(Solved):   Consider the following three elements of \( \mathrm{SL}(2, \mathbf{Z}) \) : \[ A=\left(\be ...

 

Consider the following three elements of \( \mathrm{SL}(2, \mathbf{Z}) \) : \[ A=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right) \quad B=\left(\begin{array}{cc} 0 & -1 \\ 1 & 1 \end{array}\right) \quad C=\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right) \] For each element determine the order and prove your answer. (An exhaustive calculation is a good proof that an element has finite order, but you might need a different proof method to prove something has infinite order.) Part b: Prove or disprove by counter-example: If \( x \) and \( y \) are finite order elements of a group \( G \), then their product \( x y \) is also finite order.