4
$\begingroup$

I was thinking about the question Is a HNN extension of a virtually torsion-free group virtually torsion-free? and I started wondering whether or not $G = \mathbb{Z}\times C_2$ was a counterexample to the question. In the process, I came across the following question:

Is $\mathbb{Z}* C_2=\left< t,x|x^2=1 \right>$ virtually torsion-free?

A group is said to be virtually torsion-free if it has a finite index subgroup which does not contain elements of finite order.

I suspect that the answer is no because words of the form $t^nxt^{-n}$ all have order two within $\mathbb{Z}*C_2$. If $\mathbb{Z}*C_2$ has a finite index torsion-free subgroup $T$ then the cosets $t^nxt^{-n}T$ cannot all be distinct.

$\endgroup$
1
  • $\begingroup$ Yes. Indeed, every group with a single defining relation is virtually torsion free. $\endgroup$
    – user1729
    May 2, 2019 at 20:32

1 Answer 1

4
$\begingroup$

Yes, the subgroup $N := \langle t,t^x \rangle$ is free and has index $2$ in $G$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .