What is the use of partitioning a group into cosets

Question

An equivalence relation on a set partitions that set.Now what is the use of this partition?

In specific in group Theory we partition the group into cosets, now what is the use of it? Does it help me to study the group in parts?

Like if I'm studying a group on whole there are lots of elements, but when I Partition it to cosets, in a given coset, the elements of that coset have similar properties, so studying one element from a coset is itself enough.

Is my intuition correct?

Answer 1

In some cases (in particular, if the subgroup is normal), you can define a group structure on the set of cosets, forming what's called a quotient group. Being smaller, the quotient group is easier to study, and can yield important information about the group.

Answer 2

Algebra, at least in the classical sense, is about solving equations using "algebraic" transformations. For instance, we can solve the equation $2x+1=5$ by first subtracting 1 and then multiplying by $\frac12$. Both operations transform the (true) equation into another (true) equation in an algebraic way. Both operations are also homomorphisms! One is a homomorphism of the affine structure, the second is a ring homomorphism. So homomorphisms transform true equations into other true equations algebraically.

This also works for more complicated things. For instance, the equation $(x-2)^2=4$ can be solved by taking the square root, but also keeping in mind that we can add a negative or positive sign, so we have two solutions. This, too, can be expressed in the language of homomorphisms: the equation can be obtained by applying a homomorphism of the multiplicative group (squaring) to one of two equations: $x-2=2$ or $x-2=-2$. So we get a collection of multiple equations which, if they are true, imply that the original equation is true, which is exactly what we want. In this process, take note of what happened with the right side of the equation: it can take all the values which are preimages of 4 under the squaring operation. So it's a preimage of a homomorphism. But preimages of homomorphisms are cosets of the homomorphism's kernel! So to solve the equation we have to find the coset $2\ker\varphi$, where $\varphi:x\mapsto x^2$.

In modern algebra, homomorphisms appear everywhere, and we care about their preimages, among others because of the reason explained above. And their preimages are cosets of their kernels. You'll probably very rarely stumble upon cosets without an accompanying homomorphism to whose kernel they belong. For instance, when constructing quotient groups, you'll probably examine the natural projection homomorphism. The cosets forming a quotient group are exactly the cosets of that homomorphism's kernel.