Nilpotent length of a finite group admitting a frobenius group of automorphisms with fixed-point-free kernel

Suppose that a finite group G admits a Frobenius group FH of automorphisms with kernel F and complement H such that the fixed-point subgroup of F is trivial, i.e., CG(F) = 1, and the orders of G and H are coprime. It is proved that the nilpotent length of G is equal to the nilpotent length of CG(H) and the Fitting series of the fixed-point subgroup CG(H) coincides with a series obtained by taking intersections of CG(H) with the Fitting series of G.

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Correspondence to E. I. Khukhro.

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Dedicated to Yu. L. Ershov on the occasion of his 70th birthday

Translated from Algebra i Logika, Vol. 49, No. 6, pp. 819–833, November-December, 2010.

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Khukhro, E.I. Nilpotent length of a finite group admitting a frobenius group of automorphisms with fixed-point-free kernel. Algebra Logic 49, 551–560 (2011). https://doi.org/10.1007/s10469-011-9117-x

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Keywords

  • Frobenius group
  • automorphism
  • finite group
  • soluble group
  • nilpotent length
  • Fitting series