# Stabilizer codes: why is $N(S)/S$ homomorphic to the Pauli group on $k$ qubits

Given an abelian subgroup $$S$$ of the Pauli group on $$n$$ qubits with $$r$$ independent generators we obtain an $$2^{n-r}$$-dimensional subspace stabilized by $$S$$. By picking a basis for this space, it is easy to build a representation of the Pauli group on $$k$$ qubits generated by some set of matrices $$\overline{X_i}, \overline{Z_i}$$. However, I fail to see how to do so in a way that ensures that $$\overline{X_i}, \overline{Z_i}$$ are Pauli matrices in the original basis. The common way to build these logicals is to consider the quotient $$\mathcal{N}(S)/S$$, and note that it corresponds to the Pauli group on $$k$$ qubits, but I cannot quite prove it.