Classes of groups which resemble varieties (Q762272)

This paper studies certain classes of groups defined in a uniform although rather complicated way. The purpose is to provide a general setting which encompasses cases of known interest such as certain classes of torsion-free generalized soluble groups. The main results of the paper show that the classes enjoy certain closure properties. Starting with a given set of words, the author defines for each group G and each ordinal \(\alpha\) a fully-invariant subgroup \(\xi_{\alpha}(G)\) of G. Then \(X_{\alpha}\) is defined to be the class of all groups G for which \(\xi_{\alpha}(G)=1\), and X is the union of the classes \(X_{\alpha}\) where \(\alpha\) is finite. It is shown that each \(X_{\alpha}\) is subgroup closed and that \(X_{\alpha}\) is the class of all groups having a descending series of subgroups which has length \(\alpha\) and where the factors are of a certain type. It is proved that \(X_{\alpha}\) is residually closed and that when \(\alpha\) is finite \(X_{\alpha}\) is locally closed. When certain extra conditions hold X is poly closed. One example of the general construction gives X as the class of all ''completely infinite'' groups, previously studied by the author [Proc. Lond. Math. Soc., III. Ser. 28, 595-613 (1974; Zbl 0288.20034)]. In another example, corresponding to a given prime number p, a finitely generated group G belongs to \(X_{\omega}\) if and only if G is residually a finite p-group.