The symmetries of a physical system manifest themselves through its conservation laws; they are encoded in the Hamiltonian, the generator of time translations. In Quantum Mechanics, the hermitian operators which commute with the Hamiltonian generate the symmetries of the system.

Similar considerations apply to local Quantum Field Theory. There, the main ingredient is the Dirac-Feynman path integral taken over the exponential of the action functional. In local relativistic field theory, the action is the space-time integral of the Lagrange density, itself a function of fields, local in space-time, which represent the basic excitations of the system. In theories of fundamental interactions, they correspond to the elementary particles. Through E. Noether's theorem, the conservation laws are encoded in the symmetries of the action (or Lagrangian, assuming proper boundary conditions at infinity).

Physicists have identified four different forces in Nature. The force of gravity, the electromagnetic, weak, and strong forces. All four are described by actions which display stunningly similar mathematical structures, so much so that the weak and electromagnetic forces have been experimentally shown to stem from the same theory. Speculations of further syntheses abound, unifying all three forces except gravity into a Grand Unified Theory, or even including gravity in Superstring or M Theories!

In his remarkable 1939 James Scott lecture, Dirac speaks of the mathematical quality of Nature and even advocates a principle of mathematical beauty in seeking physical theories!

Earlier, we saw that finite groups can be taken apart through the composition series, in terms of simple finite groups, that is groups without normal subgroups. Remarkably the infinitude of simple groups is amenable to a complete classification. Indeed, most simple groups can be understood as finite elements of Lie groups, with parameters belonging to finite Galois fields. Their construction relies on the Chevalley basis of the Lie algebra, as well as on the topology of its Dynkin diagram. The remaining simple groups do not follow this pattern; they are the magnificent 26 sporadic groups. A singular achievement of modern mathematics was to show this classification to be complete.

So far, this beautiful subject has found but a few applications in physics. We feel nevertheless that physicists should acquaint themselves with its beauty. In this mostly descriptive chapter we introduce the necessary notions from number theory, and outline the construction of the Chevalley groups as well as that of some sporadic groups. We begin by presenting the two smallest non-Abelian simple finite groups.

A5 is simple

The 60 even permutations of the alternating group A5 are 3-ply or triply transitive. We can use this fact to prove something startling about alternating groups.

By definition, all even permutations are generated by the product of two transpositions, which can be reduced to three-cycles or the product of three-cycles.

Symmetric objects are so singular in the natural world that our ancestors must have noticed them very early. Indeed, symmetrical structures were given special magical status. The Greeks' obsession with geometrical shapes led them to the enumeration of platonic solids, and to adorn their edifices with various symmetrical patterns. In the ancient world, symmetry was synonymous with perfection. What could be better than a circle or a sphere? The Sun and the planets were supposed to circle the Earth. It took a long time to get to the apparently less than perfect ellipses!

Of course most shapes in the natural world display little or no symmetry, but many are almost symmetric. An orange is close to a perfect sphere; humans are almost symmetric about their vertical axis, but not quite, and ancient man must have been aware of this. Could this lack of exact symmetry have been viewed as a sign of imperfection, imperfection that humans need to atone for?

It must have been clear that highly symmetric objects were special, but it is a curious fact that the mathematical structures which generate symmetrical patterns were not systematically studied until the nineteenth century. That is not to say that symmetry patterns were unknown or neglected, witness the Moors in Spain who displayed the seventeen different ways to tile a plane on the walls of their palaces!

Évariste Galois in his study of the roots of polynomials of degree larger than four, equated the problem to that of a set of substitutions which form that mathematical structure we call a group. In physics, the study of crystals elicited wonderfully regular patterns which were described in terms of their symmetries. In the twentieth century, with the advent of Quantum Mechanics, symmetries have assumed a central role in the study of Nature.