A Characterization of Those Automata That Structurally Generate Finite Groups

  • Ines Klimann
  • Matthieu Picantin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)


Antonenko and Russyev independently have shown that any Mealy automaton with no cycle with exit—that is, where every cycle in the underlying directed graph is a sink component—generates a finite (semi)group, regardless of the choice of the production functions. Antonenko has proved that this constitutes a characterization in the non-invertible case and asked for the invertible case, which is proved in this paper.


automaton groups Mealy automata finiteness problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akhavi, A., Klimann, I., Lombardy, S., Mairesse, J., Picantin, M.: On the finiteness problem for automaton (semi)groups. Int. J. Algebra Comput. 22(6), 26p. (2012)Google Scholar
  2. 2.
    Antonenko, A.S.: On transition functions of Mealy automata of finite growth. Matematychni Studii 29(1), 3–17 (2008)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Antonenko, A.S., Berkovich, E.L.: Groups and semigroups defined by some classes of Mealy automata. Acta Cybernetica 18(1), 23–46 (2007)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bartholdi, L., Reznykov, I., Sushchanskiĭ, V.: The smallest Mealy automaton of intermediate growth. J. Algebra 295(2), 387–414 (2006)Google Scholar
  5. 5.
    Bartholdi, L., Silva, P.: Groups defined by automata (2010), arXiv:cs.FL/1012.1531Google Scholar
  6. 6.
    Bondarenko, I., Bondarenko, N., Sidki, S., Zapata, F.: On the conjugacy problem for finite-state automorphisms of regular rooted trees (with an appendix by Raphaël M. Jungers). Groups Geom. Dyn. 7(2), 323–355 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Cain, A.: Automaton semigroups. Theor. Comput. Sci. 410(47-49), 5022–5038 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    D’Angeli, D., Rodaro, E.: Groups and Semigroups Defined by Colorings of Synchronizing Automata (2013), arXiv:math.GR/1310.5242Google Scholar
  9. 9.
    De Felice, S., Nicaud, C.: Random generation of deterministic acyclic automata using the recursive method. In: Bulatov, A.A., Shur, A.M. (eds.) CSR 2013. LNCS, vol. 7913, pp. 88–99. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  10. 10.
    Dixon, J.D.: The probability of generating the symmetric group. Math. Z. 110, 199–205 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Gillibert, P.: The finiteness problem for automaton semigroups is undecidable. arXiv:cs.FL/1304.2295 (2013)Google Scholar
  12. 12.
    Grigorchuk, R.: On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen. 14(1), 53–54 (1980)MathSciNetGoogle Scholar
  13. 13.
    Grigorchuk, R., Nekrashevich, V., Sushchanskiĭ, V.: Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova 231, 134–214 (2000)Google Scholar
  14. 14.
    Jaikin-Zapirain, A., Pyber, L.: Random generation of finite and profinite groups and group enumeration. Ann. of Math. (2) 173(2), 769–814 (2011)Google Scholar
  15. 15.
    Klimann, I.: The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable. In: Proc. 30th STACS. LIPIcs, vol. 20, pp. 502–513 (2013)Google Scholar
  16. 16.
    Klimann, I., Mairesse, J., Picantin, M.: Implementing computations in automaton (semi)groups. In: Moreira, N., Reis, R. (eds.) CIAA 2012. LNCS, vol. 7381, pp. 240–252. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  17. 17.
    Maltcev, V.: Cayley automaton semigroups. Int. J. Algebra Comput. 19(1), 79–95 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Mintz, A.: On the Cayley semigroup of a finite aperiodic semigroup. Int. J. Algebra Comput. 19(6), 723–746 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Nekrashevych, V.: Self-similar groups. Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence (2005)zbMATHGoogle Scholar
  20. 20.
    Russyev, A.: Finite groups as groups of automata with no cycles with exit. Algebra and Discrete Mathematics 9(1), 86–102 (2010)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Savchuk, D., Vorobets, Y.: Automata generating free products of groups of order 2. J. Algebra 336(1), 53–66 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Sidki, S.: Automorphisms of one-rooted trees: growth, circuit structure, and acyclicity. J. Math. Sci. 100(1), 1925–1943 (2000); Algebra, 12Google Scholar
  23. 23.
    Silva, P., Steinberg, B.: On a class of automata groups generalizing lamplighter groups. Int. J. Algebra Comput. 15(5-6), 1213–1234 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Steinberg, B., Vorobets, M., Vorobets, Y.: Automata over a binary alphabet generating free groups of even rank. Int. J. Algebra Comput. 21(1-2), 329–354 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Šuniḱ, Z., Ventura, E.: The conjugacy problem in automaton groups is not solvable. Journal of Algebra 364, 148–154 (2012)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ines Klimann
    • 1
  • Matthieu Picantin
    • 1
  1. 1.Sorbonne Paris Cité, LIAFA, UMR 7089 CNRSUniv Paris DiderotParisFrance

Personalised recommendations