Finite and nilpotent strongly verbally closed groups

TL;DR

This paper investigates the properties of finite groups that are strongly verbally closed, demonstrating that such groups have centers that are direct factors if they are retracts of larger groups.

Contribution

It establishes a new structural property of finite strongly verbally closed groups, linking retracts and the central direct factors.

Findings

Centers of such groups are direct factors when they are retracts

Provides conditions under which finite groups are strongly verbally closed

Enhances understanding of subgroup embeddings in finite groups

Abstract

We show, in particular, that, if a finite group $H$ is a retract of any finite group containing $H$ as a verbally closed subgroup, then the centre of $H$ is a direct factor of $H$.