A characterization of root classes of groups

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.