= Solution
The $n$-qubit <Pauli group> is
$$
G_n=\{i^rP_1\otimes\cdots\otimes P_n:r\in\{0,1,2,3\},\ P_j\in\{I,X,Y,Z\}\}.
$$
A <stabilizer group> $S$ is an abelian subgroup of $G_n$ excluding $-I$. Its associated <stabilizer code> is the common positive <eigenspace>
$$
\boxed{\mathcal X_S=\{|\psi\rangle:s|\psi\rangle=|\psi\rangle\text{ for every }s\in S\}.}
$$
The exclusion of $-I$ is necessary for a nonzero code, since no nonzero vector has <eigenvalue> $+1$ under $-I$.
Every $s\in S$ is a signed Hermitian <tensor product> of <Pauli matrices>. An imaginary-phase Pauli element would have $s^2=-I$, contradicting closure of $S$ and exclusion of $-I$; hence $s^2=I$ and $s^\dagger=s$. If $s\ne I$, it cannot be a nontrivial scalar Pauli element, and at least one of its local factors is $X,Y$ or $Z$. Each of these has trace zero. Using $\operatorname{Tr}(A\otimes B)=\operatorname{Tr}(A)\operatorname{Tr}(B)$ gives $\operatorname{Tr}s=0$.
The <spectral theorem> for Hermitian operators and $s^2=I$ show that all <eigenvalues> are $+1$ or $-1$. If their multiplicities are $m_+$ and $m_-$, then
$$
m_++m_-=2^n,\qquad m_+-m_-=\operatorname{Tr}s=0.
$$
Thus the <balanced spectrum of a nonidentity stabilizer> is
$$
\boxed{m_+=m_-=2^{n-1}.}
$$
This concerns a nonidentity stabilizer acting on the entire physical space; on the code every stabilizer acts with <eigenvalue> $+1$.
Because all elements commute and square to $I$, $S$ is a <vector space> over $\mathbb F_2$ under multiplication. Its independent <stabilizer generators> $g_1,\ldots,g_r$ satisfy that no nonempty product is $I$; all $2^r$ products are distinct. The <stabilizer-projector formula> is
$$
P_S=\frac1{|S|}\sum_{s\in S}s=\prod_{j=1}^r\frac{I+g_j}{2}.
$$
To verify it, group multiplication gives $P_S^2=P_S$, Hermiticity gives $P_S^\dagger=P_S$, and $sP_S=P_S$ for every $s\in S$. Conversely every code vector is fixed by this average, so its range is exactly $\mathcal X_S$. All nonidentity stabilizers have trace zero by the preceding proof. Therefore
$$
\dim\mathcal X_S=\operatorname{Tr}P_S=\frac{2^n}{|S|}=2^{n-r}.
$$
For a code of dimension $2^k$, the number of independent generators is consequently $\boxed{r=n-k}$. One can list additional redundant generators, but an independent generating set has exactly this size.
For a correctable Pauli error $E_\alpha$, define the syndrome bit $s_j(\alpha)$ to be zero when $E_\alpha$ commutes with $g_j$ and one when it anticommutes. On an encoded state,
$$
g_jE_\alpha|\psi\rangle=(-1)^{s_j(\alpha)}E_\alpha|\psi\rangle.
$$
Measure the commuting <stabilizer generators>; their <eigenvalues> reveal the <error syndrome> without revealing logical amplitudes. In a <nondegenerate quantum code>, the distinct specified correctable errors have orthogonal images of the code and distinct syndromes. A syndrome therefore identifies the error representative, and applying the inverse <Pauli operator> $E_\alpha^\dagger$ restores the encoded state.
If an actual error operator is a coherent linear combination of the specified errors, its state is a sum of these orthogonal syndrome components. Syndrome measurement selects one component, followed by the same inverse operation. The <linearity of coherent quantum error correction> ensures the logical state is recovered in every outcome. The same procedure handles noise <Kraus operators> in that error span; it does not require a probabilistic Pauli-noise assumption.
A Pauli error commuting with every stabilizer has the all-zero syndrome and preserves the code space. If it is a stabilizer up to global phase, this silent action is harmless. If it lies in the <centralizer of a stabilizer group> but is not a stabilizer up to phase, it acts as a nontrivial logical operator, and syndrome measurements cannot detect it. More generally, a non-scalar component $P_SEP_S$ of an error is an undetectable logical action. This distinguishes harmful undetected errors from scalar actions that satisfy <quantum error detection> algebraically.
The three-qubit <bit-flip repetition code> has basis $|000\rangle,|111\rangle$ and generators
$$
\boxed{g_1=Z_1Z_2,\qquad g_2=Z_2Z_3.}
$$
A single phase flip $Z_1$ commutes with both and has zero syndrome, but
$$
Z_1(\alpha|000\rangle+\beta|111\rangle)
=\alpha|000\rangle-\beta|111\rangle.
$$
It is a nontrivial logical phase operation. In particular, $P_SZ_1P_S$ is not scalar, so the error set $\{I,Z_1\}$ violates the <Knill--Laflamme condition> and cannot be corrected. Thus \b[the code corrects single bit flips but not single phase flips]; its distance against arbitrary quantum errors is one.
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