The trivial stabilizer code is defined by $$T=\{|0\rangle^{\otimes(n-k)}\otimes|\Psi\rangle:|\Psi\rangle\in(\mathbb{C}^{2})^{k}\}\tag{1}$$ which is stabilized by the Pauli operators $Z_1, ...., Z_{n-k}$.
How can we prove Claim 2 on page 2 of this note?
It seems we need to apply the discussion on page 4, but how? In other words, if $g_1,...,g_{n-k}$ are the stabilizer generators of code $S$, we need to find a unitary operator $u$ such that $ug_iu^{\dagger}=Z_i$ for all $i$.
