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Engel-like conditions in fixed points of automorphisms of profinite groups

  • Cristina AcciarriEmail author
  • Danilo Silveira
Article
  • 16 Downloads

Abstract

Let q be a prime and A an elementary abelian q-group acting as a coprime group of automorphisms on a profinite group G. We show that if A is of order \(q^2\) and some power of each element in \(C_G(a)\) is Engel in G for any \(a\in A^{\#}\), then G is locally virtually nilpotent. Assuming that A is of order \(q^3\), we prove that if some power of each element in \(C_G(a)\) is Engel in \(C_G(a)\) for any \(a\in A^{\#}\), then G is locally virtually nilpotent. Some analogues of quantitative nature for finite groups are also obtained.

Keywords

Profinite groups Automorphisms Centralizers Engel-like conditions 

Mathematics Subject Classification

Primary 20E18 20E36 Secondary 20F45 20F40 20D45 20F19 

Notes

References

  1. 1.
    Acciarri, C., Shumyatsky, P.: Profinite groups and the fixed points of coprime automorphisms. J. Algebra 452, 188–195 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Acciarri, C., Shumyatsky, P., Silveira, D.S.: On groups with automorphisms whose fixed points are Engel. Ann. Mater. Pura Appl. 197, 307–316 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bahturin, Y.A., Zaicev, M.V.: Identities of graded algebras. J. Algebra 205, 1–12 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bastos, R., Shumyatsky, P.: On profinite groups with Engel-like conditions. J. Algebra 427, 215–225 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burns, R.G., Macedonska, O., Medvedev, Yu.: Groups satisfying semigroup laws, and nilpotent-by-Burnside varieties. J. Algebra 195, 510–525 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D.: Analytic Pro-p Groups. Cambridge University Press, Cambridge (1991)zbMATHGoogle Scholar
  7. 7.
    Gruenberg, K.W.: The Engel structure of linear groups. J. Algebra 3, 291–303 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Huppert, B., Blackburn, N.: Finite Groups II. Springer, Berlin (1982)CrossRefzbMATHGoogle Scholar
  9. 9.
    Kelly, J.L.: General Topology. Van Nostrand, Toronto (1955)Google Scholar
  10. 10.
    Khukhro, E.I., Shumyatsky, P.: Bounding the exponent of a finite group with automorphisms. J. Algebra 212, 363–374 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lazard, M.: Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. Éc. Norm. Supér. 71, 101–190 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lazard, M.: Groupes analytiques \(p\)-adiques. Publ. Math. Inst. Hautes Études Sci. 26, 389–603 (1965)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Linchenko, V.: Identities of Lie algebras with actions of Hopf algebras. Comm. Algebra 25, 3179–3187 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lubotzky, A., Mann, A.: Powerful p-groups. I, II. J. Algebra 105, 484–515 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mal’cev, A .I.: Nilpotent semigroups. Ivanov. Gos. Ped. Inst. Uč. Zap. Fiz. Mat. Nauki 4, 107–111 (1953)MathSciNetGoogle Scholar
  16. 16.
    Neumann, B.H., Taylor, T.: Subsemigroups of nilpotent groups. Proc. R. Soc. Lond. Ser. A 274, 1–4 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Robinson, D.J.S.: A Course in the Theory of Groups. Springer, New York (1996)CrossRefGoogle Scholar
  18. 18.
    Shumyatsky, P.: Centralizers in groups with finiteness conditions. J. Group Theory 1, 275–282 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shumyatsky, P.: Applications of Lie ring methods to group theory. In: Costa, R., et al. (Eds.) Nonassociative Algebra and Its Applications, pp. 373–395. Marcel Dekker, New York (2000)Google Scholar
  20. 20.
    Shumyatsky, P.: Coprime automorphisms of profinite groups. Q. J. Math. 53, 371–376 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shumyatsky, P.: Positive laws in fixed points. Trans. Am. Math. Soc. 356, 2081–2091 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shumyatsky, P., Silveira, D.S.: On finite groups with automorphisms whose fixed points are Engel. Arch. Math. 106, 209–218 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wilson, J.S.: On the structure of compact torsion groups. Monatsh. Math. 96, 57–66 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wilson, J.S.: Profinite Groups. Clarendon Press, Oxford (1998)zbMATHGoogle Scholar
  25. 25.
    Wilson, J.S., Zelmanov, E.I.: Identities for Lie algebras of pro-\(p\) groups. J. Pure Appl. Algebra 81, 103–109 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zelmanov, E.I.: Lie Methods in the Theory of Nilpotent Groups, in Groups ’93 Galaway/ St Andrews, pp. 567–585. Cambridge University Press, Cambridge (1995)Google Scholar
  27. 27.
    Zelmanov, E.I.: Nil Rings and Periodic Groups. The Korean Math. Soc. Lecture Notes in Math, Seoul (1992)Google Scholar
  28. 28.
    Zelmanov, E.I.: On periodic compact groups. Isr. J. Math. 77, 83–95 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zelmanov, E.I.: Lie algebras and torsion groups with identity. J. Comb. Algebra 1, 289–340 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BrasiliaBrasília-DFBrazil
  2. 2.Department of MathematicsFederal University of GoiásCatalãoBrazil

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