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I'm really looking for a "cute" way of showing that $SL_2(\mathbb{F}_5)$ is a double cover of $A_5$. The sort of action I am looking for is something like the action of $GL_2(\mathbb{F}_3)$ on $\mathbb{P}^1(\mathbb{F}_3)$, which shows $GL_2(\mathbb{F}_3)$ is a double cover of $S_4$. Now that's cute.

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    Here's a related thread you may be interested in: http://math.stackexchange.com/q/93762/2012-03-29

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