  We study the commutation structure within the Pauli groups built on all
decompositions of a given Hilbert space dimension $q$, containing a square,
into its factors. The simplest illustrative examples are the quartit ($q=4$)
and two-qubit ($q=2^2$) systems. It is shown how the sum of divisor function
$\sigma(q)$ and the Dedekind psi function $\psi(q)=q \prod_{p|q} (1+1/p)$ enter
into the theory for counting the number of maximal commuting sets of the qudit
system. In the case of a multiple qudit system (with $q=p^m$ and $p$ a prime),
the arithmetical functions $\sigma(p^{2n-1})$ and $\psi(p^{2n-1})$ count the
cardinality of the symplectic polar space $W_{2n-1}(p)$ that endows the
commutation structure and its punctured counterpart, respectively. Symmetry
properties of the Pauli graphs attached to these structures are investigated in
detail and several illustrative examples are provided.
