Decomposition of Pauli groups via weak central products.

For any $n,m \ge 1$ and prime power $\mathtt{p}^m$, we show a result of decomposition for Pauli groups $\mathcal{P}_{n,\mathtt{p}^m}$ in terms of weak central products. This can be used to describe the underlying structure of Pauli groups on $n$ qudits of dimension $\mathtt{q} = \mathtt{p}^m$ and enables us to identify abelian subgroups of $\mathcal{P}_{n,\mathtt{p}^m}$. 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.