Let H be the subgroup of consisting of all matrices of the form , where . I have to prove that Z(H) is isomorphic to and that is isomorphic to .
I'm really not sure how to begin with this. I started by taking two arbitrary matrices h and k from H and doing hk = kh to see what a matrix in Z(H) would have to look like, but I didn't really get anywhere with that. My initial instinct would be to just define a mapping from Z(H) to , but I'm not sure how to do that, since I can't figure out what's in Z(H). Is there a better way to do this?