Isomorphism between $D_{2n}/Z(D_{2n})$ and $D_n$

Question

I need some help with the following task:

Show that for all $n\geq2$ is $D_{2n}/Z(D_{2n})\cong D_n$

$D_n$ is the dihedrial group with $2n$ elements and looks like $D_n=\{id,r,...,r^{n-1},s,sr,...,sr^{n-1}\}$

Sadly I have no idea how $D_{2n}/Z(D_n)$ looks like but I guess I need to know that to show that it is a homomorphism and bijectiv.

Thanks for helping me

Is that $Z(D_n)$ (the center of $D_n$) or $\Bbb Z(D_n)$ (which I don't know how to interpret)? — Ben Grossmann, Nov 12 '20 at 15:31
Also, I suspect that it should be $D_{2n}/Z(D_{\color{red}{2n}})$, is that right? — Ben Grossmann, Nov 12 '20 at 15:35
Im really sorry for that. Normally I double check my posts. Sorry.... I changed it — Walter Frosch, Nov 12 '20 at 15:42
No need to apologize. The fact that you wrote your question clearly and explained your thoughts makes your question much better than the ones that I'm used to seeing, even if there were a few typos — Ben Grossmann, Nov 12 '20 at 15:48
The claim is proved in the duplicate: Claim. If $n\geq 3$ and $n$ is even, then $D_{2n}/Z(D_{2n})\cong D_n$. For $n\ge 3$ odd, $D_{2n}$ has trivial center. — Dietrich Burde, Nov 12 '20 at 15:49

Ben Grossmann · Accepted Answer · 2020-11-12T16:25:21.443

2

Hint: By the first isomorphism theorem, it is equivalent to find a homomorphism $f:D_{2n} \to D_n$ with $\ker(f) = Z(D_{2n})$.

Let $s_n,r_n$ denote the generators of $D_{n}$ and let $s_{2n},r_{2n}$ denote the generators of $D_{2n}$. Define the map $f:D_{2n} \to D_n$ by $$ f(s_{2n}^j r_{2n}^k) = s_{n}^jr_n^k. $$ Verify that $f$ is a homomorphism and that $\ker(f) = Z(D_{2n})$. In verifying that $f$ is a homomorphism, you might find the identity $ sr^ks = r^{-k} $ to be useful.

edited Nov 12 '20 at 16:25

answered Nov 12 '20 at 15:42

Ben Grossmann

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1

Do you mean $f(s_{2n}^j r_{2n}^k) = f(s_{n}^jr_n^k)$ or just $f(s_{2n}^j r_{2n}^k) = s_{n}^jr_n^k$ ? – MAS Nov 12 '20 at 15:52
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@Why The latter; thanks for pointing that out – Ben Grossmann Nov 12 '20 at 16:25

Isomorphism between $D_{2n}/Z(D_{2n})$ and $D_n$

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