On semiconcise words
Let \(w=w(x_1, \ldots, x_n)\) be a group-word in the variables \(x_1, \ldots, x_n\). If \(G\) is a group, the set \(G_{w}=\{ w(g_{1}, \ldots, g_n) \mid g_1, \ldots, g_n \in G \}\) of all \(w\)-values of \(G\) is a normal subset of \(G\). The verbal subgroup of \(G\) corresponding to \(w\) is the normal subgroup \(w(G)=\langle G_{w} \rangle\). A word \(w\) is called concise if the verbal subgroup \(w(G)\) is finite in each group \(G\) such that \(G_w\) is finite. In the paper under review, a word \(w\) is called \textit{semiconcise} if the finiteness of \(G_w\) for a group \(G\) always implies the finiteness of \([w(G),G]\). A subgroup \(H\) of a group \(G\) is FC-embedded in \(G\) if \(x^{H}=\{x^{h} \mid h \in H \}\) is finite for all \(x\in G\); \(H\) is BFC-embedded in \(G\) if \(x^{H}\) is finite and \(|x^{H}|\) is bounded by a constant that does not depend on the choice of \(x\). For any group-word \(w\), a group \(G\) is said to be an \(\mathrm{FC}(w)\)-group if \(x^{G_{w}}\) is finite for all \(x \in G\) and a \(\mathrm{BFC}(w)\)-group if \(x^{G_w}\) is finite for all \(x \in G\) and \(|x^{G_{w}}|\) is bounded by a constant that does not depend on the choice of \(x\). The main results of this paper are the following theorems. Theorem A: Let \(w\) be a semiconcise word, and let \(G\) be an \(\mathrm{FC}(w)\)-group. Then \([w(G),G]\) is FC-embedded in \(G\). Theorem B: Let \(w\) be a semiconcise word, and let \(G\) be an \(\mathrm{BFC}(w)\)-group. Then \([w(G),G]\) is BFC-embedded in \(G\).
scientific article; zbMATH DE number 7216584