For all $n > 0$ express $D_{2n}$ as a semidirect product $\mathbb Z_n \rtimes_\theta \mathbb Z_2$, finding $\theta$ explicitly.

Question

For all $n > 0$ express $D_{2n}$ as a semidirect product $\mathbb Z_n \rtimes_\theta \mathbb Z_2$, finding $\theta$ explicitly.

I am not sure how to go about finding $\theta: \mathbb Z_2 \to \mathrm{Aut}(\mathbb Z_n)$ explicitly.

The map is determined by where $1 \in \mathbb Z_2$ maps to. Based on playing with $\mathbb Z_2 \to \mathrm{Aut}(\mathbb Z_4)$, I predict $\theta$ will map to the automorphism sending $1 \mapsto -1$ in $\mathbb Z_n$.

However, I don't want to just guess where to send $1$ to.

How would I begin to systematically find $\theta$ explicitly?

Answer 1

There is one very simple automorphism of $\Bbb Z_n$ (for $n>2$) which has order 2 and is described basically the same way no matter what $n$ is. That's what $\theta(1)$ is.

And guessing isn't that bad. The real work lies in showing that $\theta$ works anyways, not in finding it.