Aut(Aut $\displaystyle \mathbb{Z}_{8}$) is isomorphic to which group? I came out with $\displaystyle \mathbb{Z}$ but I'm not sure.
Thanks.
Hi,
$\displaystyle Aut(\mathbb{Z}_n)\cong\mathbb{Z}_n^\ast$ where $\displaystyle \mathbb{Z}_n^\ast$ is the multiplicative group of integers relatively prime to n.
So $\displaystyle Aut(\mathbb{Z}_8)\cong\mathbb{Z}_2\oplus\mathbb{Z} _2$
Thus $\displaystyle Aut(Aut(\mathbb{Z}_8))\cong\text{GL}(2,2)$ where $\displaystyle \text{GL}(2,2)$ is the group of non-singular 2 by 2 matrices over the field with 2 elements; $\displaystyle \text{GL}(2,2)$ has order 6. This is definitely not the group of integers.