Automorphisms of Klein group

Question

I have a problem I can't solve I need to show all the automorphism in Klein group and their order and after that, I need to find that what is the group that these automorphisms are isomorph with.

I have half of it I have the automorphisms in the group I used representatives like $Id=e$ $(12)(34)=x$ $\;\;(13)(24)=y$ $\;\;(14)(23)=z$ and I found $6$ automorphism

1. $xyz\rightarrow xyz$

2. $xyz\rightarrow xzy$

3. $xyz\rightarrow yxz$

4. $xyz\rightarrow yzx$

5. $xyz\rightarrow zxy$

6. $xyz\rightarrow zyx$

And my question is about the second part. How do I get the order each of these automorphisms and with what group this automorphismgroup isomorph with?

Thank you in advance.

Answer 1

Here may be a useful way of thinking about it. Note any automorphism will send $e \to e$. What about the three other elements? Well, as I think you have shown above, we can map $x$ to any one of the elements, i.e. we are just shuffling around the $x,y,z$ any way that we can. Hence what you found is that the automorphism group is permuting $3$ things. What well known group has order $6$ and permutes three things? This will also tell you the order of all automorphisms.

Answer 2

Hint: let $V=\{1,a_1(=a_2a_3),a_2(=a_1a_3),a_3(=a_1a_2)\}$ be Klein's $4$-group. Prove that the map $\varphi\colon S_3\to\operatorname{Aut}(V)$, defined by:

\begin{alignat}{1} &\varphi(\sigma)(1)=1 \\ &\varphi(\sigma)(a_i)=a_{\sigma(i)} \end{alignat}

is an isomorphism (good-definiteness, injectivity, surjectivity, operation-preservation [homomorphism]).