Frobenius groups of low rank

Abstract

Let G be a finite Frobenius group of degree n. We show, by elementary means, that n is a power of some prime p provided the rank {mathrm{rk}}(G)le 3+sqrt{n+1}. Then the Frobenius kernel of G agrees with the (unique) Sylow p-subgroup of G. So our result implies the celebrated theorems of Frobenius and Thompson in a special situation.