sphere corresponds to the density matrix $$\frac{1}{2}\left(\begin{array}{cc}1+z & x-iy \\ x+iy & 1-z
\end{array} \right) \in {\mathbb C}^{2\times 2}.$$ For some special states the points on the Bloch sphere are shown in Figure

\[Figur Blochkugel\]. We now extend the Bloch sphere representation for states to a representation of POVM operators of rank one. Note that each pure state is a projection $\rho={|\Psi\rangle}{\langle \Psi|}$ of rank one. By rescaling a POVM operator $A_j =
{|\Psi_j\rangle}{\langle \Psi_j|}$ to a density matrix we can identify $A_j$ with a point on the Bloch sphere using the correspondence for pure states.

A symmetry of a POVM can be defined by a symmetry of the corresponding points on the Bloch sphere. We are interested in POVMs with a finite symmetry group for the points on the Bloch sphere, i.e., we consider finite subgroups of $SO(3)$ [@Sternberg]. There are two infinite families of finite subgroups, namely the cyclic groups $C_m$ and the dihedral groups $D_m$ for $m\geq 2$ as symmetry groups of an $m$-sided regular polygon (with the special case $m=2$). Furthermore, we have the symmetry groups of the five platonic solids (tetrahedron, cube, octa-, dodeca-, and icosahedron). As a restriction for the latter symmetry groups, we 