Uspekhi Matematicheskikh Nauk
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Uspekhi Mat. Nauk: Year: Volume: Issue: Page: Find

 Uspekhi Mat. Nauk, 2010, Volume 65, Issue 5(395), Pages 5–60 (Mi umn9376)

The Burnside problem and related topics

Steklov Mathematical Institute, Russian Academy of Sciences

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.

DOI: https://doi.org/10.4213/rm9376

Full text: PDF file (1254 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2010, 65:5, 805–855

Bibliographic databases:

UDC: 512.54+512.54.0+512.543
MSC: Primary 20F50; Secondary 01A65

Citation: S. I. Adian, “The Burnside problem and related topics”, Uspekhi Mat. Nauk, 65:5(395) (2010), 5–60; Russian Math. Surveys, 65:5 (2010), 805–855

Citation in format AMSBIB
\Bibitem{Adi10} \by S.~I.~Adian \paper The Burnside problem and related topics \jour Uspekhi Mat. Nauk \yr 2010 \vol 65 \issue 5(395) \pages 5--60 \mathnet{http://mi.mathnet.ru/umn9376} \crossref{https://doi.org/10.4213/rm9376} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2767907} \zmath{https://zbmath.org/?q=an:1230.20001} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2011RuMaS..65..805A} \elib{https://elibrary.ru/item.asp?id=20423077} \transl \jour Russian Math. Surveys \yr 2010 \vol 65 \issue 5 \pages 805--855 \crossref{https://doi.org/10.1070/RM2010v065n05ABEH004702} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000286623200001} \elib{https://elibrary.ru/item.asp?id=16978879} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79955652897} 

• http://mi.mathnet.ru/eng/umn9376
• https://doi.org/10.4213/rm9376
• http://mi.mathnet.ru/eng/umn/v65/i5/p5

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. I. Adian, “Once More on Periodic Products of Groups and on a Problem of A. I. Maltsev”, Math. Notes, 88:6 (2010), 771–775
2. Atabekyan V.S., “On CEP-subgroups of $n$-periodic products”, J. Contemp. Math. Anal., 46:5 (2011), 237–242
3. V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2011, no. 3, 62–64
4. V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Sb. Math., 204:2 (2013), 182–189
5. A. R. Chekhlov, Ml. V. Agafontseva, “Ob abelevykh gruppakh s tsentralnymi kvadratami kommutatorov endomorfizmov”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2013, no. 4(24), 54–59
6. Atabekyan V.S., “The Groups of Automorphisms Are Complete for Free Burnside Groups of Odd Exponents N >= 1003”, Int. J. Algebr. Comput., 23:6 (2013), 1485–1496
7. V. S. Atabekyan, “The automorphism tower problem for free periodic groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2013, no. 2, 3–7
8. Daria V. Lytkina, Victor D. Mazurov, “Groups with given element orders”, Zhurn. SFU. Ser. Matem. i fiz., 7:2 (2014), 191–203
9. V. S. Atabekyan, “Splitting Automorphisms of Order $p^k$ of Free Burnside Groups are Inner”, Math. Notes, 95:5 (2014), 586–589
10. S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71
11. S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110
12. Movsisyan Yu.M., “Hyperidentities and Related Concepts, I”, Armen. J. Math., 9:2 (2017), 146–222
13. 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
14. 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
•  Number of views: This page: 1600 Full text: 467 References: 120 First page: 49