Towards a group theoretical proof of the Frobenius theorem

Darafsheh, Mohammad Reza

doi:10.22103/jmmr.2025.25316.1804

Towards a group theoretical proof of the Frobenius theorem

Document Type : Research Paper

Author

Mohammad Reza Darafsheh

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

10.22103/jmmr.2025.25316.1804

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 $H$.

Keywords

complement

kernel

sylow subgroup

Subjects