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Izv. Akad. Nauk SSSR Ser. Mat., 1968, Volume 32, Issue 2, Pages 251–524 (Mi izv2475)  

This article is cited in 56 scientific papers (total in 58 papers)

Infinite periodic groups. II

P. S. Novikov, S. I. Adian


Abstract: In this paper we construct an example of an infinite periodic group with a finite number of generators, in which the orders of all the elements are bounded by a specified number. This is a solution of the well-known Burnside problem.

Full text: PDF file (35403 kB)

English version:
Mathematics of the USSR-Izvestiya, 1968, 2:2, 241–479

Bibliographic databases:

Document Type: Article
UDC: 519.4
MSC: 20F50, 20F05, 20F10, 20Cxx
Received: 23.10.1967

Citation: P. S. Novikov, S. I. Adian, “Infinite periodic groups. II”, Izv. Akad. Nauk SSSR Ser. Mat., 32:2 (1968), 251–524; Math. USSR-Izv., 2:2 (1968), 241–479

Citation in format AMSBIB
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\by P.~S.~Novikov, S.~I.~Adian
\paper Infinite periodic groups.~II
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1968
\vol 32
\issue 2
\pages 251--524
\mathnet{http://mi.mathnet.ru/izv2475}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=240179}
\zmath{https://zbmath.org/?q=an:0194.03301}
\transl
\jour Math. USSR-Izv.
\yr 1968
\vol 2
\issue 2
\pages 241--479
\crossref{https://doi.org/10.1070/IM1968v002n02ABEH000640}


