Fixed points of Frobenius groups of automorphisms

A finite group admits a Frobenius automorphisms group FH with a kernel and complement H such that the fixed-point subgroup of F is trivial. It is further proved that every FH-invariant elementary Abelian section of G is a free module for an appropriate prime p. The exponent of a group is bounded with a metacyclic Frobenius group of automorphisms and it is supposed that a finite Frobenius group FH with cyclic kernel F and complement H acts on a finite group G. Bounds for the nilpotency class of groups and Lie rings admitting a metacyclic Frobenius group of automorphisms with fixed-point free kernel are obtained. It is also found that a locally nilpotent torsion-free group G admits a finite Frobenius group of automorphisms FH with cyclic kernel F and complement H of order q