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.

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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 $U=\ker~\vartheta$
  • $V=W\oplus U$
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