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On finite soluble groups with almost fixed-point-free automorphisms of noncoprime order

Abstract

It is proved that if a finite p-soluble group G admits an automorphism φ of order p n having at most m fixed points on every φ-invariant elementary abelian p′-section of G, then the p-length of G is bounded above in terms of p n and m; if in addition G is soluble, then the Fitting height of G is bounded above in terms of p n and m. It is also proved that if a finite soluble group G admits an automorphism ψ of order p a q b for some primes p and q, then the Fitting height of G is bounded above in terms of |ψ| and |C G (ψ)|.

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References

  1. Brauer R. and Fowler K. A., “On groups of even order,” Ann. Math. (2), 62, 565–583 (1955).

    MATH  MathSciNet  Article  Google Scholar 

  2. Thompson J. G., “Finite groups with fixed-point-free automorphisms of prime order,” Proc. Nat. Acad. Sci., 45, 578–581 (1959).

    MATH  MathSciNet  Article  Google Scholar 

  3. Hartley B., “A general Brauer-Fowler theorem and centralizers in locally finite groups,” Pacific J. Math., 152, 101–117 (1992).

    MATH  MathSciNet  Article  Google Scholar 

  4. Higman G., “Groups and rings which have automorphisms without non-trivial fixed elements,” J. London Math. Soc., 32, No. 2, 321–334 (1957).

    MATH  MathSciNet  Article  Google Scholar 

  5. Kreknin V. A., “Solvability of Lie algebras with regular automorphisms of finite period,” Soviet Math. Dokl., 4, 683–685 (1963).

    MATH  Google Scholar 

  6. Kreknin V. A. and Kostrikin A. I., “Lie algebras with regular automorphisms,” Dokl. Akad. Nauk SSSR, 149, 249–251 (1963).

    MathSciNet  Google Scholar 

  7. Khukhro E. I., “Lie rings and groups admitting an almost regular automorphism of prime order,” Math. USSR-Sb., 71, No. 1, 51–63 (1992).

    MATH  MathSciNet  Article  Google Scholar 

  8. Kovács L. G., “Groups with regular automorphisms of order four,” Math. Z., Bd 75, 277–294 (1960/1961).

    Article  Google Scholar 

  9. Khukhro E. I. and Makarenko N. Yu., “Finite groups with an almost regular automorphism of order four,” Algebra and Logic, 45, No. 5, 326–343 (2006).

    MathSciNet  Article  Google Scholar 

  10. Alperin J. L., “Automorphisms of solvable groups,” Proc. Amer. Math. Soc., 13, 175–180 (1962).

    MATH  MathSciNet  Article  Google Scholar 

  11. Khukhro E. I., “Finite p-groups admitting an automorphism of order p with a small number of fixed points,” Math. Notes, 38, No. 5, 867–870 (1985).

    MATH  MathSciNet  Article  Google Scholar 

  12. Shalev A., “On almost fixed point free automorphisms,” J. Algebra, 157, 271–282 (1993).

    MATH  MathSciNet  Article  Google Scholar 

  13. Khukhro E. I., “Finite p-groups admitting p-automorphisms with few fixed points,” Russian Acad. Sci., Sb. Math., 80, 435–444 (1995).

    MathSciNet  Google Scholar 

  14. Medvedev Yu., “p-Divided Lie rings and p-groups,” J. London Math. Soc. (2), 59, 787–798 (1999).

    MATH  MathSciNet  Article  Google Scholar 

  15. Jaikin-Zapirain A., “On almost regular automorphisms of finite p-groups,” Adv. Math., 153, 391–402 (2000).

    MATH  MathSciNet  Article  Google Scholar 

  16. Thompson J., “Automorphisms of solvable groups,” J. Algebra, 1, 259–267 (1964).

    MATH  MathSciNet  Article  Google Scholar 

  17. Turull A., “Fitting height of groups and of fixed points,” J. Algebra, 86, 555–566 (1984).

    MATH  MathSciNet  Article  Google Scholar 

  18. Hartley B. and Isaacs I. M., “On characters and fixed points of coprime operator groups,” J. Algebra, 131, 342–358 (1990).

    MATH  MathSciNet  Article  Google Scholar 

  19. Bell S. D. and Hartley B., “A note on fixed-point-free actions of finite groups,” Quart. J. Math. Oxford Ser. (2), 41, No. 162, 127–130 (1990).

