Decomposition of Pauli groups via weak central products

For any $m\ge 1$ and odd prime power $\mathtt{q}={\mathtt{p}}^m$, for $\mathtt{q}=2$, and for any $n\ge 1$, we show a result of decomposition for Pauli groups ${\mathcal{P}}_{n,\mathtt{q}}$ in terms of weak central products. This can be used to describe the underlying structure of Pauli groups on $n$ qudits of dimension $\mathtt{q}$ and enables us to identify abelian subgroups of ${\mathcal{P}}_{n,\mathtt{q}}$. As a consequence of our main results, we show a similar factorisation for the so–called `lifted’ Pauli groups, recently introduced by Gottesman and Kuperberg in the context of error-correcting codes in quantum information theory.