Let $\widetilde{\mathcal{P}}_n = \langle X_1,X_2,\dots,X_n,Z_1,\dots,Z_n\rangle$ together with all the phases $\{\pm 1, \pm i\}$ the regular Pauli group, and $N = \langle \pm i I\rangle $. I would like to understand better the group theoretic construction:
$$\mathcal{P}_n := \widetilde{\mathcal{P}}_n / N$$
In this work, arXiv, the authors mention that this group corresponds to the Pauli operators with only $+1$ phase. However, in group theoretic terms, this construction generates sets as elemens instead of Pauli operators. In effect, the elements of $G/N$ in group theoretic 'jargon' should be left/right cosets.
However, the authors proceed to calculate, for instance $$\sum_{P \in \mathcal{P}_n} \langle \psi \vert P \vert \psi \rangle $$ for any $\vert \psi \rangle$. How can I understand the construction of the set $\mathcal{P}_n$, if possible, in general and not only for this specific case?