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?

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- Jun 8th 2009, 07:03 AMSwlabrNon-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?

- Jun 8th 2009, 02:51 PMsiclar
Is the identity a nontorsion element?

- Jun 8th 2009, 03:28 PMNonCommAlg
**siclar**has already answered the question. a less trivial question is to find examples of two torsion elements $\displaystyle a,b$ of a group such that $\displaystyle ab$ becomes non-torsion. - Jun 8th 2009, 06:20 PMJose27
- Jun 8th 2009, 10:37 PMSwlabr
- Jun 8th 2009, 11:44 PMNonCommAlg
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 $\displaystyle (ab)^4=I.$