Paracyclic and quasi-cyclic groups
type: Item
instance of: scholarly article
title: Paracyclic and quasi-cyclic groups
zbMATH Open document ID: 0618.20018
DOI: 10.1007/BF01790544
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1986-01-01T00:00:00Z
review text: Let G be a finite group. The authors define the multiplicity \(\gamma_ G\) of G to be the maximum of \(| (x\in G:\) \(x^ d=1)| /d\) taken over all divisors d of \(| G|\), and the cocyclicity \(\iota_ G\) as the least index in G of any cyclic subgroup of G. In earlier work [\textit{R. Dvornicich} and \textit{M. Forti}, J. Algebra 91, 499-519 and 520- 535 (1984; Zbl 0581.20022 and Zbl 0558.20018)] the authors have shown that the ratio \(\iota_ G/\gamma_ G\) is unbounded but grows no faster than \(\gamma_ G^{\epsilon}\) for any \(\epsilon >0\). The main result of the present paper is that this ratio is at most 1 when G is metacyclic. Results are also obtained in the case where G has at most one noncyclic Sylow subgroup.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 4002366
zbMATH Keywords: finite group
zbMATH Keywords: multiplicity
zbMATH Keywords: cocyclicity
zbMATH Keywords: cyclic subgroup
zbMATH Keywords: metacyclic
author: Roberto Dvornicich
author: Marco Forti
MaRDI profile type: Publication
cites work: Finite groups with few dth roots of 1
cites work: Large cyclic subgroups of finite groups
cites work: The characterization of finite groups with dihedral Sylow 2-subgroups. I, II
cites work: Endliche Gruppen I
cites work: On Finite Groups with Cyclic Sylow Subgroups for all Odd Primes
cites work: On finite groups with semi-dihedral Sylow 2-subgroups
description: scientific article; zbMATH DE number 4002366