Math Help - 2.2.7, need help

1. 2.2.7, need help

this problem is on dummit, "abstract Algebra",
it says
Let $n\in N$, with $n\geq3$. Prove the following:
(a). Z( $D_{2n})=1$ if n is odd.

(b). Z( $D_{2n})={ 1,r^{k} }$ if n=2k.

2. Originally Posted by mancillaj3
this problem is on dummit, "abstract Algebra",
it says
Let $n\in N$, with $n\geq3$. Prove the following:
(a). Z( $D_{2n})=1$ if n is odd.

(b). Z( $D_{2n})={ 1,r^{k} }$ if n=2k.
According to my word processor I've spent the last 52 minutes trying to solve this by induction. Which is silly.

I gave a short bit of thought before this, wondering if there was an easier, more subtle proof. I have just realised that there is. So here is a definition and a hint:

We can define the group thus: $D_{2n} = <\alpha, \beta | \alpha^n=\beta=1, \beta \alpha \beta = \alpha^{-1}>$.

Now, when does the equality $a \equiv -a \text{ mod } n$ hold, and why is this relevant here?