Why is the fact that a quotient group is a group relevant?
There are several reasons:
- Sometimes, a group is too complicated to study as a whole. So we look at the quotient group, which is smaller. This can give us information about the original group structure.
An example to illustrate this:
If $Z(G)$ is the center of a group $G$, and the quotient group $G/Z(G)$ is cyclic, then the group $G$ itself is abelian. This fact can be used to prove that every group of order $p²$ (p prime) is abelian.
- Some quotient groups are extremely important in mathematics. Think about the group $\mathbb{Z}/n\mathbb{Z}$, which is strongly related with modular arithmetic, number theory and cryptography.
Even when you were a kid, you were very familiar with the group $\mathbb{Z}/12 \mathbb{Z}$, without even realising: when you look at the clock, you start counting at $0$ again when it is $12$ o' clock.
Also, every cyclic group of order $n$ is isomorphic to this group, so knowing this group will allow you to fully understand a cyclic group, but there is more:
We can write every finite abelian group (up to isomorphism) as the direct product of quotient groups of the form $\mathbb{Z}/n\mathbb{Z}$. So understanding this one particular quotient group allows us to understand every abelian group!
In group theory, we are interested in building new groups using existing groups. Quotient groups are one way to build new (smaller) groups from an existing group. Other manners are direct products, semidirect products, etc.
Linking finite groups with quotient groups yields interesting methods to count the order of a group. For example, it is well known that $$sgn: (S_n, \circ) \to (\{-1,1\},.)$$ is a group homomorphism with kernel $A_n$
By the first isomorphism theorem, it follows that:
$$S_n/A_n \cong \{-1,1\} $$
Hence, $$|A_n| = \frac{|S_n|}{|\{-1,1\}|} = \frac{n!}{2}$$
In the same way, combinatorical identities can be proven.
Suppose you have an abstract group $G$ which you are not familiar with, but you manage to find an isomorphism $G \cong H/N$ where $H$ is a group you are familiar with. Then, because you know $H$ well, you also know the quotient group well (the operation on the quotient is the one induced by the operation on the group), and hence you have translated the information of this abstract group $G$ to something you can easily work with.
Quotient groups provide a way to show that all normal subgroups of a group $G$ are exactly the kernels of group homomorphisms $G \to H$.
Indeed, it is well known that the kernel of a group morphism is always a normal subgroup, and if $N$ is a normal subgroup of $G$, it is the kernel of the canonical epimorphism $G \to G/N: g \mapsto \overline{g}$
- In abstract algebra, terms like quotient rings and quotient modules pop up all the same for several reasons analoguous to what was already written earlier. These things are in the first place quotient groups.
Some of the first groups you're introduced to are $\Bbb Z/n \Bbb Z$. This is a quotient group. By the theorem about finitely generated abelian groups all finitely generated abelian groups are built up from these quotient groups.
The single biggest reason is that studying the quotients of a particular group can actually unlock information about the group itself. Concrete examples
1.) The proof of the 1st Sylow theorem (at least the ones I've seen) rely on looking at a quotient of a particular group. The third one as well.
2.) If $N$ is normal and solvable and $G/N$ is solvable, then $G$ is solvable.
3.) If $H$ is normal in $G$ and $gcd(|x|,|G/H|) = 1$ then $x \in H$
4.) If $G/Z(G)$ is cyclic, then $G$ is abelian
As you progress through algebra you will see it pop up again and again, and their usefulness will become clearer. The concept of a quotient group is one of the single most important developments in the history of mathematics.