Nilpotent Length of a Finite Solvable Group with a Frobenius Group of Automorphisms

Abstract

We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C _{C G (F)}(h) = 1 for all nonidentity elements h ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.

Key Words:

• Automorphisms
• Frobenius group
• Nilpotent length
• Solvable group

2010 Mathematics Subject Classification:

• 20D10
• 20D15
• 20D45

ACKNOWLEDGMENT

The authors would like to thank Evgeny Khukhro for pointing out an error in an earlier version of the article, and the anonymous referee for his/her valuable suggestions which made the text more readable.

Notes

Communicated by A. Turull.