$\mathcal{C}$-Hereditarily conjugacy separable groups and wreath products

TL;DR

This paper characterizes when wreath products are hereditarily conjugacy separable within a certain class of finite groups, and constructs new examples of such groups, including the Grigorchuk group and lamplighter groups.

Contribution

It provides a necessary and sufficient condition for wreath products to be $ ext{C}$-hereditarily conjugacy separable and constructs new examples of such groups with arbitrary derived length.

Findings

Wreath products are $ ext{C}$-hereditarily conjugacy separable under specific conditions.

The Grigorchuk group is 2-hereditarily conjugacy separable.

Lamplighter groups and $ ext{Z} ext{wr} ext{Z}$ are hereditarily conjugacy separable.

Abstract

We provide a necessary and sufficient condition for the restricted wreath product $A\wr B$ to be $\mathcal{C}$-hereditarily conjugacy separable where $\mathcal{C}$ is an extension-closed pseudovariety of finite groups. Moreover, we prove that the Grigorchuk group is 2-hereditarily conjugacy separable. As an application, we demonstrate that the lamplighter groups and $\mathbb{Z} \wr \mathbb{Z}$ are hereditarily conjugacy separable (but not $p$-conjugacy separable for any prime $p$) which provides infinitely many new examples of solvable, non-polycyclic hereditarily conjugacy separable groups. Furthermore, we study wreath products of cyclic subgroup separable groups and the derived length of iterated wreath products of solvable groups with an abelian base group and, as an application, we give an explicit construction of non-polycyclic hereditarily conjugacy separable groups of arbitrary derived length as an iterated wreath products of abelian groups.