Abstract:
Considering a group with unique roots (i.e., an R-group), we give a sufficient condition for the existence of a positive (constructive) enumeration with respect to which the isolator of the commutant is computable. Basing on it, we prove the constructivizability of an R-group that admitting a positive enumeration for which the dimension of the commutant is finite. We obtain a necessary and sufficient condition of constructivizability for a torsion-free nilpotent group for which the dimension of the commutant is finite.
Keywords:R-group, positive (constructive) group, positivizable (constructivizable) group, commutant, center of a group, dimension of a group, computably enumerable (computable) group.
\Bibitem{Khi12}
\by N.~G.~Khisamiev
\paper On positive and constructive groups
\jour Sibirsk. Mat. Zh.
\yr 2012
\vol 53
\issue 5
\pages 1133--1146
\mathnet{http://mi.mathnet.ru/smj2335}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3057932}
\elib{https://elibrary.ru/item.asp?id=17897071}
\transl
\jour Siberian Math. J.
\yr 2012
\vol 53
\issue 5
\pages 906--917
\crossref{https://doi.org/10.1134/S0037446612050151}
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https://www.mathnet.ru/eng/smj2335
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This publication is cited in the following 3 articles:
N. Kh. Kasymov, R. N. Dadazhanov, S. K. Zhavliev, “Uniform m-equivalences and numberings of classical systems”, Sib. elektron. matem. izv., 19:1 (2022), 49–65
N. G. Khisamiev, I. V. Latkin, “On constructive nilpotent groups”, Computability and Complexity: Essays Dedicated to Rodney G. Downey on the Occasion of His 60Th Birthday, Lecture Notes in Computer Science, 10010, eds. A. Day, M. Fellows, N. Greenberg, B. Khoussainov, A. Melnikov, F. Rosamond, Springler, 2017, 324–353
M. K. Nurizinov, R. K. Tyulyubergenev, N. G. Khisamiev, “Computable torsion-free nilpotent groups of finite dimension”, Siberian Math. J., 55:3 (2014), 471–481
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