A characterization of root classes of groups

TL;DR

This paper characterizes root classes of groups, showing they are precisely those closed under subgroups and Cartesian wreath products, and explores residual properties of certain free products within these classes.

Contribution

It provides a complete characterization of root classes of groups and applies this to analyze residual properties of generalized free products of nilpotent groups.

Findings

Root classes are exactly those closed under subgroups and Cartesian wreath products.

Nontrivial root classes closed under quotients include residual properties of free products.

Generalized free products of nilpotent C-groups are residually solvable within the class.

Abstract

A class of groups C is root in a sense of K. W. Gruenberg if it is closed under taking subgroups and satisfies the Gruenberg condition: for any group X and for any subnormal sequence Z \leqslant Y \leqslant X with factors in C, there exists a normal subgroup T of X such that T \leqslant Z and X/T \in C. We prove that a class of groups is root if, and only if, it is closed under subgroups and Cartesian wreath products. Using this result we prove also that, if C is a nontrivial root class of groups closed under taking quotient groups and G = <A*B; H=K, \varphi> is the generalized free product of two nilpotent C-groups A and B possessing \varphi-compartible central series, then G is residually a solvable C-group.