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

$${\pi }^{\prime }(v)=\frac{1}{|G|}\sum _{g\in G}{g}^{-1}\pi (gv)$$

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 $W$.∎