[Math] Infinite Dihedral Group.

Question

Let $D_{\infty}= \langle x,y \mid x^2=y^2=1\rangle$ be the infinite dihedral group. Are the following statements true?

1. Since $G$ is not torsionfree, $\mathbb{Q}[G]$ is not a domain.

2. $D_{\infty}$ is an infinite subgroup of $G= \langle x,y \mid x^2=y^2\rangle$.

3. The infinite dihedral group has a free abelian subgroup $F$ generated by $\langle x y \rangle$ of rank one and index $2$, and thus normal. $F$ is also subgroup of $G$.

Answer 1

1. Yes, it is not a domain.

2. No, it is a factor group.

3. Yes.