A residual property of certain groups

Abstract:

An element $a$ of a group $G$ is called conjugacy distinguished or c.d. in $G$ if and only if given any element $b$ of $G$ either $a$ is conjugate to $b$ or there is a homomorphism $\xi$ from $G$ onto a finite group $H$ such that $\xi (a)$ and $\xi (b)$ are not conjugate in $H$. Following A. Mostowski, a group $G$ is called conjugacy separable or c.s. if every element of $G$ is c.d. A. Mostowski remarks that a direct product of c.s. groups is c.s. and proves that the conjugacy problem can be solved for finitely presented c.s. groups. N. Blackburn proves that finitely generated nilpotent groups are c.s. In this paper it is proven that a free product of c.s. groups is c.s., a free group is c.s. and that every element of infinite order in a finite extension of a free group is c.d.