What is the concept of quotient space and quotient groups?

[Coolphreak]

I just wanted to know if someone can explain to me the basic concept of a quotient space and quotient groups.

 

[radou]

Well, let ~ be an equivalence relation on S. For some x from S, the equivalence class of x is the class of all elements from S which are equivalent to x, i.e. [x] = {y from S : y ~ x}. The class of all equivalence classes in S is called the quotient class of S by ~, and is denoted S/~. This gives an outline of the idea - you need to have an equivalence relation defined.

So, if you have a subgroup H of a group G, and some elements a, b from G, define the relation "~r (~l) to be right (left) congruent modulo H", so that a~r b (mod H) if ab^-1 is in H (a~l b (mod H) is a^-1 b is in H). One can easily show that right (or left) congruence modulo H is an equivalence relation on G, and you may look at the equivalence classes of that relation (usually called cosets of H in G and denoted Ha (aH) for right (left)congruence modulo H). Now, if you have a normal subgroup N of G (left and right cosets coincide, eg the relations ~r and ~l coincide), it can be shown that G/N is a group (under a specific binary operation), and it's called the quotient group of G by N.

So, we started off with an equivalence relation, took its equivalence classes and defined a binary operation with aNbN = abN, and arrived at a (quotient) group G/N.

Edit:

It's even easier to do this for a vector space: let V be a vector space, and M some subspace of V. Define the relation ~ on V with (for x, y in V) x ~ y iff x - y is in M. It's easily verified that ~ is an equivalence relation. Again, look at the equivalence classes of this relation, and take the class of all the equivalence classes, eg V/~ = {[x] : x in in V} is called the quotient (vector) space with operations on equivalence classes defined with [x] + [y] = [x + y] and a[x] = [ax], where a is any scalar.