# Every torsion-free group is an abelian group?

Is it true that every torsion-free group (i.e. a group where the only element of finite order is the identity) is abelian?

It is torsion-free because $$\begin{pmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{pmatrix}^n = \begin{pmatrix} 1 & na & {n \choose 2}ab+nc\\ 0 & 1 & nb\\ 0 & 0 & 1\\ \end{pmatrix}$$
It is not abelian because $(AB)_{13} \ne (BA)_{13}$.