Isaacs, $\textit{Character Theory of Finite Groups}$, Theorem(1.9)

Let $G$ be a finite group and $F$ a field whose characteristic does not divide $|G|$. Then every $F[G]$-module is completely reducible.

Pf: Let $V$ be an $F[G]$-module with submodule $W$ and $V=W\oplus U_0$. 

• $\varphi$ the projection map of $V$ onto $W$ with respect to $U_0$
• $\vartheta(v)=\frac{1}{|G|}\sum_{g\in G}\varphi(vg)g^{-1}$ and $\vartheta$ is an $F[G]$-homomorphism from $V$ to $W$
• $W=im~\vartheta$ and $U=\ker~\vartheta$
• $V=W\oplus U$