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Abelian Extensions and Solvable Loops

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Abstract

Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on one hand, and abelian extensions and congruence solvability on the other hand. In particular, we show that a loop is congruence solvable (that is, an iterated abelian extension of commutative groups) if and only if it is not Boolean complete, reaffirming the connection between computational complexity and solvability. Finally, we briefly discuss relations between nilpotence and solvability for loops and the associated multiplication groups and inner mapping groups.

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References

  1. Aichinger E., Mudrinski N.: Some applications of higher commutators in Mal’cev algebras. Algebra Univ. 63(4), 367–403 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bergman C.: Universal algebra: fundamentals and selected topics. Chapman & Hall/CRC Press, London (2011)

    Google Scholar 

  3. Bruck R.H.: Contributions to the theory of loops. Trans. Am. Math. Soc. 60, 245–354 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bruck R.H.: A survey of binary systems. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1958)

    Google Scholar 

  5. Bruck R.H., Paige L.J.: Loops whose inner mappings are automorphisms. Ann. Math. 2(63), 308–323 (1956)

    Article  MathSciNet  Google Scholar 

  6. Bulatov A.: On the number of finite Mal’tsev algebras. Contrib. Gen. Algebra 13, 41–54 (2001)

    MathSciNet  Google Scholar 

  7. Csörgő P.: Abelian inner mappings and nilpotency class greater than two. Eur. J. Combin. 28(3), 858–867 (2007)

    Article  Google Scholar 

  8. Drápal A., Vojtěchovský P.: Small loops of nilpotency class three with commutative inner mapping groups. J. Group Theory 14(4), 547–573 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Freese R., McKenzie R.: Commutator theory for congruence modular varieties. London Mathematical Society Lecture Note Series 125. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  10. The GAP Group: GAP—groups, algorithms, and programming, Version 4.5.5. http://www.gap-system.org (2012). Accessed 12 Feb 2014

  11. Kinyon, M., Kunen, K., Phillips, J.D., Vojtěchovský, P.: The structure of automorphic loops. Trans. Am. Math. Soc. (2014, to appear)

  12. Kinyon, M., Veroff, R., Vojtěchovský, P.: Loops with abelian inner mapping groups: an application of automated deduction. In: Bonacina, M.P., Stickel, M.E. (eds.) Automated Reasoning and Mathematics, Essays in Memory of William McCune. Lecture Notes in Artificial Intelligence, vol. 7788, pp. 151–164. Springer, New York (2013)

  13. Lemieux F., Moore C., Thérien D.: Polyabelian loops and Boolean completeness. Comment. Math. Univ. Carol. 41(4), 671–686 (2000)

    MATH  Google Scholar 

  14. Maurer W.D., Rhodes J.: A property of finite simple non-abelian groups. Proc. Am. Math. Soc. 16, 552–554 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mazur M.: Connected transversals to nilpotent groups. J. Group Theory 10(2), 195–203 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Moore, C., Thérien, D., Lemieux, F., Berman, J., Drisko, A.: Circuits and expressions with nonassociative gates. In: 12th Annual IEEE Conference on Computational Complexity (Ulm, 1997) (published in J. Comput. Syst. Sci. 60(2), part 2, 368–394) (2000)

  17. McKenzie, R., Snow, J.: Congruence modular varieties: commutator theory and its uses. In: Structural theory of automata, semigroups, and universal algebra. NATO Science Series II: Mathematics, Physics and Chemistry Series, vol. 207, pp. 273–329. Springer, Dordrecht (2005)

  18. Nagy, G.P., Vojtěchovský, P.: LOOPS: computing with quasigroups and loops in GAP, version 2.2.0. http://www.math.du.edu/loops. Accessed 12 Feb 2014

  19. Niemenmaa M.: Finite loops with nilpotent inner mapping groups are centrally nilpotent. Bull. Aust. Math. Soc. 79(1), 109–114 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pflugfelder, H.O.: Quasigroups and Loops: Introduction. Helder520 mann, Berlin (1990)

  21. Smith J.D.H.: Mal’cev varieties. Lecture Notes in Mathematics 554. Springer, Berlin (1976)

    Google Scholar 

  22. Stanovský D., Vojtěchovský P.: Commutator theory for loops. J. Algebra 399, 290–322 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Straubing, H.: Representing functions by words over finite semigroups. In: Technical Report #838. Université de Montréal, Montreal (1992)

  24. Vesanen A.: Solvable groups and loops. J. Algebra 180(3), 862–876 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wright C. R.B.: On the multiplication group of a loop. Ill. J. Math. 13, 660–673 (1969)

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675, Praha 8, Czech Republic

    David Stanovský

  2. Department of Mathematics, University of Denver, 2280 S Vine St, Denver, CO, 80208, USA

    Petr Vojtěchovský

Authors
  1. David Stanovský
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  2. Petr Vojtěchovský
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Corresponding author

Correspondence to Petr Vojtěchovský.

Additional information

Research partially supported by GAČR grant 13-01832S (David Stanovský) and by Simons Foundation Collaboration Grant 210176 (Petr Vojtěchovský).

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Stanovský, D., Vojtěchovský, P. Abelian Extensions and Solvable Loops. Results. Math. 66, 367–384 (2014). https://doi.org/10.1007/s00025-014-0382-6

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