TITLE: Normal Groups and Quotient Groups
QUESTION [11 upvotes]: These concepts are currently confusing me. My reading first defined a normal subgroup as one that is the kernel of a group homomorphism. Then it introduced the terms "left coset" and "right coset," which seem straightforward enough. 
The next definition was that of the index of a subgroup $[G:H]$. Then, it gave another definition of a normal subgroup (?), namely, one invariant under conjugation ($gNg^{-1} = N \,\,\forall \, g \in G$). 
Finally, it gave the following tortuous definition of a quotient group: The set of all left cosets of a normal subgroup $N$ with the law of composition $(gN)(hN) = (gh)N$ and having order $[G:N]$.
Why are those definitions equivalent? And what does that last horrendous definition mean? I'd really appreciate it if someone could provide an intuitive/simple explanation of these terms.

REPLY [2 votes]: I share your confusion. I once found the definition of normal subgroup unnatural. Why $g N g^{-1} = N$, right?
This is how I usually prefer to see normal subgroups: $N$ is normal if and only if the operation on cosets of $N$ is closed. This also gives me a better intuition why a normal subgroup is precisely a kernel of some homomorphism.
Here's a proof that the alternative definition is equivalent to the original definition.
Given two cosets $gN$ and $hN$, when will $(gN)(hN)$ be another coset of $N$? Well, if $N$ is normal, we can write
$$
gNhN = g(hNh^{-1})hN = ghNN = ghN.
$$
This proves that $N$ being normal is sufficient for the binary operation on cosets of $N$ to be closed.
It is in fact necessary too, for if $(gN)(hN) = kN$, we can pick the identity in the first $N$ of $gNhN$ to get $ghN = kN$. Then we can put $h = g^{-1}$ to get $(gNg^{-1})N = N$, which in turn implies $gNg^{-1} \subseteq N$.