\({\mathfrak X}C\)-elements in groups and Dietzmann classes
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.
scientific article; zbMATH DE number 1549701