On groups with BFC-covered word values (Q6612165)

Let \(G\) be a group, \(n \in \mathbb{N}\) and \(B_{n}(G)=\big | \big \{ x \in G \mid |x^{G}| \leq n \big \} \big |\). If \(s \geq 1\) and \(w\) is a group word, then \(G\) satisfies the \((n,s)\)-covering condition with respect to the word \(w\) if there exists a subset \(S \subseteq G\) such that \(|S| \leq s\) and all \(w\)-values of \(G\) are contained in \(B_{n} (G)S\).\N\NIn the paper under review, the authors obtain the following results:\N\NTheorem 1.1: Let \(w\) be a multilinear commutator word on \(k\) variables and let \(G\) be a group satisfying the \((n,s)\)-covering condition with respect to the word \(w\). Then \(G\) has a soluble subgroup \(T\) such that \(|G : T|\) and the derived length of \(T\) are both \((k, n,s)\)-bounded.\N\NTheorem 1.2: Let \(k \geq 1\) and \(G\) be a group satisfying the \((n,s)\)-covering condition with respect to the word \(\gamma_{k}\). Then \N\begin{itemize}\N\item[(1)] \(\gamma_{2k-1}(G)\) has a subgroup \(T\) such that \(|\gamma_{2k-1}(G) : T|\) and \(|T '|\) are both \((k, n, s)\)-bounded and \N\item[(2)] \(G\) has a nilpotent subgroup \(U\) such that \(|G : U|\) and the nilpotency class of \(U\) are both \((k,n,s)\)-bounded.\N\end{itemize}