  In the present paper we show that it is possible to obtain the well known
Pauli group $P=\langle X,Y,Z \ | \ X^2=Y^2=Z^2=1, (YZ)^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 $\mathbb{Z}(4)$ 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.
