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Maschke's theorem
Let $G$ be a finite group, and $k$ a field of characteristic not dividing $|G|$ . Then any representation $V$ of $G$ over $k$ is completely reducible.
Proof. We need only show that any subrepresentation has a complement, and the result follows by induction.
Let $V$ be a representation of $G$ and $W$ a subrepresentation. Let $\pi:V\to W$ be an arbitrary projection, and let $$ \pi'(v)=\frac 1{|G|}\sum_{g\in G} g^{-1}\pi (gv) $$ This map is obviously $G$ -equivariant, and is the identity on $W$ , and its image is contained in $W$ , since $W$ is invariant under $G$ . Thus it is an equivariant projection to $W$ , and its kernel is a complement to $W$ . 
Maschke's theorem is owned by bwebste.
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