TITLE: Why is the fact that a quotient group is a group relevant?
QUESTION [13 upvotes]: I'm studying the basics of quotient groups. I understand that if you build a quotient set from cosets of a group and the subgroup you are using to build them is normal then you end up with a group.
I fail to see why the fact that we can define operations in the quotient set and make it into a group is meaningful.
What is the motivation to do it?
Either historical perspective, practical perspective or any other to understand why we care about this is welcome.

REPLY [3 votes]: It is relevant by the first isomorphism theorem.
If $\phi: G \rightarrow H$ is a surjective homomorphism of groups, and $N$ is the kernel of $\phi$, then $N$ is a normal subgroup of $G$, and the function $\bar{\phi}: G/N \rightarrow H$ defined by $\bar{\phi}(xN) = \phi(x)$ is an isomorphism of groups.
I would say the reason we care that $G/N$ has a group structure when $N$ is a normal subgroup, is that for surjective homomorphisms as above, we can forget about the object $H$ entirely, identifying it with the quotient group.
Example: let $S^1$ be the group of complex numbers with absolute value $1$ (the unit circle), and define $\phi: \mathbb{R} \rightarrow S^1$ by $\phi(x) = \cos 2 \pi x + i \sin 2\pi x$.  Then $\phi$ is surjective with kernel $\mathbb{Z}$, so $\phi$ induces an isomorphism $\bar{\phi}: \mathbb{R}/\mathbb{Z} \rightarrow S^1$ by $\bar{\phi}(x + \mathbb{Z}) = \cos 2\pi x + i \sin 2\pi x$.  The isomorphism $\bar{\phi}$ allows us to identify the circle with $\mathbb{R}/\mathbb{Z}$.
This techically applies to all normal subgroups of a group $G$, since a normal subgroup is exactly the same thing as the kernel of a surjective homomorphism with domain $G$.