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

This article is cited in 14 scientific papers (total in 14 papers)

The Burnside problem and related topics

S. I. Adian

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.


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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
Received: 02.08.2010

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
\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
\jour Russian Math. Surveys
\yr 2010
\vol 65
\issue 5
\pages 805--855

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    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  mathnet  crossref  crossref  mathscinet  isi
    2. Atabekyan V.S., “On CEP-subgroups of $n$-periodic products”, J. Contemp. Math. Anal., 46:5 (2011), 237–242  crossref  mathscinet  zmath  isi  scopus
    3. V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2011, no. 3, 62–64  mathnet
    4. V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Sb. Math., 204:2 (2013), 182–189  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    7. V. S. Atabekyan, “The automorphism tower problem for free periodic groups”, Uch. zapiski EGU, ser. Fizika i Matematika, 2013, no. 2, 3–7  mathnet
    8. Daria V. Lytkina, Victor D. Mazurov, “Groups with given element orders”, Zhurn. SFU. Ser. Matem. i fiz., 7:2 (2014), 191–203  mathnet
    9. V. S. Atabekyan, “Splitting Automorphisms of Order $p^k$ of Free Burnside Groups are Inner”, Math. Notes, 95:5 (2014), 586–589  mathnet  crossref  crossref  mathscinet  isi  elib
    10. S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71  mathnet  crossref  crossref  isi  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. Movsisyan Yu.M., “Hyperidentities and Related Concepts, I”, Armen. J. Math., 9:2 (2017), 146–222  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref  crossref  mathscinet
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