# Automorphism

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• Mar 19th 2008, 07:46 AM
hercules
Automorphism
I need help with the following proof:
Prove that Aut( $\mathbb{Z}_{4}) \approx \mathbb{Z}_{2}$

Thanks
• Mar 19th 2008, 08:03 AM
ThePerfectHacker
Note, $\mathbb{Z}_4 = \{ [0],[1],[2],[3] \} = \left< [1]\right>$. Any homomorphism $\phi: \mathbb{Z}_4 \mapsto \mathbb{Z}_4$ is completely determined by $[1]$ (since it is the generator). Thus, there are four possibilities, $\phi([1]) = [0],\phi([1]) = [1], \phi([1]) = [2],\phi([1])= [3]$. Which of these give rise to automorphisms?