The theory of groups has found widespread application. It was used to determine the possible structure of crystals, and it has deep implications for the theory of molecular vibration. The conservation laws of physics, such as conservation of energy and conservation of electric charge, all arise from the symmetries in the equations of physics. And one of the simplest groups, known as "the multiplicative group modulo N" is used every time you send secure information over the Internet.

For an introduction to groups requiring little mathematics background, see Groups and Symmetry by David Farmer.

What is a Lie group?
Lie groups lie at the intersection of two fundamental fields of mathematics: algebra and geometry. A Lie group is first of all a group. Secondly it is a smooth manifold which is a specific kind of geometric object. The circle and the sphere are examples of smooth manifolds. Finally the algebraic structure and the geometric structure must be compatible in a precise way.

Informally, a Lie group is a group of symmetries where the symmetries are continuous. A circle has a continuous group of symmetries: you can rotate the circle an arbitrarily small amount and it looks the same. This is in contrast to the hexagon, for example. If you rotate the hexagon by a small amount then it will look different. Only rotations that are multiples of one-sixth of a full turn are symmetries of a hexagon.

Lie groups were studied by the Norwegian mathematician Sophus Lie at the end of the 19th century. Lie was interested in solving equations. At that time techniques for solving equations were basically a bag of tricks. A typical tool was to make a clever change of variables which would make one of the variable drop out of the equations. Lie's basic insight was that when this happened it was due to an underlying symmetric of the equations, and that underlying this symmetry was what is now called a Lie group.

Lie groups are ubiquitous in mathematics and all areas of science. Associated to any system which has a continuous group of symmetries is a Lie group.

The basic building blocks of Lie groups are simple Lie groups. The classification of these groups starts with the classification of the complex, simple Lie algebras. These were classified by Wilhelm Killing and Elie Cartan in the 1890s.

Cartan constructed all the simple Lie algebras, which correspond to the simple root systems: An, Bn, Cn, and Dn (n=1,2,3....) and the exceptional ones: G2, F4, E6, E7, and E8. The exceptional ones have dimensions 14, 52, 78, 133 and 248, respectively.

For a Lie group, the subscript n is called the rank of the group, which is a measure of how large the group is. In a sense E8 is the most complicated Lie group. Although A1000, for example, is certainly larger than E8, mathematicians know how to describe the representations of An for every n, so there is nothing special about A1000. All of An, Bn, Cn, and Dn are well-understood, so the remaining challenge is to tackle the exceptional groups. All 5 of the exceptional groups need to be treated separately, and E8 is the most complicated of these.

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