Existence of Hermitian form . I came across this question in my notes : If $G$ is finite group and $V$ is finite dimensional $CG$ module ( $C$ complex ) , how do i show that there exist a positive definite Hermitian Form $(.,.)$ for $($gv, gw) for all  $g$ and $v$,$w$. 
And show that V is semisimple without using Maschke's theorem. 
 A: The basic idea is that you know that $V$ admits some inner product $\langle -,-\rangle_0$. Then, just normalize it by definiing 
$$\langle u,v\rangle=\frac{1}{|G|}\sum_{g\in G}\langle gu,gv\rangle_0$$
You can then prove Maschke's theorem which says that $\mathbb{C}[G]$ is semisimple by showing that every $\mathbb{C}[G]$-submodule $W\leqslant V$ has a complement by showing that $W^\perp$ (with respect to $\langle-,-\rangle$) is such a subspace--which of course proves the semisimplicity of $\mathbb{C}[G]$ since $V$ was arbitrary.
A: Another interesting fact is the following. Let $V$ be an arbitrary representation of $G$ (it could be infinite-dimensional). A given $v\in V$ is contained in a finite-dimensional subrepresentation:
$$V_v=\mbox{span}\{g\cdot v:g\in G\}$$
(it is finite dimensional since $G$ is a finite group). The answer of Alex Youcis proves that a finite-dimensional representation is semisimple. Hence, it follows that every $v\in V$ is contained in a semisimple subrepresentation, and so we can conclude that $V$ itself is semisimple. Therefore, all representations of a finite group (even infinite-dimensional) is semisimple.
