On $\mathfrak{X}$-transitive groups and conjugate separable $\mathfrak{X}$-subgroups

TL;DR

This paper develops a systematic theory of certain classes of groups related to a variety nd explores their structural properties, connections with equational domains, and residual properties, establishing that finite SX-groups belong to nd analyzing their interplay.

Contribution

It introduces a comprehensive framework for SX- and T-groups, extending previous ideas, and proves that finite SX-groups are contained within the variety or the first time.

Findings

Finite SX-groups are in 

Structural properties of SX- and T-groups are characterized

Connections with equational domains and residual properties are established

Abstract

For a given variety of groups $\X$, we develop a systematic theory of $\CSX$-groups and $\XT$-groups, extending ideas proposed in \cite{Shah}. We analyze the interplay between these classes, describe their structural properties, and examine their connections with equational domains and residually $A$-free groups. Furthermore, we prove by elementary means that every finite $\CSX$-group lies in $\X$.