Isomorphism of dihedral with a semi-direct product

[AllRelative]

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

 

[fresh_42]

I suggest to write it in terms of $r$ and $s$ first, and then see how it translates to $\mathbb{Z}_m$, resp. $\mathbb{Z}_2$. And I will write the representatives, aka elements of $\mathbb{Z}_m$ as $[k] = k+m\mathbb{Z}$ which is more convenient and easier to read.

In general a semidirect product $G = N \rtimes_\varphi H$ goes by $(n_1\, , \,h_1)\cdot (n_2\, , \,h_2) \stackrel{(*)}{=} (n_1 \cdot \varphi(h_1)(n_2)\, , \,h_1\cdot h_2)$, see e.g. https://en.wikipedia.org/wiki/Semidirect_product. We have $\varphi\, : \,H \longrightarrow \operatorname{Aut}(N)$.

Here we have products $P =(r^k\; , \;s^\varepsilon)\cdot (r^n\; , \;s^\eta) = r^k\cdot s^\varepsilon \cdot r^n \cdot s^\eta$ with $\varepsilon \in \{\,0,1\,\}$.
Now write $P$ in the form $P=r^l \cdot s^\mu=(r^l,s^\mu)$ and compare that with $(*)$ to see how $\varphi(s^\varepsilon)(r^n)$ has to be defined.
If you're done, you can write this as $\varphi([\varepsilon])([n])$.

You have only defined $\varphi([1])([1])$. What are $\varphi([0])$ and $\varphi([1])([n])$ if $n>1$ or short: What is $\varphi([\varepsilon])$ with $\varepsilon \in \{\,0,1\,\} = \{\,[0],[1]\,\}\,?$