Nilpotency of finite groups with Frobenius groups of automorphisms

Abstract

Suppose that a finite group G admits a Frobenius group of automorphisms BC of coprime order with kernel B and complement C such that C G (C) is abelian. It is proved that if B is abelian of rank at least two and \({[C_G(u), C_G(v),\dots,C_G(v)]=1}\) for any \({u,v\in B{\setminus}\{1\}}\), where C G (v) is repeated k times, then G is nilpotent of class bounded in terms of k and |C| only. It is also proved that if B is abelian of rank at least three and C G (b) is nilpotent of class at most c for every \({b \in B{\setminus}\{1\}}\), then G is nilpotent of class bounded in terms of c and |C|. The proofs are based on results on graded Lie rings with many commuting components.

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Correspondence to P. Shumyatsky.

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Communicated by J.S. Wilson.

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Khukhro, E.I., Shumyatsky, P. Nilpotency of finite groups with Frobenius groups of automorphisms. Monatsh Math 163, 461–470 (2011). https://doi.org/10.1007/s00605-010-0198-8

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Keywords

  • Frobenius group of automorphisms
  • Centralizer
  • Nilpotent
  • Graded Lie ring

Mathematics Subject Classification (2000)

  • Primary 20D45
  • Secondary 17B70
  • 20D15
  • 20F40