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.

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关于半简洁的词

摘要 让 w 是一个组词。对于群 G，令 G w {G_{w}} 表示 G 中所有 w 值的集合，w ⁢ ( G ) {w(G)} 表示 G 对应于 w 的语言子群。如果子群 [ w ⁢ ( G ) , G ] {[w(G),G]} 是有限的，只要 G w {G_{w}} 是有限的，那么单词 w 就是半简洁的。群 G 是一个 FC ⁢ ( w ) {\mathrm{FC}(w)} -群，如果共轭集合 x G w {x^{G_{w}}} 对于所有 x ∈ G {x\在 G} 中。我们证明如果 w 是一个半简词，G 是一个 FC ⁢ ( w ) {\mathrm{FC}(w)} -群，那么子群 [ w ⁢ ( G ) , G ] {[w(G), G]} 是 FC {\mathrm{FC}} -嵌入 G，即交集 CG ⁢ ( x ) ∩ [ w ⁢ ( G ) , G ] {C_{G}(x)\cap[w( G),G]} 在 [ w ⁢ ( G ) , G ] {[w(G),G]} 中对于所有 x ∈ G {x\in G} 具有有限索引。类似的结果适用于 BFC ⁢ ( w ) {\mathrm{BFC}(w)} -groups，即集合 x G w {x^{G_{w}}} 是有界有限的群。我们还表明，如果 w 不是半简洁的，则这不再成立。