Maschke's Theorem

Group Ring

Let $G$ is a finite group with order $n$, and $k$ is a field of characteristic prime to $G$. Then group ring $kG$ is semi simple.

Proof

It is enougth to show that $kG$ satisfies the SS3.

Let $F⊲E$ be $kG$-module. Since $k$ is a field, there exists $k$-linear projection $\pi :E\to F$ such that $\pi (f)=f$ for $f\in F$. Take the average¹

$${\pi }^{\prime }(e):=\frac{1}{n}\sum _{g\in G}g\pi (g^{-1}e)$$

Clearly, ${\pi }^{\prime }$ is $kG$-linear, i.e. ${\pi }^{\prime }(g_{0}e)=g_0{\pi }^{\prime }(e)$² and also the projection. Therefore, $E$ can be composed as $E=Im{\pi }^{\prime }\oplus Ker{\pi }^{\prime }$.

Module Case

If $V\subset W$ as $kG$-modules, and $char(k)\nmid \left|G\right|$, then $\exists V^{\prime }$ such that $W=V\oplus V^{\prime }$, i.e. $W$ is a Semi Simple.

Footnotes

1. Here $n$ is adding $1_k$ $n$ times. ↩

2. Think about the finite group table that we learnt in the basic abstract algebra coruse. ↩