Isomorphism without being told mapping

[Myriadi]

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.

 

[Hurkyl]

Among your options are:

• Guess.
• Experiment with the arithmetic in G to understand it better.
• Invoke theorems about homomorphisms from Z.

 

[NruJaC]

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!

 

[Myriadi]

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!

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. :)