I just did this problem:
"Prove that if $G$ is a finite group and $H$ is a proper normal subgroup of largest order, then $G/H$ is simple."
And I am currently working on this problem:
"Suppose that $p$ is the smallest prime that divides $|G|$. Show that any subgroup of index $p$ in $G$ is normal in $G$."
If $p$ is the smallest prime that divides $|G|$, then 'proper normal subgroup of largest order' and 'normal subgroup of index $p$' are equivalent, aren't they? Based on this, I've been trying to start with the quotient, which is of order $p$ and must be simple, and work backward. I don't know that this is necessarily a good approach.
I'm not making too much progress toward a proof, and any hints (not answers) would be much appreciated.