A new approach to character-free proof for Frobenius theorem

Document Type : Original Article


1 Department of Mathematics, Stony Brook University, Stony Brook, New York, USA

2 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 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.


Main Subjects

[1] K. Corradi and E. Horv ´ ath ´ , Steps towards an elementary proof of Frobenius’ theorem, Comm. Algebra, 24 (1996), pp. 2285–2292.
[2] P. Flavell, A note on Frobenius groups, J. Algebra, 228 (2000), pp. 367–376.
[3] G. Frobenius, Uber einen Fundamentalsatz der Gruppentheorie. ¨ , Berl. Ber., 1903 (1903), pp. 987–991.
[4] L. C. Grove, Groups and characters, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1997. A Wiley-Interscience Publication.
[5] B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, SpringerVerlag, Berlin-New York, 1967.
[6] B. Huppert and N. Blackburn, Finite groups. III, vol. 243 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, 1982. [7] H. Kurzweil and B. Stellmacher, The theory of finite groups, Universitext, Springer-Verlag, New York, 2004. An introduction, Translated from the 1998 German original.
[8] D. Passman, Permutation groups, W. A. Benjamin, Inc., New York-Amsterdam, 1968.
[9] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys., 3 (1892), pp. 265–284.