Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Frobenius groups as groups of automorphisms

Authors: N. Yu. Makarenko and Pavel Shumyatsky
Journal: Proc. Amer. Math. Soc. 138 (2010), 3425-3436
MSC (2010): Primary 20D45, 17B70
Published electronically: May 20, 2010
MathSciNet review: 2661543
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that if $ GFH$ is a double Frobenius group with ``upper'' complement $ H$ of order $ q$ such that $ C_G(H)$ is nilpotent of class $ c$, then $ G$ is nilpotent of $ (c,q)$-bounded class. This solves a problem posed by Mazurov in the Kourovka Notebook. The proof is based on an analogous result on Lie rings: if a finite Frobenius group $ FH$ with kernel $ F$ of prime order and complement $ H$ of order $ q$ acts on a Lie ring $ K$ in such a way that $ C_K(F)=0$ and $ C_K(H)$ is nilpotent of class $ c$, then $ K$ is nilpotent of $ (c,q)$-bounded class.

References [Enhancements On Off] (What's this?)

  • 1. Chong-yun Chao, Some characterizations of nilpotent Lie algebras, Math. Z. 103 (1968), 40–42. MR 0223415
  • 2. Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
  • 3. P. Hall, Some sufficient conditions for a group to be nilpotent, Illinois J. Math. 2 (1958), 787–801. MR 0105441
  • 4. Graham Higman, Groups and rings having automorphisms without non-trivial fixed elements, J. London Math. Soc. 32 (1957), 321–334. MR 0089204
  • 5. E. I. Huhro, Fixed points of 𝑝-automorphisms of finite 𝑝-groups, Algebra i Logika 14 (1975), no. 6, 697–703, 727–728 (Russian). MR 0427472
  • 6. Evgenii I. Khukhro, Nilpotent groups and their automorphisms, de Gruyter Expositions in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1993. MR 1224233
  • 7. E. I. Khukhro, Graded Lie rings with many commuting components and an application to 2-Frobenius groups, Bull. Lond. Math. Soc. 40 (2008), no. 5, 907–912. MR 2439656, 10.1112/blms/bdn075
  • 8. E. I. Khukhro and V. D. Mazurov (Eds.), Unsolved Problems in Group Theory. The Kourovka Notebook, no. 17, Institute of Mathematics, Novosibirsk, 2010.
  • 9. V. A. Kreknin, Solvability of Lie algebras with a regular automorphism of finite period, Dokl. Akad. Nauk SSSR 150 (1963), 467–469 (Russian). MR 0157990
  • 10. V. A. Kreknin and A. I. Kostrikin, Lie algebras with regular automorphisms, Dokl. Akad. Nauk SSSR 149 (1963), 249–251 (Russian). MR 0146230
  • 11. V. D. Mazurov, Recognition of the finite simple groups 𝑆₄(𝑞) by their element orders, Algebra Logika 41 (2002), no. 2, 166–198, 254 (Russian, with Russian summary); English transl., Algebra Logic 41 (2002), no. 2, 93–110. MR 1922988, 10.1023/A:1015356614025
  • 12. Aner Shalev, Automorphisms of finite groups of bounded rank, Israel J. Math. 82 (1993), no. 1-3, 395–404. MR 1239058, 10.1007/BF02808121
  • 13. John Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 578–581. MR 0104731

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20D45, 17B70

Retrieve articles in all journals with MSC (2010): 20D45, 17B70

Additional Information

N. Yu. Makarenko
Affiliation: Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
Address at time of publication: Laboratoire de Mathématiques, Informatique et Application, Université de Haute Alsace, Mulhouse, 68093, France

Pavel Shumyatsky
Affiliation: Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900, Brazil

Keywords: Automorphisms, centralizers, associated Lie rings
Received by editor(s): November 13, 2009
Published electronically: May 20, 2010
Additional Notes: The first author was supported in part by the Programme of Support of Leading Scientific Schools of the Russian Federation.
The second author was supported by CNPq-Brazil.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2010 American Mathematical Society