Do these form a subgroup? That is to say, as the inverse of a non-torsion element (an element of non-finite order) is non-torsion, basically the question is: If we multiply two non-torsion elements together do we get another non-torsion element?
Do these form a subgroup? That is to say, as the inverse of a non-torsion element (an element of non-finite order) is non-torsion, basically the question is: If we multiply two non-torsion elements together do we get another non-torsion element?
still no! for example in the group of $\displaystyle 2 \times 2$ invertible matrices over $\displaystyle \mathbb{Q}$ let $\displaystyle a=\begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix}$ and $\displaystyle b=\begin{pmatrix}1 & 0 \\ -2 & 1 \end{pmatrix}.$ then both $\displaystyle a,b$ are non-torsion but $\displaystyle ab=\begin{pmatrix}-1 & 1 \\ -2 & 1 \end{pmatrix}$ is torsion because $\displaystyle (ab)^4=I.$