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Groups with bounded verbal conjugacy classes. (English) Zbl 1108.20035

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.

MSC:

20F24 FC-groups and their generalizations
20E45 Conjugacy classes for groups
20E10 Quasivarieties and varieties of groups
20E05 Free nonabelian groups
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References:

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