Frobenius groups of automorphisms and their fixed points

Suppose that a finite group G admits a Frobenius group of automorphisms FH with kernel F and complement H such that the fixed-point subgroup of F is trivial: CG(F ) = 1. In this situation various properties of G are shown to be close to the corresponding properties of CG(H). By using Clifford’s theorem it is proved that the order |G| is bounded in terms of |H| and |CG(H)|, the rank of G is bounded in terms of |H| and the rank of CG(H), and that G is nilpotent if CG(H) is nilpotent. Lie ring methods are used for bounding the exponent and the nilpotency class ofG in the case of metacyclic FH. The exponent of G is bounded in terms of |FH| and the exponent of CG(H) by using Lazard’s Lie algebra associated with the Jennings–Zassenhaus filtration and its connection with powerful subgroups. The nilpotency class of G is bounded in terms of |H| and the nilpotency class of CG(H) by considering Lie rings with a finite cyclic grading satisfying a certain ‘selective nilpotency’ condition. The latter technique also yields similar results bounding the nilpotency class of Lie rings and algebras with a metacyclic Frobenius group of automorphisms, with corollaries for connected Lie groups and torsion-free locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for non-metacyclic Frobenius groups of automorphisms.