- #1

- 42

- 2

## Homework Statement

Let m ≥ 3. Show that $$D_m \cong \mathbb{Z}_m \rtimes_{\varphi} \mathbb{Z}_2 $$

where $$\varphi_{(1+2\mathbb{Z})}(1+m\mathbb{Z}) = (m-1+m\mathbb{Z})$$

## Homework Equations

I have seen most basic concepts of groups except group actions. Si ideally I should not use them for this problem.

## The Attempt at a Solution

So I've been thinking about this problem for a couple of days and I just can't seem to arrive to the proof I am looking for.

And so we have that <s> is cyclical of order 2 and <r> is cyclical of order m. Therefore, $$\langle s \rangle \cong \mathbb{Z}_2$$

and $$\langle r \rangle \cong \mathbb{Z}_m$$

I feel I have to use the fact that <r> is normal in D

_{m}and that <r>∩<s> = {e}. I unsure where to go from there

Thanks for the help