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Periodic products of groups. (English. Russian original) Zbl 1377.20026
J. Contemp. Math. Anal., Armen. Acad. Sci. 52, No. 3, 111-117 (2017); translation from Izv. Nats. Akad. Nauk Armen., Mat. 52, No. 3, 3-15 (2017).
Summary: In this paper, we provide an overview of the results relating to the \(n\)-periodic products of groups that have been obtained in recent years by the authors of the present paper, as well as some results obtained by other authors in this direction. The periodic products were introduced by the first author [Proc. Steklov Inst. Math. 142, 1–19 (1976; Zbl 0424.20020); translation from Tr. Mat. Inst. Steklov 142, 3–21 (1976)] to solve the Maltsev’s well-known problem. It was shown that the periodic products are exact, associative and hereditary for subgroups. They also possess some other important properties such as the Hopf property, the \(C*\)-simplicity, the uniform non-amenability, the \(SQ\)-universality, etc. It was proved that the \(n\)-periodic products of groups can uniquely be characterized by means of certain quite specific and simply formulated properties. These properties allow to extend to \(n\)-periodic products of various families of groups a number of results previously obtained for free periodic groups \(B(m,n)\). In particular, we describe the finite subgroups of \(n\)-periodic products, Also, we analyze and extend the simplicity criterion of \(n\)-periodic products obtained previously by the first author [Math. Notes 88, No. 6, 771–775 (2010; Zbl 1230.20025); translation from Mat. Zametki 88, No. 6, 803–810 (2010)].

MSC:
20F50 Periodic groups; locally finite groups
20F05 Generators, relations, and presentations of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F28 Automorphism groups of groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
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