# Math Help - help with automorphisms

1. ## help with automorphisms

I have two problems:

1) Identify Aut(Aut( $\mathbb{Z}_8$))
I know that Aut( $\mathbb{Z}_8$) is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ but what about Aut(Aut( $\mathbb{Z}_8$))?

2) Identify Aut G, where G=({1, 3, 5, 7}, $._8$).
{1, 3, 5, 7} are generators of $\mathbb{Z}_8$ so I'm guessing I should head in that direction, but don't know how.

Please help.

2. Originally Posted by georgel
I have two problems:

1) Identify Aut(Aut( $\mathbb{Z}_8$))
I know that Aut( $\mathbb{Z}_8$) is isomorphic to $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ but what about Aut(Aut( $\mathbb{Z}_8$))?
.
Since $\text{Aut}(\mathbb{Z}_2\times \mathbb{Z}_2) \simeq S_3$ it means $\text{Aut}(\text{Aut}(\mathbb{Z}_8)) \simeq S_3$.

3. Originally Posted by georgel

2) Identify Aut G, where G=({1, 3, 5, 7}, $._8$).

{1, 3, 5, 7} are generators of $\mathbb{Z}_8$ so I'm guessing I should head in that direction, but don't know how.
G is a group of order 4 and every non-identity element of G has order 2. so $G \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2$ and hence $\text{Aut}(G) \cong S_3.$

4. Thank you so much! It all makes sense now.

I'm just starting to learn Algebra so I'll probably have another question or two in the next few days (weeks? ), I hope that's okay.

Thanks again!