Maschke’s theorem

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Any submodule $V \leq W \in {}_{G}{\mathsf{Mod}}$ has a $G{\hbox{-}}$ invariant complement. Proof: choose $\pi:W\to V$ a projection and define \begin{align*} \pi_{G}(x)=\frac{1}{|G|} \sum_{g \in G} g \cdot \pi\left(g^{-1} \cdot x\right). \end{align*}

Alternative statement: $kG \in {}_{k} \mathsf{Alg} $ is semisimple.

Maschke's theorem

Maschke’s theorem