# 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 $W$. ∎

Title Maschke’s theorem MaschkesTheorem 2013-03-22 13:21:16 2013-03-22 13:21:16 bwebste (988) bwebste (988) 9 bwebste (988) Theorem msc 20C15