Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Isomorphism without being told mapping

  1. Nov 26, 2009 #1
    Given:

    G is the group of matrices of the form:

    1 n
    0 1

    Where n is an element of Z, and G is a group under matrix multiplication.

    I must show that G is isomorphic to the group of integers Z. I do not know how to do this, since all examples we covered gave us the specific mapping from one group to the other.

    Any help would be appreciated.
     
  2. jcsd
  3. Nov 26, 2009 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Among your options are:
    • Guess.
    • Experiment with the arithmetic in G to understand it better.
    • Invoke theorems about homomorphisms from Z.
     
  4. Nov 26, 2009 #3
    Basically, you need to come up with a mapping yourself. Here's a hint, create a notation such as G(n) represents the matrix with n in Z, in the first row second column. The ideal isomorphism should pop out at you now. Let me know if that helps!
     
  5. Nov 26, 2009 #4
    I managed to figure the problem out not too long ago. That is exactly what I decided to do. Thanks for confirming for me! Problem solved. :)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook