Solvable groups having almost relatively prime distinct irreducible character degrees
type: Item
instance of: scholarly article
title: Solvable groups having almost relatively prime distinct irreducible character degrees
zbMATH Open document ID: 0833.20011
author: Mark L. Lewis
published in: Journal of Algebra
publication date: 1996-03-12T00:00:00Z
review text: Suppose that the finite solvable group \(G\) has the property that for every \(m,n\in\{\chi(1)\mid\chi\in\text{Irr}(G)\}\) with \(m\neq n\), it holds that the greatest common divisor of \(m\) and \(n\) equals either 1 or a prime. Then the main theorem of this paper provides the bound \(\#\{\chi(1)\mid\chi\in\text{Irr}(G)\}\leq 14\). The proof is a little tedious but clear, as it needs a lot of preliminary results all proved here. It is worthwhile to study these results on their own account. There are some theorems derived from the main theorem, for instance when \(K\) is a normal subgroup of \(G\) with \(G/K\) a nonabelian \(p\)-group for some prime \(p\), then the bound 14 can be replaced by 7. The paper deserves a serious study by the interested reader as there is a lot to learn from it.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 783861
zbMATH Keywords: irreducible characters
zbMATH Keywords: character degrees
zbMATH Keywords: finite solvable groups
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
full work available at URL: jabr.1995.1124
OpenAlex ID: W2037492649
DOI: 10.1006/JABR.1995.1124
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description: scientific article; zbMATH DE number 783861