A weird proposition in Lang's "Algebra" It goes like this:
Let $G$ be a group, $H < G$, and $N \lhd G$ be the smallest normal subgroup containing $H$. Then any $f \in \operatorname{Hom}(G,G')$ such that $H$ is in its kernel uniquely factors through $G/N$.
It's a simple proposition, but it's so weirdly formulated that I spent the whole day picking it apart.
Basically, it says two things:
1) $H < N \lhd G$, and $N < \operatorname{Ker}f$ (so $N \lhd \operatorname{Ker}f$), which is trivial to prove.
2) Any hom uniquely factors through a quotient by any normal group that is a subgroup of its kernel.
So why combine two natural propositions into a single one that is so weird? Is it used in some important place or does it have any significance in the group theory by itself? Or am I mistaken and the second proposition I put forward is generally false?
 A: I suspect you find Lang's statement weird because you are thinking of it as a technique for generating homomorphisms, and then wondering why Lang is putting a strange emphasis on these particular kinds of normal subgroups.  
But to appreciate the proposition, you should think from a different point of view.  E.g. suppose that you are given a homomorphism $f: G \to G'$, and that
you are told that $f(h) = 1$ for every $h \in H$.  Then what can you conclude about $ker(f)$?   Well, it certainly contains $H$, since this is what you are told.  But since the kernel is normal, it in fact contains the normal subgroup of $G$ generated by $H$, which is what Lang calls $N$. 
As a side remark: although $N$ is correctly described as the smallest normal subgroup containing $H$, it is probably better (from a psychological point of view) to think of $N$ as the normal subgroup of $G$ generated by $H$; this helps bring out the role of $H$.   
A: Well, I don't know Lang's reason, so I can only guess. First of all, it is a concise and easy enough statement and I don't see what's so particularly weird about it.
That said, I think the main reason is that the proposition as stated by Lang accurately describes the categorical image $\langle\langle H \rangle\rangle_{G}$ (the normal closure of $H$ in $G$, or as your proposition states the kernel of the cokernel) of the inclusion $H \hookrightarrow G$ and its categorical cokernel $G/\langle\langle H \rangle\rangle_{G}$ when the inclusion is viewed as a morphism in the category of groups, see e.g. here.
As for your last question, your second proposition is correct and follows from the usual homomorphism theorem.
Added much later: It seems that $\langle H^{G} \rangle$ is the usual notation for the normal closure which I denoted $\langle\langle H \rangle\rangle_{G}$ — some people insisted on that by editing it in — but in my opinion this is really ugly and, much worse, quite ambiguous.
