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 CG(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 CG(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 CG(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.