Topological decompositions of the Pauli group and their influence on dynamical systems

In the present paper we show that it is possible to obtain the well known Pauli group $P = \langle X,Y,Z \mid X^2 = Y^2 = Z^2 = 1, {(Y Z)}^4 = {(ZX)}^4 = {(XY )}^4 = 1\rangle$ of order 16 as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere $S^3$. The first of these spaces of orbits is realized via an action of the quaternion group $Q_8$ on $S^3$; the second one via an action of the cyclic group of order four on $S^3$. We deduce a result of decomposition of $P$ of topological nature and then we find, in connection with the theory of pseudo-fermions, a possible physical interpretation of this decomposition.