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Nov 18, 2023 at 19:27 comment added stupid boy When $p=2$, the group ${\rm SL}_2(\mathbb{Z})$ (and hence ${\rm SL}_2(\mathbb{Z}_2)$ ) has an element of order $4$ (resp. $6$). In particular, your claim is not true for $p=2$.
Jun 1, 2023 at 4:12 history answered Kenta Suzuki CC BY-SA 4.0