Torsion in \(\mathrm{PSL}(2,I_2)\) (Q1239267)

It is well known that the free product of a group of order 3 with a group of order 2 is isomorphic to \(\mathrm{PSL}(2,\mathbb{Z})\). In particular every element of finite order in \(\mathrm{PSL}(2,\mathbb{Z})\) is a conjugate of \(x,y\), or \(y^{-1}\) where \(x= \left(\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix}\right)\) and \(y= \left(\begin{matrix} 0 & 1 \\ -1 & 1 \end{matrix}\right)\). Denote by \(I_{2}\) the 2-adic integers. The author generalizes the above result to: Theorem. Every element of order 3 in \(\mathrm{PSL}(2,I_{2})\) is a conjugate of \(y\) or \(y^{-1}\).