Towards a group theoretical proof of the Frobenius theorem

Document Type : Research Paper

Author

School of mathematics, statistics and computer science, College of science, University of Tehran, Tehran, Iran

Abstract

The Frobenius finite group was defined by Frobenius more than 100 years ago to be a group with a non-trivial proper subgroup $H$ whose intersection with all of its conjugates is the trivial group. In 1901, Frobenius proved that $H$ has a normal complement $K$ in the group, but in his proof, the character theory was used. Since then providing a character free proof was a challenging problem. Although the existence of $K$ is proved by imposing extra conditions on $H$, but up to present time no general proof is known. In this note, we prove this theorem using elementary group theory by setting a certain condition on $H$.

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References

[1] K. Corradi & E. Horvath, (1996). Steps towards on elementary proof of the Frobeniuse theorem, Comm. Alg. 24(7), 2285-2292.
[2] F. G. Frobenius, (1901). Uber auflosbare Groppen IV, S'ber. Akad. Wiss. Berlin, 1216-1230; Ges, Abh. III, 189-203.
[3] D. Gorenstein, (1968). Finite groups, Harper & Row, Publications, New York, Evanston, and New York.
[4] W. R. Scott, (1964). Group theory, Dover publishing Inc., New York.
[5] M. Suzuki, (1982). Group theory I, Springer-Verlay, Berlin, Heidelberg, New York. 
Journal of Mahani Mathematical Research
Volume 15, Issue 1 - Serial Number 33
January 2026
Pages 317-319

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History

  • Receive Date: 23 May 2025
  • Revise Date: 30 July 2025
  • Accept Date: 30 August 2025

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