Order of Automorphism Group of Dihedral Group

Theorem

Let $D_n$ denote the dihedral group of order $n$.

Let $\Aut {D_n}$ denote the automorphism group of $D_n$.

Then:

$\order {\Aut {D_n} } = n \cdot \map \phi n$

where:

$\order {\, \cdot \,}$ denotes the order of a group

$\map \phi n$ is the Euler $\phi$ function.

Proof

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Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64 \delta$