On finite \(p\)-groups which have only two conjugacy lengths
type: Item
instance of: scholarly article
title: On finite \(p\)-groups which have only two conjugacy lengths
zbMATH Open document ID: 1003.20022
DOI: 10.1007/BF02773158
author: Kenta Ishikawa
published in: Israel Journal of Mathematics
publication date: 2002-10-28T00:00:00Z
review text: Let \(G\) be a nonabelian finite group. If all noncentral conjugacy classes of \(G\) have the same size \(n\), then, according to \textit{N. Itô} [Nagoya Math. J. 6, 17-28 (1953; Zbl 0053.01202)], \(n=p^\alpha\) for some prime \(p\) and \(G=P\times A\), where \(P\) is a Sylow \(p\)-subgroup of \(G\) and \(A\) is Abelian. Hence, the further study of such groups is reduced to that of \(p\)-groups. Itô has proved that \(G\) has an Abelian normal subgroup \(A\) such that \(G/A\) is of exponent \(p\) (through many years Isaacs proved that even \(\exp(G/Z(G))=p\)). Itô wrote: ``It may be of use to investigate whether there exists a group of type \((p^\alpha,1)\) ... of arbitrary high class''. As Isaacs proved, if \(p=2\) then \(\Phi(G)\leq Z(G)\), i.e., \(G\) is of class \(2\). It follows from Isaacs' result that \(G'\) is elementary Abelian. The author proves in this note that if \(p>2\), then the class of \(G\) does not exceed \(3\). The proof is highly computational.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1760710
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: conjugacy classes
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1760710