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),…,C G (v)]=1 for any u,v∈B∖{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∈B∖{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.