I am currently studying torsion groups and I am playing around with defintions to get used to them. An element $g \in G$ is a torsion element, if there exists $n \in \mathbb{N}$ so that $g^n = e$, where $e$ denotes the identity element of the group.
I was looking at a group $G$ and its set of torsion elements $G_t$. For a non-abelian group, I was able find an example where the set of torsion elements is no subgroup of the abelian group.
Now I am trying to find examples the other way round. Assume $G$ is nonabelian and I want the torsion elements set to be a (proper) subgroup of $G$. All I could come up with were trivial subgroups like the quaternions. Do you have any examples where the torsion elements form a nontrivial subgroup?
The set of torsion elements of an abelian group is certainly a subgroup.
For the example you're looking for in non abelian groups, consider a free group $F$ on two elements, which has no nontrivial torsion elements; then consider any finite nontrivial group $G$; then $F\times G$ will give you the example.