Let $P_n$ be the Pauli group on $n$ qubits. Let $H \subseteq P_n$ s.t. $H$ is commuting and consist of independent elements. it is well known that $|H| \leq n$. However I have not been able to find a proper reference of this result explicitly (even though this is folklore). Does anyone know where this result first appeared?
I also have a similar question for pairwise anticommuting subset $G \subseteq P$, i.e. the elements of $G$ can be paired up so that they anticommute with each other but commute with everything else. It is again well-known that $|G| \leq 2n$. Where does this result first appear?
an Abelian subgroup of the Pauli group over $n$ qubits corresponds (modulo phase) with a linear, self-orthogonal (under a symplectic inner product) code in $\mathbb{F}_{2}^{2 n}$. Such codes cannot have dimension larger than $n$. See here for an accessible exposition
As for your second question, let $n = 1$ and consider $\{X, Y, Z\}$. These pairwise anti-commute and violate the bound reported above.