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The small index property and the cofinality of the automorphism group

Abstract

Within the general model-theoretical framework, we study the small index property and representation of the automorphism group as the union of an increasing chain of proper subsets of a special form.

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References

  1. 1.

    J. D. Dixon, P. M. Neumann, and S. Thomas, “Subgroups of small index in infinite symmetric groups,” Bull. London Math. Soc. 18, 580 (1986).

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    M. Droste, W. C. Holland, and H. D. Macpherson, “Automorphism groups of infinite semilinear orders. II,” Proc. LondonMath. Soc., Ser. III 58, 479 (1989).

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    M. Droste and J. K. Truss, “The uncountable cofinality of the automorphism group of the countable universal distributive lattice,” DemonstratioMath. 44, 473 (2011).

    MathSciNet  MATH  Google Scholar 

  4. 4.

    D. M. Evans, “Subgroups of small index in infinite general linear groups,” Bull. London Math. Soc. 18, 587 (1986).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    C. Gourion, “À propos du groupe des automorphismes de (Q, =),” C. R. Acad. Sci. Paris, Sér. I Math. 315, 1329 (1992) [in French].

    MathSciNet  Google Scholar 

  6. 6.

    W. Hodges, I. M. Hodkinson, D. Lascar, and S. Shelah, “The small index property for ω-stable ω-categorical structures and for the random graph,” J. London Math. Soc., Ser. II 48, 204 (1993).

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    K. Zh. Kudaĭbergenov, “Groups of automorphisms and chains of closed subgroups,” Mat. Trudy 4, 128 (2001) [Siberian Adv. Math. 11, 40 (2001)].

    MathSciNet  MATH  Google Scholar 

  8. 8.

    K. Zh. Kudaĭbergenov, “On the definition of the small index property,” Mat. Trudy 17, 123 (2014) [Siberian Adv. Math. 25, 206 (2015)].

    MATH  Google Scholar 

  9. 9.

    D. Lascar, “Autour de la propriétéde petit indice, ”Proc. London Math. Soc., Ser. III 62, 25 (1991) [in French].

  10. 10.

    D. Lascar and S. Shelah, “Uncountable saturated structures have the small index property,” Bull. London Math. Soc. 25, 125 (1993).

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    H. D. Macpherson and P. M. Neumann, “Subgroups of infinite symmetric groups,” J. London Math. Soc., Ser. II 42, 64 (1990).

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    G. Melles and S. Shelah. “A saturated model of an unsuperstable theory of cardinality greater than its theory has the small index property,” Proc. London Math. Soc., Ser. III 69, 449 (1994).

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    J. K. Truss, “Infinite permutation groups. II. Subgroups of small index,” J. Algebra 120, 494 (1989).

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to K. Zh. Kudaĭbergenov.

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Original Russian Text © K.Zh. Kudaĭbergenov, 2016, published in Matematicheskie Trudy, 2016, Vol. 19, No. 1, pp. 70–90.

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Kudaĭbergenov, K.Z. The small index property and the cofinality of the automorphism group. Sib. Adv. Math. 27, 1–15 (2017). https://doi.org/10.3103/S1055134417010011

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Keywords

  • small index property
  • homogeneous model
  • automorphism group