Brazil, Sergio; Krasilnikov, Alexei; Shumyatsky, Pavel Groups with bounded verbal conjugacy classes. (English) Zbl 1108.20035 J. Group Theory 9, No. 1, 127-137 (2006). Let \(F\) be a free group, \(w\in F\), \(G_w\) be the set of all \(w\)-values in \(G\) and let \(w(G)\) be the verbal subgroup of \(G\) corresponding to \(w\) (i.e., \(w(G)\) is the subgroup generated by \(G_w\)). A word \(w\) is called boundedly concise if, for each group \(G\) such that \(|G_w|\leq m\), we have \(|w(G)|\leq c\) for some integer \(c\) depending only on \(m\). The main theorem of this paper says that if \(w\) is a boundedly concise word and \(G\) is a group such that \(|x^{G_w}|\leq m\) for all \(x\in G\) then \(|x^{w(G)}|\leq d\) for all \(x\in G\) and some integer \(d\) depending only on \(m\) and \(w\). It is shown that all non-commutator words, the lower central words \(\gamma_k\), and derived words \(\delta_k\) are boundedly concise. Also it is found a group \(G\) and a word \(w\) such that \(G\) contains at most four \(w\)-values and the set \(b^{w(G)}\) is infinite. Reviewer: Alexander Ivanovich Budkin (Barnaul) Cited in 9 Documents MSC: 20F24 FC-groups and their generalizations 20E45 Conjugacy classes for groups 20E10 Quasivarieties and varieties of groups 20E05 Free nonabelian groups Keywords:verbal subgroups; FC-groups; boundedly concise words; conjugacy classes; FC-embedded subgroups PDFBibTeX XMLCite \textit{S. Brazil} et al., J. Group Theory 9, No. 1, 127--137 (2006; Zbl 1108.20035) Full Text: DOI References: [1] Franciosi S., Houston J. Math. 28 pp 683– (2002) [2] Ivanov S. V., VUZ) 33 pp 6– (1989) [3] Neumann B. H., Proc. London Math. Soc. 1 pp 178– (3) · Zbl 0043.02401 · doi:10.1112/plms/s3-1.1.178 [4] Segal D., Quart. J. Math. Oxford Ser. 50 pp 505– (2) · Zbl 0946.20011 · doi:10.1093/qjmath/50.200.505 [5] DOI: 10.1098/rspa.1957.0007 · Zbl 0077.03002 · doi:10.1098/rspa.1957.0007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.