Does every index $ p$ subgroup of $ SL(2,\mathbb{Z}_p)$ contain the principal congruence subgroup $ \Gamma(p)$ ?

Equivalently, must it be the preimage of an index $ p$ subgroup of $ SL(2,\mathbb{Z}/p\mathbb{Z})$ ?

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# Tag: $SL2\mathbb{Z}_p$

## Does every index $p$ subgroup of $SL(2,\mathbb{Z}_p)$ contain $\Gamma(p)$?

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Does every index $ p$ subgroup of $ SL(2,\mathbb{Z}_p)$ contain the principal congruence subgroup $ \Gamma(p)$ ?

Equivalently, must it be the preimage of an index $ p$ subgroup of $ SL(2,\mathbb{Z}/p\mathbb{Z})$ ?

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