Finite \(p\)-groups up to isoclinism, which have only two conjugacy lengths
type: Item
instance of: scholarly article
title: Finite \(p\)-groups up to isoclinism, which have only two conjugacy lengths
zbMATH Open document ID: 0937.20011
author: Kenta Ishikawa
published in: Journal of Algebra
publication date: 2000-03-08T00:00:00Z
review text: Let \(G\) be a finite group and let \(\text{Cl}(G)\) be the set of conjugacy classes of \(G\). Let \(\text{cl}(G)=\{|C|:C\in\text{Cl}(G)\}=\{1,n_1,\dots,n_r\}\), where \(|C|\) is the number of elements in \(C\) and \(1<n_1<\dots<n_r\). Then \(G\) is called a group of conjugacy type \(\{1,n_1,\dots,n_r\}\). The structure of \(p\)-groups of conjugacy type \(\{1,n\}\) is studied here. The classification of \(p\)-groups of conjugacy type \(\{1,p\}\) and \(\{1,p^2\}\) is given using Hall's concept of isoclinism which is an equivalence relation between groups and it preserves the conjugacy type.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 1356304
zbMATH Keywords: finite groups
zbMATH Keywords: conjugacy classes
zbMATH Keywords: \(p\)-groups
zbMATH Keywords: isoclinism
reviewed by: N. M. Kurnosenko
MaRDI profile type: Publication
full work available at URL: jabr.1999.7907
OpenAlex ID: W1981028004
cites work: Some conditions which almost characterize Frobenius groups.
cites work: The classification of prime-power groups.
cites work: Q3934684
cites work: On Finite Groups with Given Conjugate Types I
cites work: The Groups of Order p 6 (p and Odd Prime)
cites work: Some p-groups of Frobenius and extra-special type
cites work: Breadth and commutator subgroups of p-groups
DOI: 10.1006/JABR.1999.7907
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description: scientific article; zbMATH DE number 1356304