More generally speaking, the idea of quotients in mathematics reaches far beyond its specific application in group theory. A quotient is a new structure built out of an existing structure by declaring certain elements of the old structure equivalent to each other.
The first time in mathematics you probably saw a quotient was when you learned about equivalent fractions. We can define addition, subtraction, multiplication, and division on numerator-denominator pairs, but that structure isn't very interesting or useful in itself, nor do the arithmetic operations have the nice properties we expect of them. Only after taking the quotient by declaring certain numerator-denominator pairs equivalent do we get the nice and useful field of fractions.
In order to make a quotient, you need:
- some type of mathematical object
- an equivalence relation on those objects (reflexive, symmetric, and transitive)
- strictly speaking optionally, but this is what makes the quotient useful: some operations on the type of object that respect the equivalence relation
(For example, if a/b and c/d are equivalent fractions, then a/b - e/f is equivalent to c/d - e/f, and similarly e/f - a/b is equivalent to e/f - c/d; thus subtraction respects the equivalence relation.)
There are several ways to construct a quotient:
- Elements of the quotient can be taken to be the equivalence classes of elements of the original structure. This is the most common method. In the example of quotient groups, cosets are the equivalence classes.
- You can choose a representative element from each equivalence class (for example, reduced fractions).
- You can assume that quotient types exist and assume rules about how quotient types work, as is done in the Lean proof assistant.
Quotients are ubiquitous in mathematics, even in constructing basic stuff like naturals -> integers -> rational numbers -> complex numbers. We just mentioned rational numbers, which are presented as a quotient even to schoolchildren. Someone already mentioned how we can construct the complex numbers from real polynomials in a variable called i by setting i^2 = -1. Integers may be constructed from pairs of natural numbers in a manner very similar to the construction of the rationals. The common Cauchy-sequence construction of the reals starts with a subset of sequences of rational numbers, then takes sequences to be equivalent if their difference approaches zero. You can even try to construct natural numbers as equivalence classes of finite sets, where finite sets are equivalent if they can be put into one-to-one correspondence; this approach has the obvious problem of making every natural other than zero a proper class, but we can fix this by taking a representative element approach instead.
