Commutators in the infinite dihedral group

Question

I started to learn about the infinite dihedral group $D_\infty=\langle x,y\mid x^2=y^2=1\rangle.$ Can you tell me some references about this group? I need to know about it as specifically as possible. Now, I'm wondering a few things as follows.

1. $xy$ can be written a product commutator, can't you? Moreover, its commutator can be chosen by elements, which have order $2.$ I hope that $xy$ can be a commutator of elements $c,d$, and $c,d$ have order $2.$
2. How many are elements of order $2$ in $D_\infty$?

Thanks for all your support.

Answer 1

Cosider the group $(G, \circ)$ where:

• $G$ is the set of all functions $f : \mathbb{Z} \to \mathbb{Z}$ of the form $f(k) = a \pm k$, $a \in \mathbb{Z}$
• $\circ$ means function composition.

Then $D_{\infty} \cong G$ and via that isomorphism $x$ maps to $x(k) = -k$ and $y$ maps to $y(k) = 1-k$. Since $G$ is a concrete group, it should be easy to answer these questions now.