I have been trying to study groups from I.N Herstein's "Topics in Algebra" and have come across the idea of a "quotient structure".
Herstein while talking about Normal subgroups $N$ shows how the product of two right cosets of $N$ is again a right coset of $N$. From this observation he motivates the definiton of a quotient group as follows:
Suppose $N$ is a normal subgroup of G. The formula $NaNb=Nab$ is highly suggestive. Can we use this product to make a collection of right cosets into a group? Indeed we can! This type of construction often occuring in mathematics and usually called forming a quoteint structure, is of utmost importance."
From the above quote it is clear that forming a quotient structure is a general thing and not something that occurs only in groups as quotient groups.
I want to know what do the quotient structures mean in general?
What is the general idea behind the existence of such quoteint structures and how are they useful in mathematics?
Quotienting in algebra is a way to ignore some structure in order to make the remaining structure more apparent.
Take the integers as an example. Lots of things one can study with those, like all sorts of divisibility and primes, the ordering and sizes of integers, and so on.
But some times you don't want that, you just care about whether the numbers you have are even or odd. Then you can often take $N$ to be the subgroup of even integers, and quotient out by that. You now have two cosets: the even numbers and the odd numbers. Work with those cosets instead of pure integers, and whatever you do will hone in on properties that have to do with the evenness of numbers.