On semiconcise words

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