$T(G)$ may not be a subgroup? It is obvious that for an abelian group $G$; the set of all torsion elements: $$T(G)=\{x\in G|x^n=0 \text{, for some nonzero integer } n\}$$ is a subgroup of the group. I am asked to probe this fact when $G$ is not abelian. Thanks.

In $G=SL_2(\mathbb Z)$ observe that
$$\left(\begin{matrix}0&1\\-1&0\end{matrix}\right)\cdot\left(\begin{matrix}0&-1\\1&1\end{matrix}\right)=\left(\begin{matrix}1&1\\0&1\end{matrix}\right) $$
where the factors on the left ae in $T(G)$ and their product is not.
