A new characterization of frobenius complements

Let H, G be finite groups such that H acts on G and each non-trivial element of H fixes at most f elements of G. It is shown that, if G is sufficiently large, then H has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude that, if G is a finite group and A ⊆G is any non-cyclic abelian subgroup, then the order of G is bounded above in terms of the maximal order of a centralizer C _G(a) for 1≠a ∈A.