The following is an old exam question from a n introduction to group theory course:
Let $G$ be a group (possibly infinite) that acts non-trivially on a finite set $X$ - that is there are $g\in G,x\in X$ s.t $g\cdot x\neq x$.
Prove that there is a proper subgroup $N\trianglelefteq G$ of finite index.
I didn't manage to get very far - Assume by contradiction that $G$ is simple, I know that the Kernel of the operation is normal in $G$ thus is is either $G$ or $\{e\}$ but since the action is non-trivial the kernel of the operation is $\{e\}$ and the action is faithful. - this is where I'm stuck.
Can someone please hint me ?
Notice that $G/Kernel$ isomorphic to a subgroup of $Sym(X).$ As $Sym(X)$ is finite we are done.