On finite groups with the Magnus Property

Abstract

AbstractWe investigate finite groups with the Magnus Property (MP), where a group is said to have the MP if whenever two elements have the same normal closure, then they are conjugate or inverse conjugate. In particular, we observe that a finite MP group is solvable, determine the finite primitive MP groups, and determine all the possible orders of the chief factors of a finite MP group. We also determine the MP finite direct products of finite primitive groups, as well as the MP crown‐based powers of a finite monolithic primitive group.