I need help with the following proof:
Prove that Aut($\displaystyle \mathbb{Z}_{4}) \approx \mathbb{Z}_{2}$
Thanks
Note, $\displaystyle \mathbb{Z}_4 = \{ [0],[1],[2],[3] \} = \left< [1]\right>$. Any homomorphism $\displaystyle \phi: \mathbb{Z}_4 \mapsto \mathbb{Z}_4$ is completely determined by $\displaystyle [1]$ (since it is the generator). Thus, there are four possibilities, $\displaystyle \phi([1]) = [0],\phi([1]) = [1], \phi([1]) = [2],\phi([1])= [3]$. Which of these give rise to automorphisms?