Maximal groups in free Burnside semigroups

  • Alair Pereira do Lago
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1380)


We prove that any maximal group in the free Burnside semigroup defined by the equation x n =x n+m for any n ≥ 1 and any m ≥ 1 is a free Burnside group satisfying x m =1. We show that such group is free over a well described set of generators whose cardinality is the cyclomatic number of a graph associated to the ℑ-class containing the group. For n=2 and for every m ≥ 2 we present examples with 2m−1 generators. Hence, in these cases, we have infinite maximal groups for large enough m. This allows us to prove important properties of Burnside semigroups for the case n=2, which was almost completely unknown until now. Surprisingly, the case n=2 presents simultaneously the complexities of the cases n=1 and n ≥ 3: the maximal groups are cyclic of order m for n ≥ 3 but they can have more generators and be infinite for n ≤ 2; there are exactly 2¦A¦ ℑ-classes and they are easily characterized for n=1 but there are infinitely many, ℑ-classes and they are difficult to characterize for n ≥ 2.


Word Problem Local Group Short Word Congruence Class Finiteness Property 
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  1. 1.
    S. I. Adian. The Burnside problem and identities in groups, volume 95 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin-New York, 1979. Translated from the Russian by John Lennox and James Wiegold.Google Scholar
  2. 2.
    J. Brzozowski, K. Culik, and A. Gabrielian. Classification of non-counting events. J. Comp. Syst. Sci., 5:41–53, 1971.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Adyan, S. I. Problema Bernsaida i tozhdestva v gruppakh. Izdat. “Nauka”, Moscow, 1975.zbMATHGoogle Scholar
  4. 4.
    A. de Luca and S. Varricchio. On non-counting regular classes. In M.S. Paterson, editor, Automata, Languages and Programming, pages 74–87, Berlin, 1990. Springer-Verlag. Lecture Notes in Computer Science, 443.Google Scholar
  5. 5.
    A. de Luca and S. Varricchio. On finitely recognizable semigroups. Acta Inform., 29(5):483–498, 1992.CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    A. de Luca and S. Varricchio. On non-counting regular classes. Theoretical Computer Science, 100:67–104, 1992.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    A. P. do Lago. Doctoral thesis in preparation.Google Scholar
  8. 8.
    A. P. do Lago. Sobre os semigrupos de Burnside x n=x n+m. Master's thesis, Institute de Matemática e Estatística da Universidade de SÃo Paulo, November 1991.Google Scholar
  9. 9.
    A. P. do Lago. On the Burnside semigroups x n=x n+m. In I. Simon, editor, LATIN'92, volume 583 of Lecture Notes in Computer Science, pages 329–43, Berlin, 1992. Springer-Verlag.Google Scholar
  10. 10.
    A. P. do Lago. On the Burnside semigroups x n=x n+m. Int. J. of Algebra and Computation, 6(2):179–227, 1996.zbMATHCrossRefGoogle Scholar
  11. 11.
    J. A. Green and D. Rees. On semigroups in which x r=x. Proc. Cambridge. Philos. Soc., 48:35–40, 1952.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    V. S. Guba. The word problem for the relatively free semigroup satisfying t m=t m+n with m ≥ 3. Int. J. of Algebra and Computation, 2(3):335–348, 1993.CrossRefMathSciNetGoogle Scholar
  13. 13.
    V. S. Guba. The word problem for the relatively free semigroup satisfying t m=t m+n with m ≥ 4 or m=3, n=1. Int. J. of Algebra and Computation, 2(2):125–140, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S. V. Ivanov. On the Burnside problem on periodic groups. Bull. Amer. Math. Soc. (N.S.), 27(2):257–260, 1992.zbMATHMathSciNetGoogle Scholar
  15. 15.
    S. V. Ivanov. The free Burnside groups of sufficiently large exponents. Internat. J. Algebra Comput., 4(1–2):ii+308, 1994.Google Scholar
  16. 16.
    L. Kadourek, Jiríand Polák. On free semigroups satisfying x r ≃ x. Simon Stevin, 64(1):3–19, 1990.MathSciNetzbMATHGoogle Scholar
  17. 17.
    G. Lallement. Semigroups and Combinatorial Applications. John Wiley & Sons, New York, NY, 1979.zbMATHGoogle Scholar
  18. 18.
    I. G. LysËnok. Infinity of Burnside groups of period 2k for k ≥ 13. Uspekhi Mat. Nauk, 47(2(284)):201–202, 1992.zbMATHGoogle Scholar
  19. 19.
    J. McCammond. The solution to the word problem for the relatively free semigroups satisfying t a=t a+b with a ≥ 6. Int. J. of Algebra and Computation, 1:1–32, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    I. Simon. Notes on non-counting languages of order 2. manuscript, 1970.Google Scholar
  21. 21.
    A. Thue. über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I Mat. Nat. Kl., 1:1–67, 1912.zbMATHGoogle Scholar
  22. 22.
    B. Tilson. Categories as algebra: an essential ingredient in the theory of monoids. J. Pure Appl. Algebra, 48(1–2):83–198, 1987.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Alair Pereira do Lago
    • 1
  1. 1.Instituto de Matemática e EstatísticaUniversidade de SÃo PauloSÃo Paulo - SPBrazil

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