On free groups in the infinitely based varieties of S. I. Adian. (English. Russian original) Zbl 1436.20044
Izv. Math. 81, No. 5, 889-900 (2017); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 81, No. 5, 3-14 (2017).
In this very nicely written article, the authors not only provide a thorough collection of the former results, mostly built upon the varieties constructed by S. I. Adian to solve the finite basis problem for groups posed by B. H. Neumann in 1937, but they prove new important results also. In particular for any relatively free group \(\Gamma=\Gamma(m,n,\Pi)\) in the aforementioned varieties constructed by Adian, the authors prove that the centralizer of any non-identity element is cyclic, every abelian subgroup is cyclic and the center is trivial for \(m \geq 2\). Moreover, they also prove that \(\mathrm{Aut}(\mathrm{End}(\Gamma))\) canonically embeds in \(\mathrm{Aut}(\mathrm{Aut}(\Gamma))\).
Reviewer: Kıvanç Ersoy (Berlin)
MSC:
| 20E10 | Quasivarieties and varieties of groups |
| 20F50 | Periodic groups; locally finite groups |
| 20F05 | Generators, relations, and presentations of groups |
| 20E05 | Free nonabelian groups |
| 20E26 | Residual properties and generalizations; residually finite groups |
Keywords:
infinitely based variety; self-centralizing subgroup; semigroup of endomorphisms; automorphism group; free Burnside groupReferences:
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