TITLE: Second Isomorphism Theorem Notation Query
QUESTION [5 upvotes]: I'm reading about the Second Isomorphism Theorem and it seems to me as if $HN/N$ should equal $H/N$. Because, $hnN$ is the same as $hN$. I just don't understand what's wrong with my train of thought, as surely if I were correct, the book wouldn't be using the notation $HN/N$. 
I hope this doesn't sound like a silly question, it's really doing my head in.
Thanks for the help guys!

REPLY [4 votes]: In the usual framework $H$ and $N$ are subgroups of a larger group $G$.
It is usual to assume only that $H$ is a subgroup and $N$ is a normal subgroup of $G$.
In particular, $N$ may not be a subgroup of $H$ (let alone normal), so $H/N$ doesn't make sense in all cases.

REPLY [3 votes]: The issue is that $N$ may not even be a subset of $H$, let alone a subgroup, so you take $HN$ which is a subgroup of $G$ (as $N$ is normal) and then $N$ is normal in $HN$ so the quotient is well defined.