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Let $G$ be the group $ ( \mathscr{M}_{2\times2}(\mathbb{Q}) , \times ) $ of nonsingular matrices.

Let $ A = \left ( \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right ) $, the order of $A$ is $4$;

Let $ B = \left ( \begin{matrix} 0 & 1 \\ -1 & -1 \end{matrix} \right ) $, the order of $B$ is $3$.

Show that $AB$ has infinite order.


The only reasoning possible here is by contradiction as $G$ is not abelian. And so I tried, but I got stuck before any concrete development.

Any hints are welcome, Thanks.

Answers