Costantino Delizia, Pavel Shumyatsky, Antonio Tortora
March 20, 2020

### Abstract

Let w be a group-word. For a group G , let Gw{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 Gw{G_{w}} is finite. The group G is an FC(w){\mathrm{FC}(w)}-group if the set of conjugates xGw{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 CG(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 xGw{x^{G_{w}}} are boundedly finite. We also show that this is no longer true if w is not semiconcise.