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This is a question from Lang's Algebra Ch 13 number 22 that I cannot solve.

Suppose $p$ is prime and $\ge 5$. Let $G$ be a subgroup of $SL(2,\mathbb{Z}/p^{n}\mathbb{Z})$. Suppose the image of $G$ under the reduction map mod $p$ is $SL(2,\mathbb{Z}/p\mathbb{Z})$. Show $G$ equals $SL(2,\mathbb{Z}/p^{n}\mathbb{Z})$.

I considered looking at the kernel of the map but did not see anything promising. I was hoping that the kernel would be a simple group.

Mykie
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  • You can find the solution in 'Lang, Elliptic Function, Chapter 17 §4' (as suggested at the end of the exercise). I can write down a proof if you don't have access. – user10676 Aug 30 '13 at 16:35

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