Finite symmetric groups are strongly verbally closed

TL;DR

This paper proves that finite symmetric and alternating groups are retracts of any larger group in which they are verbally closed, answering a question posed by A. V. Vasil'ev.

Contribution

It establishes that finite symmetric and alternating groups are strongly verbally closed, providing a significant insight into their algebraic embedding properties.

Findings

Finite symmetric groups are retracts of any containing group where they are verbally closed.

Finite alternating groups share the same property.

Answers a longstanding question by Vasil'ev about verbal closedness.

Abstract

Answering a question of A. V. Vasil'ev, we show that each finite symmetric (or alternating) group $H$ is a retract of any group containing $H$ as a verbally closed subgroup.