# Find the number of Sylow 2-subgroups of the special linear group of order 2 on 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.

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.
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Kaya Garza
Hints: $G=SL\left(2,3\right)$ has 24 elements, hence ${n}_{2}=\mathrm{#}{\text{Syl}}_{2}\left(G\right)=1$ 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.
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