While studying group theory, I've got confused by the following statement:
For a molecule to have a permanent dipole moment, it must have an asymmetric charge distribution. The point group of the molecule not only determines whether the molecule may have a dipole moment but also in which direction(s) it may point. If a molecule has a $C_n$ axis with $n > 1,$ it cannot have a dipole moment perpendicular to the axis of rotation (for example, a $C_2$ rotation would interchange the ends of such a dipole moment and reverse the polarity, which is not allowed — rotations with higher values of $n$ would also change the direction in which the dipole points). Any dipole must lie parallel to a $C_n$ axis.
why does the dipole need to be parallel to the $C_n$ axis? If I put a fluorine atom in just one of the light gray atoms of the following molecule, wouldn't it have a molecular dipole that isn't parallel to the main axis?