\({\mathfrak X}C\)-elements in groups and Dietzmann classes
type: Item
instance of: scholarly article
title: \({\mathfrak X}C\)-elements in groups and Dietzmann classes
zbMATH Open document ID: 0965.20020
author: Rudolf Maier
author: J. R. Rogério
published in: Beiträge zur Algebra und Geometrie
publication date: 2001-01-09T00:00:00Z
full work available at URL: https://eudml.org/doc/227935
full work available at URL: 19.html
review text: Let \(\mathfrak X\) be a class of groups. An element \(x\) of a group \(G\) is said to be an \({\mathfrak X}C\)-element if the factor group \(G/C_G(\langle x\rangle^G)\) belongs to \(\mathfrak X\). The set \(Z_{\mathfrak X}(G)\) of all \({\mathfrak X}C\)-elements of \(G\) is called the \({\mathfrak X}C\)-centre of \(G\), and \(G\) is an \({\mathfrak X}C\)-group if \(Z_{\mathfrak X}(G)=G\). In particular, if \(\mathfrak F\) denotes the class of all finite groups, \({\mathfrak F}C\)-groups are precisely the groups with finite conjugacy classes of elements, and the structure of such groups has been investigated by several authors. A class \(\mathfrak D\) of groups is called a Dietzmann class if \(\mathfrak D\) is closed with respect to subgroups and homomorphic images and if \(\langle x\rangle^G\) belongs to \(\mathfrak D\) whenever \(x\) is an \({\mathfrak D}C\)-element of a group \(G\) such that \(\langle x\rangle\) is a \(\mathfrak D\)-group. Thus the classical Dietzmann Lemma just states that \(\mathfrak F\) is a Dietzmann class. Finally, a group class \(\mathfrak X\) is said to be centrally saturated if it is closed under subgroups and homomorphic images and if every central-by-\(\mathfrak X\) group is likewise an \(\mathfrak X\)-group. Clearly every centrally saturated class of groups is a Dietzmann class, while the class \({\mathfrak F}_\pi\), consisting of all finite \(\pi\)-groups for a given set of prime numbers \(\pi\), is a Dietzmann class which is not centrally saturated. This article is devoted to the study of Dietzmann classes, and of their behaviour under the action of certain operators. Among other results, the authors prove that if \(\mathfrak D\) is a Dietzmann class, then also the class \(L{\mathfrak D}\) of all local \(\mathfrak D\)-groups and the class \(Z{\mathfrak D}\) of all central-by-\(\mathfrak D\) groups are Dietzmann classes.
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 1549701
zbMATH Keywords: Dietzmann Lemma
zbMATH Keywords: conjugacy classes
zbMATH Keywords: classes of groups
zbMATH Keywords: Dietzmann classes
zbMATH Keywords: FC-groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1549701