Let $G$ be a group. The automorphism group $\mathrm{Aut}(G)$ acts on $G$ by $\psi\cdot g=\psi(g)$ for all $\psi\in\mathrm{Aut}(G)$ and all $g\in G$. Find the orbits of this action for $G=\mathbb Z_{12}$, $\mathbb Z_7$ and $D_4$.

In cases for first two, since $\mathrm{Aut}(Z_n)\cong U(n)$, by computation, the followings are orbits:

For $\mathbb Z_{12}$,

$$\mathrm{Aut}(G)\cdot0 =\{0\}$$

$$\mathrm{Aut}(G)\cdot1 =\{1,5,7,11\}$$

$$\mathrm{Aut}(G)\cdot2 =\{2,10\}$$

$$\mathrm{Aut}(G)\cdot3 =\{3,9\}$$

$$\mathrm{Aut}(G)\cdot4 =\{4,6,8\}$$

$$\mathrm{Aut}(G)\cdot6 =\{6\}$$

$$\mathrm{Aut}(G)\cdot8 =\{4,8\}$$

For $\mathbb Z_7$,

$$\mathrm{Aut}(G)\cdot0 =\{0\}$$ $$\mathrm{Aut}(G)\cdot1 =\{1,2,3,4,5,6\}$$

But, in case of $D_4$, dihedral group of order 8, I don't know what the automorphism group is. And this is an exercise for groups acting on themselves such as left multiplication and conjugation. How could I use it to solve this problem?

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    $\begingroup$ To the last question on $Aut(D_4)$ - why don't you try to google this site? See this MSE-question. $\endgroup$ – Dietrich Burde Nov 6 '16 at 16:43
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    $\begingroup$ An automorphism is completely known if one knows its action on generators. Once this established one can follow the action of the automorphism on words formed from the generators. $\endgroup$ – Marc Bogaerts Nov 7 '16 at 7:19

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