Abstract. We give an example of an infinite simple Frobenius group G without involutions, with a trivial kernel and a nilpotent complement. Nevertheless, this group is not $\omega $- stable (not even superstable), this is the "only" property missing in order to be a counterexample to the Cherlin-Zil'ber Conjecture which says that simple $\omega $- stable groups are algebraic groups.