1. ## Automorphisms

Find Aut(Z).

I understand kind of what an automorphism is (and isomorphism from a group G onto itself), and I understand what an isomorphism is (a one-to-one mapping from G to G-bar that preserves the group operation), but we were never given any examples of finding an automorphism. I feel like this shouldn't be very hard, but I can't figure it out!

2. Originally Posted by bluejay
Find Aut(Z).

I understand kind of what an automorphism is (and isomorphism from a group G onto itself), and I understand what an isomorphism is (a one-to-one mapping from G to G-bar that preserves the group operation), but we were never given any examples of finding an automorphism. I feel like this shouldn't be very hard, but I can't figure it out!

4. If $\displaystyle \phi : \mathbb{Z}\to \mathbb{Z}$ is an automorphism it is completely determined by knowing $\displaystyle \phi (1)$ since $\displaystyle 1$ generates the group. Let $\displaystyle \phi(1) = n$ where $\displaystyle n\in \mathbb{Z}$. Then $\displaystyle \phi ( x ) = nx$. It can be shown for $\displaystyle \phi$ to be one-to-one and onto we require $\displaystyle n=\pm 1$. Thus, $\displaystyle \phi_1 (x) = x$ and $\displaystyle \phi_2 (x) = -x$ are the only two automorphisms.