For each positive integer $n \geq 3$, determine the centre of $D_{2n}$.
Proof. Let $F$ be any flip and $R$ any rotation. Drawing out the effects of each operation, we find that $FR = R^{−1}F$. This is $RF$ if and only if $R= R^{−1}$. Letting $R = R_{360/n}$, we find that $R\neq R^{−1}$. Thus, no flip is central. In fact, $R = R^{−1}$ if and only if $R = R_0$ or $R = R_{180}$. If n is odd, there is no $R_{180}$, so $Z(D_{2n}) = {R_0}$. If $n$ is even, we see that $R_{180}$ commutes with every flip, and surely with every rotation, so $Z(D_{2n}) = \{R_{0}, R_{180}\}$.
About this proof I have a couple of doubts.
First:
Drawing out the effects of each operation, we find that $FR = R^{−1}F$. This is $RF$ if and only if $R= R^{−1}$
How can I determine that this is indeed generally true?
Second:
$R = R^{−1}$ if and only if $R = R_0$ or $R = R_{180}$
This is clear, but how could it be proven?
It must be said that it is not a very advanced material that I am studying. I hardly go by subgroups. Thanks in advance
For the first one: $$ R=R^{-1}\Longrightarrow R\color{blue}F=R^{-1}\color{blue}F=FR \\ RF=R^{-1}F\Longrightarrow RF\color{blue}{F^{-1}}=R^{-1}F\color{blue}{F^{-1}}=R\left( FF^{-1} \right) =R^{-1}\left( FF^{-1} \right) \Longrightarrow R=R^{-1} $$ For the second one: Assume $R_x$ satisfies $R_x=R^{-1}_x$, then $R_x=R_{(360-x)\text{ mod } 360}$, then it is obvious that $x=0$ or $x=180$ (Note $0\leq x<360$).