Subgroup of a virtually cyclic group

Question

Let $G$ be a virtually cyclic group, i.e., G has an infinite cyclic subgroup $H$ of finite index. Is it true that if $H'$ is another infinite cyclic subgroup of $G$ then $H'$ must be of finite index and it only has finitely many conjugates?

Answer 1

Firstly, the definition of "virtually cyclic" makes no presumptions about the group being infinite, so any finite group is virtually cyclic. Indeed, we can say that every finite group is virtually trivial (a cutting pun to throw at any finite group theorist!). What you describe is "virtually-$\mathbb{Z}$".

Suppose $\mathbb{Z}\cong K\leq G$ where $G$ is virtually-$\mathbb{Z}$, and write $H$ for the maximal infinite cyclic subgroup of $G$, as you have done. Now, look at $K^{\prime}=K\cap H$. This is non-trivial, because, letting $K=\langle k\rangle$ and $H=\langle h\rangle$, we have that $G/H$ is finite so there exists $i\neq j$ such that $k^iH=k^jH$, so $1\neq k^{i-j}\in H$. The fact that $K^{\prime}$ is non-trivial proves the result. Can you see why?