Why are finite groups linearly reductive?

Question

Let $G$ be a linear algebraic group contained in $GL(n)$. $G$ is linearly reductive iff every regular representation is completely reducible. Among the examples of linearly reductive groups, there are finite groups: why is that?

Answer 1

This is because for any such group $G$ and its (finite dimensional) representation $V$ we can define on $V$ a hermitian $G$-invariant inner product. Using this product we can easily check that an orthogonal compliment to any $G$-invariant subspace is $G$-invariant itself. Of course everything is in characteristic $c \neq \#G$.

To construct such inner product one can start with an arbitrary hermitian inner product $(\cdot,\cdot)$ and define $$ \langle x \, | \, y\rangle = \sum_{g \in G}(gx, gy). $$