The abstract study of symmetry is the object of mathematical group theory, which, as its name implies, describes groups defined by the specific symmetry elements they contain and the specific symmetry operations they allow. Groups are also related to each other by symmetry transformations. Group theory provides physics with a powerful means for classifying "everything" in the universe and understanding the interdependence of physical processes based on symmetry transformations.

For example, many transformations in the physical world leave the laws of physics invariant, i.e. the laws of physics are the same everywhere in the universe. This is referred to as translational invariance, and it corresponds to the law of conservation of linear momentum; similarly, the symmetry that states the laws of physics are the same at all times is equivalent to the law of conservation of energy; and the rotational invariance...