Complete irreducibility of infinitely dimensional representation

Question

This is a step to prove Maschke’s theorem. Let $G$ be a finite group, every indecomposable $G$-module is simple $\iff$ complete reducibility. For finite dimensional representation, we can prove this by induction on dimension. How to prove this for infinite dimensional case?

Answer 1

It's easy to show that complete reducibility implies that every indecomposable representation is simple.

If you can show complete reducibility for finite-dimensional representations, then you can show it for the regular representations $k[G]$ (where $k$ is the base field). But this means that $k[G]$ is a semisimple ring. Over a semisimple ring, every module is semisimple.

To see the last claim, note that infinite direct sums of semisimple modules are semisimple, so we get that every free module is semisimple. Also quotients of semisimple modules are semisimple, so as every module is a quotient of a free module, this implies the result.