On semiconcise words

Abstract

Abstract 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.