1. ## Non-torsion elements

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?

2. Is the identity a nontorsion element?

3. siclar has already answered the question. a less trivial question is to find examples of two torsion elements $a,b$ of a group such that $ab$ becomes non-torsion.

4. Originally Posted by NonCommAlg
siclar has already answered the question. a less trivial question is to find examples of two torsion elements $a,b$ of a group such that $ab$ becomes non-torsion.
$A= \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)$ and $B= \left( \begin{array}{cc} 0 & 1 \\ -1 & -1 \end{array} \right)$ then $A^4 = I = B ^3$ but $AB$ has infinite order.

5. Originally Posted by siclar
Is the identity a nontorsion element?

Sorry - do the non-torsion elements along with the identity form a subgroup?

6. Originally Posted by Swlabr
Sorry - do the non-torsion elements along with the identity form a subgroup?
still no! for example in the group of $2 \times 2$ invertible matrices over $\mathbb{Q}$ let $a=\begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix}$ and $b=\begin{pmatrix}1 & 0 \\ -2 & 1 \end{pmatrix}.$ then both $a,b$ are non-torsion but $ab=\begin{pmatrix}-1 & 1 \\ -2 & 1 \end{pmatrix}$ is torsion because $(ab)^4=I.$