Let a finite group $G$ act on a finite set $X$ and let $\rho$ be the representation of $G$ on $\mathbb{C}X$ via $\rho(g)\left(\sum_x a_xe_x\right)=\sum_x a_xe_{gx}$.
Then in my notes I have the proof of the fact that the character of this representation, $\pi_X$, must contain the trivial character $1_G$ in it's decomposition into irreducible characters as follows:
"$\langle e_{x_1}+\ldots e_{x_n}\rangle$ is a trivial $G$-subspace with $G$-invariant complement $\{ \sum_x a_xe_x\ |\ \sum_x a_x=0\}$."
I can see why this is $G$ invariant and why it has that $G$ invariant complement but I can't see why that would imply that $\pi_X$ contains $1_G$ (the trivial character) in it's decomposition into irreducible characters.