What exactly is the purpose of a coset?

Let me begin with a geometric example:

$\Bbb C^*$ be the group of non-zero complex numbers under multiplication

Let $H=\{x+iy\in \Bbb C:x^2+y^2=1\}$ which basically contains all the complex numbers lying on the unit circle.

Now consider the left coset $(3+4i)H$, Geometrically speaking this coset contains all points lying on the circle centered at the origin and radius $5$ (Why?)

In general the left coset $(a+ib)H$ contains all points on the circle centered at the origin and radius $\sqrt{a^2+b^2}$

Here the cosets of $H$ partition the punctured complex plane (i.e. $\Bbb C^*$) into concentric circles. Now isn't that amazing?

As seen in the above example:=

The essence of cosets lies in the fact that they partition the entire group into equivalence classes.

Moreover if the group is finite, each of the partitions will have the same number of elements, this is because of the way we define Right/Left cosets.

All the above observations leads to one of the most important result in Group Theory - The Lagrange's Theorem.

It is crucial to be comfortable with equivalence relations for this subject to make sense. Remember that an equivalence relation $\sim$ on a set $X$ is a relation satisfying three properties: reflexivity, symmetry, and transitivity. But the real reason why such a relation is interesting is that it is equivalent to a partition of $X$. A partition of $X$ is a collection of non-empty subsets of $X$ such that every element of $X$ belongs to exactly one of those subsets. The bijection between equivalence relations $\sim$ and partitions is given by $$a \sim b \iff a \text{ and } b \text{ are in the same subset}.$$ Here are some examples on finite sets:

1. On a set $X = \{0,1,2,3\}$ of four elements, the equivalence relation generated by $0 \sim 1 \sim 2$, $0 \nsim 3$ yields the partition $\{\{0,1,2\},\{3\}\}.$
2. On a set $X = \{0,1,2,3\}$ of four elements, the equivalence relation given by $a \sim b$ if and only if $a-b$ is a multiple of $2$ yields the partition $\{\{0,2\},\{1,3\}\}.$
3. On a set $X = \{0,1,2,3,4,5\}$ of six elements, the equivalence relation given by $a \sim b$ if and only if $a-b$ is a multiple of $3$ yields the partition $\{\{0,3\},\{1,4\},\{2,5\}\}.$
4. On a set $X = \{0,1,2,3,4,5\}$ of six elements, the equivalence relation given by $a \sim b$ if and only if $a-b$ is a multiple of $2$ yields the partition $\{\{0,2,4\},\{1,3,5\}\}.$

As you can see, the last three examples exhibit a symmetry in their partition, and their equivalence relations take a very specific form. This is typical of group quotients. If you have a subgroup $H$ of some group $G$, the cosets $gH$ partition $G$, and each coset has the same size. Moreover, if $G$ is finite, the order of each coset has to divide the order of $G$ (Lagrange theorem). The corresponding equivalence relation is given by $$g \sim h \iff g \text{ and } h \text{ are in the same coset} \iff gh^{-1} \in H.$$

The last three examples above are merely instances of this phenomenon for the finite groups $G = \mathbb{Z}/4,\mathbb{Z}/6,\mathbb{Z}/6$ and their subgroups $H = \{0,2\},\{0,3\},\{0,2,4\}$.