Advertisement

Science in China Series A: Mathematics

, Volume 52, Issue 2, pp 311–317 | Cite as

On periodic groups with prescribed orders of elements

  • Victor Danilovich Mazurov
  • WuJie ShiEmail author
Article

Abstract

This is a survey of results and open problems concerning the structure of periodic group with prescribed orders of their elements. In particular, an outline of a proof of the following recent result is given: every periodic group whose set of element orders coincides with {1, 2, 3, 4, 8} is locally finite.

Keywords

spectrum periodic group local finiteness 

MSC(2000)

20F50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Novikov P S, Adian S I. Infinite periodic groups. I, II, III (in Russian). Izvestija AN SSSR Mathematics, 32: 212–244, 251–524, 709–731 (1968)Google Scholar
  2. 2.
    Adian S I. The Burnside problem and identities in groups. Translated from the Russian by John Lennox and James Wiegold. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol 95. Berlin-New York: Springer-Verlag, 1979Google Scholar
  3. 3.
    Ivanov S V. The free Burnside groups of sufficiently large exponents. Intern J Algebra Comput, 4: 3–308 (1994)Google Scholar
  4. 4.
    Lysenok I G. Infinite Burnside groups of even period. Izv Math, 60: 453–654 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Levi F, van der Waerden B L. Über eine besondere Klasse von Gruppen. Abh Math Semin Hamburg Univ, 9: 154–158 (1932)zbMATHCrossRefGoogle Scholar
  6. 6.
    Neumann B H. Groups whose elements have bounded orders. J London Math Soc, 12: 195–198 (1937)zbMATHCrossRefGoogle Scholar
  7. 7.
    Sanov I N. Solution of Burnside’s problem for exponent 4 (in Russian). Leningrad State Univ Annals Math Ser, 10: 166–170 (1940)MathSciNetGoogle Scholar
  8. 8.
    Lytkina D V. The structure of a group with elements of order at most 4. Siberian Math J, 48: 283–287 (2007)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Hall M Jr. Solution of the Burnside problem for exponent six. Illinois J Math, 2: 764–786 (1958)MathSciNetGoogle Scholar
  10. 10.
    Newman M F. Groups of exponent dividing seventy. Math Scientist, 4: 149–157 (1979)zbMATHGoogle Scholar
  11. 11.
    Jabara E. Fixed point free action of groups of exponent 5. J Austral Math Soc, 77: 297–304 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Mazurov V D. Infinite groups with abelian centralizers of involutions. Algebra Logic, 39: 42–49 (2000)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Zhurtov A Kh, Mazurov V D. On the recognition of the finite simple groups L 2(2m) in the class of all groups. Siberian Math J, 40: 62–64 (1999)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Shi W J. A characteristic property of J 1 and PSL2(2n) (in Chinese). Adv Math, 16: 397–401 (1987)zbMATHGoogle Scholar
  15. 15.
    Gupta N D, Mazurov V D. On groups with small orders of elements. Bull Austral Math Soc, 60: 197–205 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mazurov V D. On groups of exponent 60 with prescribed orders of elements. Algebra Logic, 39: 189–198 (2000)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Brandl R, Shi W J. Finite groups whose element orders are consecutive integers. J Algebra, 143: 388–400 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lytkina D V, Kuznetsov A A. Recognizability by spectrum of the group L 2(7) in the class of all groups. Siberian Electronic Mathematical Reports, 4: 136–140 (2007)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Shi W J. A characteristic property of PSL2(7). J Austral Math Soc Ser A, 36: 354–356 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    The GAP Group. GAP — Groups, Algorithms, and Programming, Version 4.4.10; 2008 (http://www.gapsystem.org)
  21. 21.
    Neumann B H. Groups with automorphisms that leave only the neutral element fixed. Arch Math, 7: 1–5 (1956)zbMATHCrossRefGoogle Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsSiberian Branch of Russian Academy of SciencesNovosibirskRussia
  2. 2.School of Mathematical SciencesSuzhou UniversitySuzhouChina

Personalised recommendations