Let $ n > 1 $. Show that there exists a proper subgroup $ H $ of $ S_{n} $ such that $ [S_n : H] \leq n. $
From what i learn i can only deduce that $[S_n:H] = \frac{|S_n|}{|H|}$ because $S_n$ and $H$ is finite so its a consequence of Lagrange. And since $|S_n| = n!$, $|H|$ divides $n!$.
From then i can't seem to get to the problem. Please help.