1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Help with quotient space?

  1. Aug 19, 2007 #1
    I just wanted to know if someone can explain to me the basic concept of a quotient space and quotient groups.
  2. jcsd
  3. Aug 19, 2007 #2


    User Avatar
    Homework Helper

    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.


    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.
    Last edited: Aug 19, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Help with quotient space?
  1. Quotient space (Replies: 0)