The symmetry of ammonia can be understood in terms of the molecular orbitals of the molecule. The nitrogen atom has five valence electrons, and these electrons occupy three sp3 hybrid orbitals. The three sp3 hybrid orbitals are arranged in a tetrahedral shape, and the three hydrogen atoms are bonded to the nitrogen atom through these orbitals.
The fourth valence electron of the nitrogen atom occupies a p orbital. This p orbital is perpendicular to the plane of the three sp3 hybrid orbitals, and it is responsible for the lone pair of electrons on the nitrogen atom.
The symmetry of ammonia can also be understood in terms of the molecular point group. The point group of ammonia is C3v, which means that the molecule has a three-fold axis of rotation, three vertical mirror planes, and one horizontal mirror plane.
The symmetry of ammonia has a number of important consequences. For example, the symmetry of the molecule means that the three hydrogen atoms are all equivalent, and they all have the same bond length and bond angle. The symmetry of the molecule also means that the molecule has a dipole moment, which is a measure of the polarity of the molecule.