A new approach to character-free proof for Frobenius theorem

Let G be a Frobenius group. Using Character theory, it is proved that the Frobenius kernel of G is a normal subgroup of G, which is well-known as the Frobenius Theorem. There is no known character-free proof for this Theorem. In our note, we prove it by assuming that Frobenius groups are non-simple. Also, we prove that whether K is a subgroup of G or not, Sylow 2-subgroups of G is either cyclic or generalized quaternion group. In addition, by assuming some extra arithmetical hypotheses on G, we prove the Frobenius Theorem. We should mention that our proof is character-free.