Let $G$ be a group, $a \in G$, $|a|$ is infinite. If $|\langle a \rangle ^G : \langle a \rangle |$ is finite ($>1$). I want know whether or not that $|x|$ is infinite for every non-trivial element in $\langle a \rangle^G$?


We hope to investigate the influence of normal closure, so any reference about this situation will be appociated.


Added: By the answer of Kevin, I see that there exists example that $\langle a \rangle ^G$ is not torsion-free. Hence the case that $\langle a \rangle ^G$ is torsion-free is more atractive to me. Thanks the answer of Richard, we know it will be cyclic in this case. But I can't find a non-trivial example. Anyone can provide an example?



About normal closure of cyclic subgroup
No. Consider the semi-direct product $(\mathbf{Z}\times \mathbf{Z}/2)\rtimes \mathbf{Z}/2$, where the rightmost factor acts by $(1,0)\mapsto (1,1)$ and fixes $(0,1)$. Then the normal closure of $\mathbf{Z}\times 0\times0$ is $\mathbf{Z}\times\mathbf{Z}/2\times 0$.


