Homomorphism and isomorphism

  • Thread starter hsong9
  • Start date
  • #1
80
1

Homework Statement


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}.

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?
 

Answers and Replies

  • #2
316
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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.
 
  • #3
80
1
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).. ?
 
  • #4
316
0
Yes, there are 16 different pairs of elements in G.
 

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