Let $G$ be a subgroup of the group of permutations on a finite set $X$. Let $F$ be the $\mathbb{C}$-vector-space of all the functions from $X$ to $\mathbb{C}$. $G$ acts on $F$ by $(g \cdot f) : x \mapsto f(g^{-1}(x))$. Show that there is a $\phi \in F$ such that $g \cdot \phi = \phi$ for every $g \in G$. Show that there is a subspace $F'$ of $F$ such that $F = F' \oplus \mathbb{C}\langle \phi \rangle$ and such that $g \cdot f \in F'$ for every $g \in G$ and $f \in F'$.