Abstract:
This paper gives a survey of results related to the famous Burnside problem on periodic groups. A negative solution of this problem was first published in joint papers of P. S. Novikov and the author in 1968. The theory of transformations of words in free periodic groups that was created in these papers and its various modifications give a very productive approach to the investigation of hard problems in group theory. In 1950 the Burnside problem gave rise to another problem on finite periodic groups, formulated by Magnus and called by him the restricted Burnside problem. Here it is called the Burnside–Magnus problem. In the Burnside problem the question of local finiteness of periodic groups of a given exponent was posed, but the Burnside–Magnus problem is the question of the existence of a maximal finite periodic group R(m,n) of a fixed period n with a given number m of generators. These problems complement each other. The publication in a joint paper by the author and Razborov in 1987 of the first effective proof of the well-known result of Kostrikin on the existence of a maximal group R(m,n) for any prime n, together with an indication of primitive recursive upper bounds for the orders of these groups, stimulated investigations of the Burnside–Magnus problem as well. Very soon other effective proofs appeared, and then Zel'manov extended Kostrikin's result to the case when n is any power of a prime number. These results are discussed in the last section of this paper.
Bibliography: 105 titles.
Keywords:
Burnside problem, infinite periodic group, finiteness, periodic word, simultaneous induction, identities in groups, Burnside–Magnus problem, Lie algebras, Engel condition.
\Bibitem{Adi10}
\by S.~I.~Adian
\paper The Burnside problem and related topics
\jour Russian Math. Surveys
\yr 2010
\vol 65
\issue 5
\pages 805--855
\mathnet{http://mi.mathnet.ru/eng/rm9376}
\crossref{https://doi.org/10.1070/RM2010v065n05ABEH004702}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2767907}
\zmath{https://zbmath.org/?q=an:1230.20001}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2011RuMaS..65..805A}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000286623200001}
\elib{https://elibrary.ru/item.asp?id=20423077}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79955652897}
Linking options:
https://www.mathnet.ru/eng/rm9376
https://doi.org/10.1070/RM2010v065n05ABEH004702
https://www.mathnet.ru/eng/rm/v65/i5/p5
This publication is cited in the following 27 articles:
Dhiraj K. Pandey, Antonio R. Nicolosi, “Pseudorandom Function from Learning Burnside Problem”, Mathematics, 13:7 (2025), 1193
Anton Beletskiy, Ilya Ivanov-Pogodaev, “Combinatorial Estimations on Burnside Type Problems”, Mathematics, 12:5 (2024), 665
Dhiraj K. Pandey, Antonio R. Nicolosi, Lecture Notes in Computer Science, 14534, Innovative Security Solutions for Information Technology and Communications, 2024, 178
Vladimir I. Senashov, “m-aperiodic words on three-letter alphabet”, Siberian Aerospace Journal, 25:2 (2024), 176
I. A. Ivanov-Pogodaev, “A semigroup of paths on a sequence of uniformly elliptic complexes”, Funct. Anal. Appl., 57:2 (2023), 117–142
Figelius M., Lohrey M., Zetzsche G., “Closure Properties of Knapsack Semilinear Groups”, J. Algebra, 589 (2022), 437–482
Atabekyan V.S., Gevorkyan G.G., “Central Extensions of N-Torsion Groups”, J. Contemp. Math. Anal.-Armen. Aca., 57:1 (2022), 26–34
V. S. Atabekyan, G. G. Gevorgyan, “Tsentralnye rasshireniya n-kruchenykh grupp”, Proceedings of NAS RA. Mathematics, 2022, 19
V. S. Atabekyan, L. D. Beklemishev, V. S. Guba, I. G. Lysenok, A. A. Razborov, A. L. Semenov, “Questions in algebra and mathematical logic. Scientific heritage of S. I. Adian”, Russian Math. Surveys, 76:1 (2021), 1–27
I. A. Ivanov-Pogodaev, A. Ya. Kanel-Belov, “Finitely presented nilsemigroups: complexes with the property of uniform ellipticity”, Izv. Math., 85:6 (2021), 1146–1180
Ville Salo, Lecture Notes in Computer Science, 12286, Cellular Automata and Discrete Complex Systems, 2020, 111
Ivanov-Pogodaev I., Malev S., Sapir O., “A Construction of a Finitely Presented Semigroup Containing An Infinite Square-Free Ideal With Zero Multiplication”, Int. J. Algebr. Comput., 28:8, SI (2018), 1565–1573
Movsisyan Yu.M., “Hyperidentities and Related Concepts, I”, Armen. J. Math., 9:2 (2017), 146–222
S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71
S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110
George M. Bergman, Universitext, An Invitation to General Algebra and Universal Constructions, 2015, 45
Daria V. Lytkina, Victor D. Mazurov, “Groups with given element orders”, Zhurn. SFU. Ser. Matem. i fiz., 7:2 (2014), 191–203
V. S. Atabekyan, “Splitting Automorphisms of Order $p^k$ of Free Burnside Groups are Inner”, Math. Notes, 95:5 (2014), 586–589
V. L. Popov, “Jordan groups and automorphism groups of algebraic varieties”, Springer Proc. Math. Statist., 79 (2014), 185–213
V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Sb. Math., 204:2 (2013), 182–189
Citation in format AMSBIB
Contact us: