Show that $S_4$ has a unique subgroup $H$ of index 2.
What I tried:
Of course, $A_4$ has index 2. And since any subgroup of index 2 must be normal, I need to show that $A_4$ is the only normal subgroup of $S_4$.
Let $H≠A_4$ from index 2. Then 6 of the elements must be even and 6 of the elements must be odd. If this would be a subgroup, then it would also be normal, so I think I need to show that this can't be a subgroup. I think I need to show that $\langle H \rangle$ must be $S_4$ or something like that.
Any hints ?