    MATH  MathSciNet  Article  Google Scholar 

  20. Dade E. C., “Carter subgroups and Fitting heights of finite solvable groups,” Illinois J. Math., 13, 449–514 (1969).

    MATH  MathSciNet  Google Scholar 

  21. Mazurov V. D. and Khukhro E. I. (Eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 18th ed., Sobolev Inst. Math., Novosibirsk (2014). http://math.nsc.ru/alglog/18kt.pdf

    Google Scholar 

  22. Hartley B. and Turau V., “Finite soluble groups admitting an automorphism of prime power order with few fixed points,” Math. Proc. Cambridge Philos. Soc., 102, 431–441 (1987).

    MATH  MathSciNet  Article  Google Scholar 

  23. Rae A., “Sylow p-subgroups of finite p-soluble groups,” J. London Math. Soc. (2), 71, 117–123 (1973).

    MathSciNet  Article  Google Scholar 

  24. Hartley B. and Rae A., “Finite p-groups acting on p-soluble groups,” Bull. London Math. Soc., 5, 197–198 (1973).

    MATH  MathSciNet  Article  Google Scholar 

  25. Kurzweil H., “Eine Verallgemeinerung von fixpunktfreien Automorphismen endlicher Gruppen,” Arch. Math. (Basel), 22, 136–145 (1971).

    MATH  MathSciNet  Article  Google Scholar 

  26. Meixner T., “The Fitting length of solvable Hpn-groups,” Israel J. Math., 51, No. 1–2, 68–78 (1985).

    MATH  MathSciNet  Article  Google Scholar 

  27. Wilson J. S., “On the structure of compact torsion groups,” Monatsh. Math., 96, No. 1, 57–66 (1983).

    MATH  MathSciNet  Article  Google Scholar 

  28. Khukhro E. I. and Shumyatsky P., “Words and pronilpotent subgroups in profinite groups,” Austral. Math. Soc. J., 97, No. 3, 343–364 (2014).

    MATH  MathSciNet  Article  Google Scholar 

  29. Hall P. and Higman G., “The p-length of a p-soluble group and reduction theorems for Burnside’s problem,” Proc. London Math. Soc. (3), 6, 1–42 (1956).

    MATH  MathSciNet  Article  Google Scholar 

  30. Hoare A. H. M., “A note on 2-soluble groups,” J. London Math. Soc., 35, 193–199 (1960).

    MATH  MathSciNet  Article  Google Scholar 

  31. Berger T. R. and Gross F., “2-Length and the derived length of a Sylow 2-subgroup,” Proc. London Math. Soc. (3), 34, 520–534 (1977).

    MATH  MathSciNet  Article  Google Scholar 

  32. Bryukhanova E. G., “Connection between the 2-length and the derived length of a Sylow 2-subgroup of a finite solvable group,” Math. Notes, 29, No. 2, 85–90 (1981).

    MATH  MathSciNet  Article  Google Scholar 

  33. Hartley B., “Automorphisms of finite soluble groups. Preliminary version,” MIMS EPrint: 2014.52. http://eprints.ma.man.ac.uk/2188/01/covered/MIMSep201452.pdf.

  34. Hartley B., “Some theorems of Hall-Higman type for small primes,” Proc. London Math. Soc. (3), 41, 340–362 (1980).

    MATH  MathSciNet  Article  Google Scholar 

  35. Curtis C. W. and Reiner I., Representation Theory of Finite Groups and Associative Algebras, Interscience, New York and London (1962).

    MATH  Google Scholar 

  36. Khukhro E. and Mazurov V., “Automorphisms with centralizers of small rank,” in: Groups St. Andrews 2005. Vol. II. Selected Papers of the Conference, St. Andrews, UK, July 30–August 6, 2005, London Math. Soc. Lecture Note Ser., vol. 340, Cambridge Univ. Press, Cambridge, 2007, pp. 564–585.

    Google Scholar 

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Correspondence to E. I. Khukhro.

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Original Russian Text Copyright © 2015 Khukhro E.I.

The author was supported by the Russian Science Foundation (Grant 14-21-00065).

To Yuriĭ Leonidovich Ershov on the occasion of his 75th birthday.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 3, pp. 682–692, May–June, 2015; DOI: 10.17377/smzh.2015.56.317.

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Khukhro, E.I. On finite soluble groups with almost fixed-point-free automorphisms of noncoprime order. Sib Math J 56, 541–548 (2015). https://doi.org/10.1134/S0037446615030179

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  • DOI: https://doi.org/10.1134/S0037446615030179

Keywords

  • finite soluble group
  • automorphism
  • p-length
  • Fitting height