On groups and Lie algebras admitting a Frobenius group of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms FH of coprime order with kernel F and complement H such that CG(H) is nilpotent of class c. We prove that if F is abelian of rank at least two and [CG(a),CG(b),…,CG(b)]=1 for any a,b∈F∖{1}, where CG(b) is repeated k times, then G is nilpotent of class bounded in terms of c, k and |H| only. It is also proved that if F is abelian of rank at least three and CG(a) is nilpotent of class at most d for every a∈F∖{1}, then G is nilpotent of class bounded in terms of c, d and |H|. The proofs are based on results on graded Lie rings.