Document Type : Original Article
Department of Mathematics, Stony Brook University, Stony Brook, New York, USA
Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
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.