1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Homomorphism and isomorphism

  1. Mar 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that G = {[1 0 [-1 0 [0 -1 [0 1
    0 1], 0 -1], 1 0], -1 0]} is a subgroup of GL2[/SUB(Z) isomorphic to {1,-1,i,-i}.

    3. The attempt at a solution

    I am clearly sure each element in G can be denoted as {1,-1,i,-i}.
    (I can explain why {1,-1,i,-i}, but I will not explain at here.)
    so G -> {1,-1,i,-i} is a bijection, so isomorphism.

    Is it too simple?
  2. jcsd
  3. Mar 9, 2009 #2
    Not every bijection is a group isomorphism. For a map f:G->H, G and H are groups, to be a group isomorphism it needs to be a bijection and have the property that f(ab)=f(a)f(b) for all a,b in G.
    You will need to choose your bijection carefully so that this property is satisfied.
  4. Mar 9, 2009 #3
    How do I show that f(ab) = f(a)f(b)..?
    Shows everything such that f(1*-1) = f(1)f(-1), f(i*-i)=f(i)*f(-i).. ?
  5. Mar 10, 2009 #4
    Yes, there are 16 different pairs of elements in G.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook