Prime index of normal subgroup I can do all of this question except the last part: Let $N$ be a normal subgroup of a finite group $G$ of prime index $p$. Show that if $H$ is a subgroup of $G$ then $H\cap N$ is a normal subgroup of $H$ of index $1$ or $p$. Could I have some help please?
I've shown that $H\cap N$ is a subgroup using the one-step test, and I've shown it's normal by conjugating it with an element of $H$. 
I'm stuck on identifying the index of $H\cap N$. I'm guessing it involves Lagrange's Theorem, and I tried to come up with a homomorphism from $H$ to something with kernel $H\cap N$, but with no success. 
I'd prefer hints to a full solution, if that's alright.
Thank you very much for all your help.

Even without the 2nd isomorphism theorem, it could be you've already studied that
$$[H:H\cap N]=[NH:N]$$
and since the last one divides the index of $\;N\;$ in G we're done.
To prove the above equality you could try the following: define
$$f:H\to NH/N\;,\;\;f(h):=hN$$
(1) Show $\;f\;$ is a homomorphism
(2) Show $\;H\cap N=\ker f\;$
(3) Show that $\;f\;$ is surjective
(4) and finally use the first isomorphism theorem ...
In fact, the above is almost-almost the proof of the 2nd isomorphism theorem...:)
