How can I prove that $$ \operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n), $$ where $\mathbb {D}_n$ is the dihedral group.
Can someone help me please? Thank you.
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Do you know how many automorphisms a dihedral group has? That might be a good start. Remember that isomorphisms must send generating sets to generating sets (To think about automorphisms, not the overall statement). – pjs36 Dec 02 '15 at 22:42
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@pjs36 I can propose that $n\phi(n)$ – Belogurow Dec 02 '15 at 22:53
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This is Theorem $7.2$ here, where it is shown that the short exact sequence $$ 1\rightarrow \mathbb{Z}_n\rightarrow \operatorname{Aut}(D_n)\rightarrow \operatorname{Aut}(\mathbb{Z}_n)\rightarrow 1 $$ splits, which just says that the middle group is a semidirect product of the outer ones.
tomasz
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Dietrich Burde
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