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Let $q\geq 3$ be a prime number and $D_q$ be the Dihedral Group of order $2q$. Find all automorphism of $D_q$ of order $2$.

I tried this using a 'generic' automorphism $\varphi$ such that $\varphi(r)=sr^k$ and $\varphi(s)=sr^j$ for some $k$ and $j$, but I don't know what else I should do.

Yaosheng Deng
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Thomas García Villar
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  • An automorphism preserves the order of an element, so a rotation of order $q$ should map onto another... – Jyrki Lahtonen Jun 19 '24 at 03:22
  • $D_q$ is centerless, hence equal to its inner automorphism group. There's $q+1$ elements of order $2.$ So are their associated inner automorphisms. Usually $D_n $ has some outer automorphisms. It remains to see if there are any of order $2.$ – suckling pig Jun 19 '24 at 04:09
  • Sorry, there's $q$ inner automorphisms of order $2.$ Since $q$ is odd, none of the rotations are order $2.$ – suckling pig Jun 19 '24 at 04:27
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    You have been given a number of hints now. What developments did they give you? – Jyrki Lahtonen Jun 19 '24 at 05:25

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Hint: You can also use, that the automorphism group of $D_n$ for $n\ge 3$ is isomorphic to $$ {\rm Aut}(D_n)\cong C_n\rtimes U(n), $$ of order $n\phi(n)$, where $U(n)=(\Bbb Z/n\Bbb Z)^{\times}$ is the group of units of the ring $\Bbb Z/n\Bbb Z$. This has been shown in the reference below:

Prove that $| \operatorname{Aut}(D_n)|= n\phi(n)$.

For $n=q>2$ prime, $U(q)$ is cyclic of order $\phi(q)=q-1$. Hence we have $$ {\rm Aut}(D_q)\cong C_q\rtimes C_{q-1}. $$

Dietrich Burde
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