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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.
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