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    This publication is cited in the following articles:
    1. P. S. Novikov, S. I. Adian, “Defining relations and the word problem for free periodic groups of odd order”, Math. USSR-Izv., 2:4 (1968), 935–942  mathnet  crossref  mathscinet  zmath
    2. P. S. Novikov, S. I. Adian, “On abelian subgroups and the conjugacy problem in free periodic groups of odd order”, Math. USSR-Izv., 2:5 (1968), 1131–1144  mathnet  crossref  mathscinet  zmath
    3. S. I. Adian, “Infinite irreducible systems of group identities”, Math. USSR-Izv., 4:4 (1970), 721–739  mathnet  crossref  mathscinet  zmath
    4. S. I. Adian, “On some torsion-free groups”, Math. USSR-Izv., 5:3 (1971), 475–484  mathnet  crossref  mathscinet  zmath
    5. Joseph G Rosenstein, “ℵ0-Categoricity of groups”, Journal of Algebra, 25:3 (1973), 435  crossref
    6. Robert McNaughton, Yechezkel Zalcstein, “The Burnside problem for semigroups”, Journal of Algebra, 34:2 (1975), 292  crossref
    7. V. L. Shirvanyan, “Independent systems of defining relations for a free periodic group of odd exponent”, Math. USSR-Sb., 29:1 (1976), 119–122  mathnet  crossref  mathscinet  zmath  isi
    8. A. I. Sozutov, V. P. Shunkov, “On a generalization of Frobenius' theorem to infinite groups”, Math. USSR-Sb., 29:4 (1976), 441–451  mathnet  crossref  mathscinet  zmath  isi
    9. Yu. A. Medvedev, “Local finiteness of periodic subloops of an alternative PI-ring”, Math. USSR-Sb., 32:2 (1977), 265–271  mathnet  crossref  mathscinet  zmath  isi
    10. S. I. Adian, “An axiomatic method of constructing groups with given properties”, Russian Math. Surveys, 32:1 (1977), 1–14  mathnet  crossref  mathscinet  zmath
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    14. S. I. Adian, “Random walks on free periodic groups”, Math. USSR-Izv., 21:3 (1983), 425–434  mathnet  crossref  mathscinet  zmath  isi
    15. Antonio Restivo, Christophe Reutenaeur, “Rational languages and the Burnside problem”, Theoretical Computer Science, 40 (1985), 13  crossref
    16. S. I. Adian, A. A. Razborov, “Periodic groups and Lie algebras”, Russian Math. Surveys, 42:2 (1987), 1–81  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    17. Kirby A. Baker, George F. McNulty, Walter Taylor, “Growth problems for avoidable words”, Theoretical Computer Science, 69:3 (1989), 319  crossref
    18. Srećko Brlek, “Enumeration of factors in the Thue-Morse word”, Discrete Applied Mathematics, 24:1-3 (1989), 83  crossref
    19. E. I. Zel'manov, “Solution of the restricted Burnside problem for groups of odd exponent”, Math. USSR-Izv., 36:1 (1991), 41–60  mathnet  crossref  mathscinet  zmath  adsnasa
    20. A. Yu. Ol'shanskii, “Periodic factor of hyperbolic groups”, Math. USSR-Sb., 72:2 (1992), 519–541  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    21. E. I. Zel'manov, “A solution of the restricted Burnside problem for 2-groups”, Math. USSR-Sb., 72:2 (1992), 543–565  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    22. S. I. Adian, I. G. Lysenok, “On groups all of whose proper subgroups of which are finite cyclic”, Math. USSR-Izv., 39:2 (1992), 905–957  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    23. I. G. Lysenok, “Infinite Burnside groups of even exponent”, Izv. Math., 60:3 (1996), 453–654  mathnet  crossref  crossref  mathscinet  zmath  isi
    24. Vaughan-Lee M. Zel'Manov E., “Bounds in the Restricted Burnside Problem”, J. Aust. Math. Soc. A-Pure Math. Stat., 67:Part 2 (1999), 261–271  isi
    25. S. I. Adian, V. G. Durnev, “Decision problems for groups and semigroups”, Russian Math. Surveys, 55:2 (2000), 207–296  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    26. S. I. Adian, “On the centenary of the birth of Petr Sergeevich Novikov”, Russian Math. Surveys, 56:4 (2001), 793–802  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    27. do Lago A., “Local Groups in Free Groupoids Satisfying Certain Monoid Identities”, Int. J. Algebr. Comput., 12:1-2 (2002), 357–369  crossref  isi
    28. S. I. Adian, “The Burnside Problem on Periodic Groups, and Related Problems”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S2–S12  mathnet  crossref  crossref  mathscinet  zmath  isi
    29. Hammoudi L., “Burnside and Kurosh Problems”, Int. J. Algebr. Comput., 14:2 (2004), 197–211  crossref  isi
    30. V. I. Senashov, A. I. Sozutov, V. P. Shunkov, “Investigation of groups with finiteness conditions in Krasnoyarsk”, Russian Math. Surveys, 60:5 (2005), 805–848  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    31. Ivanov S. Storozhev A., “Non-Hopfian Relatively Free Groups”, Geod. Dedic., 114:1 (2005), 209–228  crossref  isi
    32. L. D. Beklemishev, I. G. Lysenok, A. A. Mal'tsev, S. P. Novikov, M. R. Pentus, A. A. Razborov, A. L. Semenov, V. A. Uspenskii, “Sergei Ivanovich Adian (on his 75th birthday)”, Russian Math. Surveys, 61:3 (2006), 575–588  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    33. I. G. Lysenok, “Burnside structures of finite subgroups”, Izv. Math., 71:5 (2007), 939–965  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    34. E. Aladova, B. Plotkin, “PI-groups and PI-representations of groups”, J. Math. Sci., 164:2 (2010), 155–162  mathnet  crossref  mathscinet  elib
    35. V. D. Mazurov, A. S. Mamontov, “On periodic groups with small orders of elements”, Siberian Math. J., 50:2 (2009), 316–321  mathnet  crossref  mathscinet  isi  elib  elib
    36. V. S. Atabekian, “On subgroups of free Burnside groups of odd exponent $n\ge 1003$”, Izv. Math., 73:5 (2009), 861–892  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    37. V. S. Atabekyan, “Monomorphisms of Free Burnside Groups”, Math. Notes, 86:4 (2009), 457–462  mathnet  crossref  crossref  mathscinet  zmath  isi
    38. An. A. Muchnik, Yu. L. Pritykin, A. L. Semenov, “Sequences close to periodic”, Russian Math. Surveys, 64:5 (2009), 805–871  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    39. V. S. Atabekyan, “The normalizers of free subgroups in free Burnside groups of odd period $n\ge1003$”, J. Math. Sci., 166:6 (2010), 691–703  mathnet  crossref  mathscinet  elib
    40. Minasyan, A, “Periodic quotients of hyperbolic and large groups”, Groups Geometry and Dynamics, 3:3 (2009), 423  isi  elib
    41. Mazurov, VD, “On periodic groups with prescribed orders of elements”, Science in China Series A-Mathematics, 52:2 (2009), 311  crossref  isi  elib
    42. Grigorchuk R.I. Ivanov S.V., “On Dehn Functions of Infinite Presentations of Groups”, Geom. Funct. Anal., 18:6 (2009), 1841–1874  crossref  isi
    43. S. I. Adian, “The Burnside problem and related topics”, Russian Math. Surveys, 65:5 (2010), 805–855  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    44. Atabekyan V.S., “Non-Phi-Admissible Normal Subgroups of Free Burnside Groups”, J. Contemp. Math. Anal.-Armen. Aca., 45:2 (2010), 112–122  crossref  isi
    45. V. S. Atabekyan, “Normal automorphisms of free Burnside groups”, Izv. Math., 75:2 (2011), 223–237  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    46. Atabekyan V.S., “On CEP-Subgroups of n-Periodic Products”, Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences, 46:5 (2011), 237–242  isi
    47. Ershov M., “Golod-Shafarevich Groups: a Survey”, Int. J. Algebr. Comput., 22:5 (2012), 1230001  crossref  isi
    48. Lytkina D.V., “Periodic Groups with Given Properties of Finite Subgroups”, Dokl. Math., 85:1 (2012), 55–56  crossref  isi
    49. Lytkina D.V., “Gruppy s zadannymi poryadkami elementov”, Matematicheskii forum (itogi nauki. yug Rossii), 6 (2012), 85–97  elib
    50. V. S. Atabekyan, “Splitting automorphisms of free Burnside groups”, Sb. Math., 204:2 (2013), 182–189  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    51. 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  isi
    52. F. A. Dudkin, K. S. Sviridov, “Complementing a subgroup of a hyperbolic group by a free factor”, Algebra and Logic, 52:3 (2013), 222–235  mathnet  crossref  mathscinet  isi
    53. S. I. Adian, V. S. Atabekyan, “The Hopfian Property of $n$-Periodic Products of Groups”, Math. Notes, 95:4 (2014), 443–449  mathnet  crossref  crossref  mathscinet  isi  elib
    54. Daria V. Lytkina, Victor D. Mazurov, “Groups with given element orders”, Zhurn. SFU. Ser. Matem. i fiz., 7:2 (2014), 191–203  mathnet
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    56. Zelmanov E., “Groups With Identities”, Note Mat., 36:1 (2016), 101–113  crossref  mathscinet  isi  scopus
    57. Przytycki J.H., “On Slavik Jablan's work on 4-moves”, J. Knot Theory Ramifications, 25:9, SI (2016), 1641014  crossref  mathscinet  zmath  isi  elib  scopus
    58. Coulon R. Hilion A., “Growth and Order of Automorphisms of Free Groups and Free Burnside Groups”, Geom. Topol., 21:4 (2017), 1969–2014  crossref  isi
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