Locally graded minimal non \(CC\)-groups are \(p\)-groups
type: Item
instance of: scholarly article
title: Locally graded minimal non \(CC\)-groups are \(p\)-groups
zbMATH Open document ID: 0778.20016
DOI: 10.1007/BF01196848
author: J. M. Peña
author: Brian Hartley
author: Javier Otal
published in: Archiv der Mathematik
publication date: 1992-09-27T00:00:00Z
review text: The group \(G\) is a \(CC\)-group if \(G/C_ G(x^ G)\) is Chernikov for all \(x\in G\). If \(G\) is not a \(CC\)-group but all its proper subgroups are \(CC\)-groups \(G\) is said to be a minimal non \(CC\)-group. \textit{J. Otal} and \textit{J. M. Peña} [Commun. Algebra 16, 1231-1242 (1988; Zbl 0644.20025)] have shown that a locally graded minimal non-\(CC\)-group is locally finite, countable and has no nontrivial finite or abelian images. From this starting point the authors adapt the methods of \textit{M. Kuzucuoglu} and \textit{R. E. Phillips} [Math. Proc. Camb. Philos. Soc. 105, 417-420 (1989; Zbl 0686.20034)] to show that a locally graded minimal non-\(CC\)- group must be a \(p\)-group. The existence of such a group remains an open question.
reviewed by: M. J. Tomkinson
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 62232
zbMATH Keywords: locally graded minimal non-\(CC\)-group
zbMATH Keywords: \(p\)-group
MaRDI profile type: Publication
cites work: Q3901697
cites work: Q5675882
cites work: Locally finite minimal non-FC-groups
cites work: Minimal non-cc-groups
cites work: Characterizations of the Conjugacy of Sylow p-Subgroups of CC-Groups
cites work: Q5654235
full work available at URL: https://doi.org/10.1007/bf01196848
OpenAlex ID: W1977846522
description: scientific article; zbMATH DE number 62232