ABSTRACT
This chapter begins with an introduction to finite group theory. It concentrates on a few simple facts that are useful in understanding the compact Lie algebras. A group is finite if it has a finite number of elements. Otherwise it is infinite. The matrices are the matrix elements of the linear operators. There is a natural multiplication law for transformations of a physical system. The unitary operators form a representation of the transformation group because the transformed quantum states represent the transformed physical system. Parity is the operation of reflection in a mirror. The integers form an infinite group under addition. In the case of parity, the linear operator representing parity is hermitian. A classical problem which is quite analogous to the problem of diagonalizing the Hamiltonian in quantum mechanics is the problem of finding the normal modes of small oscillations of a mechanical system about a point of stable equilibrium.
