Find the number of Sylow $2$-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo $3$). I think it will be $1$. But I failed to prove it using the counting principle. It has $4$ sylow $3$-subgroups.
2026-03-26 23:11:08.1774566668
Number of Sylow 2-subgroups of a special linear group
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Hints: $G=SL(2,3)$ has 24 elements, hence $n_2=\#\text{Syl}_2(G) = 1$ or $=3$. If $n_2=1$, then a Sylow 2-subgroup must be normal, which is the case indeed. Show that the Sylow $2$-subgroup is isomorphic to the quaternion group $Q$ of order 8. Write down the matrices.