Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201A new approach to character-free proof for Frobenius theorem99103486310.22060/ajmc.2022.21305.1085ENSeyedeh FatemehArfaeezarandiDepartment of Mathematics, Stony Brook University, Stony Brook, New York, USAVahidShahverdiDepartment of Mathematics, KTH Royal Institute of Technology, Stockholm, SwedenJournal Article20220416Let 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 a Frobenius theorem. There is no known character-free proof for Frobenius theorem. In this 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 are either cyclic or generalized quaternion group. Also by assuming some additional arithmetical hypothesis on G we prove Frobenius theorem. We should mention that our proof is character-free.https://ajmc.aut.ac.ir/article_4863_36ef014a4f85c1bbee9f6f358b602a34.pdf