In a proof of Maschke's theorem, my lecturer writes "If $N$ is a $\mathbb C G$-module and $N\leq M $ is a submodule, let $\pi: M \to M$ be a projection (i.e. a map with $\pi^2=\pi$) with image $N$." I am wondering how we know that such a projection exists.
In one of our problem sheets it is highlighted that the theorem sometimes does and sometimes does not apply for fields other than $\mathbb C$. In the first part of the question I have showed that there does not exist a complement for the $\mathbb F_2 C_2$-submodule $\mathbb F_2 <e+g>$ (where $C_2 = <g>$, the cyclic group of order 2.) In the second part of the question, I am to show that there does exist a complement of the $\mathbb F_3 C_2$-submodule $\mathbb F_3 <e+g>.$
I know that I could tackle the above problem just by trying to spot a complement but I feel like that's cheating somewhat, and I'm also aiming to understand the proof of Maschke's theorem, so I wanted to do the question by forming a projection $\pi : \mathbb F_3C_2 \to \mathbb F_3C_2$ with image $\mathbb F_3 <e+g>.$ I may be missing something silly, but I can't spot one. My first try $\pi(e)=e+g=\pi (f)$ gives $\pi^2(e)=2(e+g)$, so this isn't a projection.
In short I have two questions:
Why does the above projection in the proof of Maschke always exist? And what is a projection that I could use for my exercise, (if it exists) for this $\mathbb F_3 <e+g>$?