ON GROUPS WITH FINITE CONJUGACY CLASSES IN A VERBAL SUBGROUP

Let $w$ be a group-word. For a group $G$ , let $G_{w}$ denote the set of all $w$ -values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$ . The group $G$ is an $FC(w)$ -group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$ . It is known that if $w$ is a concise word, then $G$ is an $FC(w)$ -group if and only if $w(G)$ is $FC$ -embedded in $G$ , that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$ . There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$ , we show that if $G$ is an $FC(w)$ -group, then the commutator subgroup $w(G)^{\prime }$ is $FC$ -embedded in $G$ . We also establish the analogous result for $BFC(w)$ -groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly finite.