Algebra and Logic

, Volume 45, Issue 5, pp 326–343 | Cite as

Finite groups with an almost regular automorphism of order four

  • N. Yu. Makarenko
  • E. I. Khukhro
Article
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Abstract

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1.

Keywords

finite group almost regular automorphism Lie ring nilpotency class centralizer Hall-Higman type theorems characteristic subgroup 
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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • N. Yu. Makarenko
    • 1
  • E. I. Khukhro
    • 1
  1. 1.Institute ofMathematics, SiberianBranchRussian Academy of SciencesNovosibirskRussia

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