This is a problem from Herstein that I have been stuck upon for ages. I am becoming increasingly disappointed and disillusioned about my abilities due to this problem.
Let G be finite and let $p$ be the smallest prime dividing $|G|$. Let $H \le G$ be of index $p$. Prove that $H$ is a normal subgroup of $G$.
I am nowhere near getting a solution. Assume the statement is not true, so $N(H) \ne G$ then $H \le N(H) \le G$ implies that $N(H)=H$. How can I get a contradiction from this? I am trying to find a homomorphism which has H has its kernel, but getting nowhere. I would like to see, atleast a couple of different solutions for this.