I'm an undergraduate learning about group representations and Young tableaux, and have came across Maschke's theorem stating;
If $G$ is a finite group and $F$ is a field whose characteristic does not divide the order of $G$, then every finite dimensional $G$-module over $F$ is completely reducible
The many proofs I have seen, show this by proving an equivalent statement that if $G$ and $F$ are as above, and $H$ is a submodule of a $G$-module $V$, then there exists a submodule $H'$ of $V$ such that $V$ is the direct product of $H$ and $H'$. This is done by showing $H'$ is the kernal of some homomorphism of $G$-modules that maps $H$ to $H$. However, the this approach doesn't seem well-motivated in the sense of; where did this homomorphism come from?
Does anyone have a suitable alternative proof for an undergraduate? Or can possibly shed some intuition on the theorem? Thank you.