I was wondering what the order of the group $Aut(Z_{p^a} \times Z_{p^b})$ is, where $p$ is prime and $a $ and $b$ are natural numbers, eventually equal. I had found a way on internet that was using matrices, but it looks like I cannot find that topic anymore, and I think I hadn't really understood that explanation, could someone please explain how to count them?
By the way, it seems that using matrices is the best option (as I clearly remember the explanation I had found wasn't very long and looked like it didn't require many calculations), but couldn't I just calculate how many "good" possibilities there are? I mean, I would send the generator of $Z_{p^b}$ in another element of order $p^b$, then the generator of $Z_{p^a}$ in another element of order $p^a$. The problem is that the cyclic subroups generated by the images of the generators must have a trivial intersection. So I was hoping to find how many choices I had for the second generator, but here it gets tricky and I don't know how to proceed, how to count how many "good" choices I have.
Thanks for the help
