$G$-circulant Matrices and the Classical Maschke Theorem

TL;DR

This paper presents a concise proof of the Classical Maschke Theorem by leveraging the isomorphism between G-circulant matrices over a field and the group ring, simplifying the understanding of semisimplicity in group algebras.

Contribution

It introduces a novel, streamlined proof of the Maschke Theorem using the isomorphism between G-circulant matrices and the group ring, highlighting a new algebraic perspective.

Findings

Simplified proof of Maschke Theorem

Establishes isomorphism between G-circulant matrices and group ring

Provides algebraic insight into semisimplicity

Abstract

In this note, we use the isomorphism of the ring of $G$-circulant matrices over a field $k$ with the group ring $k[G]$ to derive a very short proof of the Classical Maschke Theorem.