Maschke's theorem

Remarks

Take any linear projection $p \colon V \to W$ onto $W$ and average it over the group: $$\tilde{p}(v) = \frac{1}{\abs{G}} \sum_{g \in G} g \, p(g^{-1} v).$$ The division by $\abs{G}$ is allowed by the hypothesis on the characteristic, $\tilde{p}$ is $G$-equivariant and is still a projection onto $W$, and $W' = \ker \tilde{p}$ is the required complement. For example, every finite-dimensional complex representation of a finite group is a direct sum of irreducible representations. The hypothesis is necessary: let $G = \set{1, g}$ have order $2$ and $F = \F_2$, acting on $F^2$ by swapping coordinates. The line spanned by $(1, 1)$ is invariant, but it is the only invariant line, so it has no invariant complement.

Going deeper

For a student

By the Artin–Wedderburn theorem, Maschke's theorem gives $\C[G] \cong M_{n_1}(\C) \times \cdots \times M_{n_k}(\C)$, which leads to $\abs{G} = \sum n_i^2$ and to character theory. The converse also holds: if the characteristic of $F$ divides $\abs{G}$, then $F[G]$ is not semisimple. Indeed, $N = \sum_{g \in G} g$ spans a nonzero two-sided ideal with $N^2 = \abs{G} N = 0$. A semisimple ring has no such nilpotent ideal. That case is the subject of modular representation theory. The averaging argument also works for compact groups with the Haar integral in place of the sum.