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Adyan-Lisenok groups and (U) condition. (English. Russian original) Zbl 1299.20055
J. Contemp. Math. Anal., Armen. Acad. Sci. 43, No. 5, 265-273 (2008); translation from Izv. Nats. Akad. Nauk Armen., Mat. 43, No. 5, 13-24 (2008).
Summary: A group \(G\) possesses the property (U) with respect to \(S\) if there exists a number \(M=M(G)\) such that for each generating set \(P\) of the group \(G\) there exists an element \(t\in G\) for which \(\max_{x\in S}|t^{-1}xt|_P\leq M\). It is proved that the well-known Adyan-Lisenok groups possess the property (U). In connection with the problem on finding infinite groups with the property (U), which is stated in a joint unpublished work by D. Osin and D. Sonkin, it is shown that for any odd \(n\geq 1003\) there is a continuum set of non-isomorphic, i.e. simple groups with the property (U) in the variety of groups satisfying the identity \(x^n=1\).

MSC:
20F50 Periodic groups; locally finite groups
20F05 Generators, relations, and presentations of groups
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