While I was reading a proof, there's a line that I couldn't quite understand. Let $\Gamma$ be a lattice and $\Gamma'$ a sublattice of $\Gamma$ of index $p^{n+1}$. The statement i don't understand is the following:
If $\Gamma'$ is not contained in $p\Gamma$ then the image of $\Gamma'$ in $\Gamma/p\Gamma$ is of order $p$.
I can kind of see why this is true but I don't know how to prove it rigorously. Can anyone help me out here?