I don't know where to start and I'm looking for help on the following problem.
Let V be a finite dimensional $K[G]$-Module ($G$ finite) and $\mathrm{char}(K)=0$ or coprime to $\mathrm{Ord}(G)$. Prove that if every $K[G]$-homomorphism $V \to V$ is a scalar multiple of $\mathrm{id}_{V}$ then $V$ is irreducible.
We obviously can apply Maschke's Lemma but how can we use this direct sum decomposition to deduce the irreduciblity.
As you mentioned, we know from Maschke's theorem that if $W \subset V$ is a $k[G]$-submodule, then there is a decomposition of $V$ as direct sum of $k[G]$-submodules $V = W \oplus W'$.
Consider the projection followed by the inclusion $V \to W \subset V$. This gives a $k[G]$-endomorphism of $V$. What can you say about it?