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Let w be a group-word. For a group G, let G w {G_{w}} denote the set of all w-values in G and w ⁢ ( G ) {w(G)} the verbal subgroup of G corresponding to w. The word w is semiconcise if the subgroup [ w ⁢ ( G ) , G ] {[w(G),G]} is finite whenever G w {G_{w}} is finite. The group G is an FC ⁢ ( w ) {\mathrm{FC}(w)} -group if the set of conjugates x G w {x^{G_{w}}} is finite for all x ∈ G {x\in G} . We prove that if w is a semiconcise word and G is an FC ⁢ ( w ) {\mathrm{FC}(w)} -group, then the subgroup [ w ⁢ ( G ) , G ] {[w(G),G]} is FC {\mathrm{FC}} -embedded in G, that is, the intersection C G ⁢ ( x ) ∩ [ w ⁢ ( G ) , G ] {C_{G}(x)\cap[w(G),G]} has finite index in [ w ⁢ ( G ) , G ] {[w(G),G]} for all x ∈ G {x\in G} . A similar result holds for BFC ⁢ ( w ) {\mathrm{BFC}(w)} -groups, that are groups in which the sets x G w {x^{G_{w}}} are boundedly finite. We also show that this is no longer true if w is not semiconcise.

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.

Let n be an integer. A group G is said to be n-abelian if the map phi_n that sends g to g^n is an endomorphism of G. Then (xy)^n=x^ny^n for all x,y in G, from which it follows [x^n,y]=[x,y]^n=[x,y^n]. It is also easy to see that a group G is n-abelian if and only if it is (1-n)-abelian. If nneq 0,1 and G is an n-abelian group, then the quotient group G/Z(G) has finite exponent dividing n(n-1). This implies that every torsion-free n-abelian group is abelian. We denote by B_n and C_n the classes of all groups G for which phi_n is a monomorphism and an epimorphism of G, respectively. Then B_0=C_0 contains only the trivial group, B_1=C_1 is the class of all groups, and B_-1=C_-1 is the class of all abelian groups. Furthermore, with |n|>1, G is in B_n if and only if G is an n-abelian group having no elements of order dividing |n|. Similarly, G is in C_n if and only if G is n-abelian and for every g in G there exists an element x in G such that g=x^n. We also set A_n=B_ncap C_n. In this paper we give a characterization for groups in B_n and for groups in C_n. We also obtain an arithmetic description of the set of all integers n such that a group G is in A_n.