Finite symmetric groups are strongly verbally closed

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.

4 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: An argument is simplified (thanks to A. Yu. Olshanskii)

References in corpus (5)

• Virtually free finite-normal-subgroup-free groups are strongly verbally closed
• Finite and nilpotent strongly verbally closed groups
• The Klein bottle group is not strongly verbally closed, though awfully close to being so
• Finite normal subgroups of strongly verbally closed groups
• The centre of a finitely generated strongly verbally closed group is almost always pure