Homomorphism from GL(2,N) to Z_N?

[condmatscott]

I start with the group GL(2,N), where N is prime. I want to break these elements into N classes. One way to do this would be to find a homomorphism to Z_N, does such a homomorphism exist for general N? What is it? Is there another way to break the group into classes without using a homomorphism?

 

[fzero]

I start with the group GL(2,N), where N is prime. I want to break these elements into N classes. One way to do this would be to find a homomorphism to Z_N, does such a homomorphism exist for general N? What is it? Is there another way to break the group into classes without using a homomorphism?

If the notation GL(2,N) means that this group acts on a vector space of indefinite inner product, then there is a natural SL(N) subgroup. The center of this subgroup is $\mathbb{Z}_N$.