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I need some help to show that $\text{SL}_{2}(\mathbb{Z}_{3})$ is a semi-direct product of the quaternion group and the cyclic group $C_{3}$. Any ideas? I don't want to write the elements of $\text{SL}_{2}(\mathbb{Z}_{3})$, it seems like a very ugly method.

thanks

  • 2
    Look at $SL_2(\mathbb{Z}_3)$. Find a copy of $Q_8$, $H$ say, and a copy of $C_3$, $K$ say. Do they intersect trivially? Is one normal? Does $G=KH$?2011-10-10

2 Answers 2