Unitary Group is denoted .
Theorem: for any positive integer n, is isomorphic to .
Q: Prove that every element is determined by ; meaning, results in a well-defined automorphism of .
A: let be arbitrary. Then is an isomorphism. So, to show if something is well-defined I need to show if a=c, b=d, then a*b=c*d. I am not sure what to my a,b, and c's are. I know . so, do I let and . Now, I want to show ?
I think that your analysis of the problem is a little bit too complicated. For an easy example, let us look at the group . If is an automorphism, it must send a generator to a generator. To be explicit, a number is a generator of if and only if . In our case, is a generator if and only if . This means that . So, we have the following possibilities: , , , , or . If you try composing these maps, you will find that you recover the multiplication table for your .
Give it a try. Good luck!