}}\). Functions in n are then given which bound the derived length of G, the Fitting length of G and the largest exponent m such that \(q^ m| \chi (1)\) for some \(\chi \in Irr_{{\mathbb{C}}}(G)\) and some prime q.
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4175226
zbMATH Keywords: finite solvable group
zbMATH Keywords: p-modular character degrees
zbMATH Keywords: Fong-Swan theorem
zbMATH Keywords: derived length
zbMATH Keywords: Fitting length
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4175226