# Math Help - Automorphism proof

1. ## Automorphism proof

Prove that $Aut (Z_{2}XZ_{2}) \cong S_{3}$.

2. Originally Posted by pirouette
Prove that $Aut (Z_{2}XZ_{2}) \cong S_{3}$.
Let $\phi: \mathbb{Z}_2\times \mathbb{Z}_2\to \mathbb{Z}_2\times \mathbb{Z}_2$ be an automorphism.
Obviously, $\phi(0,0) = (0,0)$.

1)Now, $\phi(0,1)$ has to be an element of order $2$ and so $\phi(0,1) = (0,1) \text{ or }(1,0)\text{ or }(1,1)$.

2)The same possibilities for $\phi(0,1)$ however except for what was used in #1.

3)Once #1 and #2 are determined then $\phi(1,1) = \phi(1,0) + \phi(0,1)$ and so $\phi(1,1)$ is determined.

There are $3$ possibilities for #2 and $2$ possibilities #3, we therefore have at most $2\cdot 3 = 6$ automorphisms.

Check that all these six instead give a raise to a hextic automorphism group.