We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

The word $w=[x_{i_{1}},x_{i_{2}},\ldots ,x_{i_{k}}]$ is a simple commutator word if $k\geq 2,i_{1}\neq i_{2}$ and $i_{j}\in \{1,\ldots ,m\}$ for some $m>1$. For a finite group $G$, we prove that if $i_{1}\neq i_{j}$ for every $j\neq 1$, then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$-values $a,b\in G$ of coprime orders. We also extend the result to a residually finite group $G$, provided that the set of all $w$-values in $G$ is finite.

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 S be a subset of a group G such that S−1 = S. Denote by gr (S) the subgroup of G generated by S, and by ls(g) the length of an element g ∈ gr(S) relative to the set S. Suppose that V is a finite subset of a free group F of countable rank such that the verbal subgroup V (F) is a proper subgroup of F. For an arbitrary group G, denote by (G) the set of values in G of all the words from the set V. In the present paper, for amalgamated products G = A *HB such that A ≠ H and the number of double cosets of B by H is at least three, the infiniteness of the set {ls(g) | g ∈ gr(S)}, where S = (G) ∪ (G)−1, is estabilished.