Irreducible subrepresentations of the conjugation representation of finite \(p\)-groups
type: Item
instance of: scholarly article
title: Irreducible subrepresentations of the conjugation representation of finite \(p\)-groups
zbMATH Open document ID: 0794.20013
DOI: 10.1007/BF02567665
author: Eberhard Kaniuth
author: Annette Markfort
published in: Manuscripta Mathematica
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/155690
review text: A finite group \(G\) acting on its group algebra \(\mathbb{C} G\) by conjugation gives rise to the conjugation representation \(\gamma_ G\) of \(G\). It was conjectured by \textit{R. L. Roth} [Pac. J. Math. 36, 515-521 (1971; Zbl 0192.357)] that \(\gamma_ G\) contains every irreducible representation of the central quotient group \(G/Z(G)\). However, using the existence of groups with outer automorphisms that preserve the conjugacy classes, \textit{E. Formanek} [Proc. Am. Math. Soc. 30, No. 3, 73-74 (1971; Zbl 0206.314)] constructed counterexamples. Let \(p\) be an arbitrary prime number. The authors construct counterexamples of order \(p^ 6\) and class 3 and of order \(p^ 7\) and class 2. They also show that these are minimal in the sense that a group of order \(p^ n\) and class \(c\) satisfies Roth's conjecture when \(n<6\) or when \(n=6\) and \(c\leq 2\).
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 60079
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: minimal counterexamples
zbMATH Keywords: finite group
zbMATH Keywords: group algebra
zbMATH Keywords: conjugation representation
zbMATH Keywords: irreducible representations
zbMATH Keywords: outer automorphisms
zbMATH Keywords: conjugacy classes
zbMATH Keywords: Roth's conjecture
reviewed by: G. E. Wall
MaRDI profile type: Publication
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OpenAlex ID: W2048613097
description: scientific article; zbMATH DE number 60079