A characterization of cyclic groups
type: Item
instance of: scholarly article
title: A characterization of cyclic groups
zbMATH Open document ID: 0636.20018
DOI: 10.1007/BF01194381
author: Jacques Thévenaz
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: If \((H)\) denotes the conjugacy class of a subgroup H of a finite group G, let \(a_{(H),(K)}=| H\setminus G/K|\), the number of double cosets. Let \(A\) be the square matrix \((a_{(H),(K)})\), indexed by the set of conjugacy classes of subgroups of \(G\). It is shown that \(\det(A)\neq 0\) if and only if \(G\) is cyclic. In fact the rank of \(A\) is the number of conjugacy classes of cyclic subgroups of \(G\). Moreover if \(G\) is cyclic of order \(n\), then \(\det (A)=\prod _{d\mid n}\varphi(d)\), where \(\varphi\) denotes the Euler function.
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4036061
zbMATH Keywords: numbers of double cosets
zbMATH Keywords: numbers of conjugacy classes of cyclic subgroups
MaRDI profile type: Publication
cites work: Artin exponent of finite groups
cites work: The Burnside algebra of a finite group
cites work: Transformation groups and representation theory
full work available at URL: https://doi.org/10.1007/bf01194381
OpenAlex ID: W1525741878
description: scientific article; zbMATH DE number 4036061