Can a measurement commuting with the stabilizer of a state disturb the state? In Nielsen and Chuang's "Quantum Computation and Quantum Information", Section 10.5.3, the authors claim the following:


  With a system in state $|\psi\rangle$ with stabilizer $g_1,...,g_n$, if a measurement $g$ that is commute with all $g_i$ is applied to the state, then either $g$ or $-g$ is a stabilizer of the system and the measurement doesn't disturb the state.


However, it seems to me that there is a simple counterexample: 


  Let $g$ be the $Z_1$ operator in the system, thus it commutes with all stabilizers as $g|\psi\rangle$ is still a valid state in the system. However neither $g$ nor $-g$ is a stabilizer of the code.


Am I missing anything or the quoted claim is indeed false?
The quoted claim from the book is true. It is not true that $Z_1$ must always commute with all stabilizers of $\lvert \psi \rangle$. For example, if $X_1X_2$ stabilizes $\lvert \psi \rangle$ (e.g., if $\lvert \psi \rangle$ was an EPR pair) then $Z_1$ does not commute with $X_1X_2$ and $Z_1$ is not a stabilizer of $\lvert \psi \rangle$.