Group Theory Tutorial

Author: Derek J. S. Robinson
Publisher:
Publication Date: 2000-12-01
Features: "A group is defined by the laws of combination of its symbols," according to a celebrated dictum of Cayley. And this is probably still a good one-line explanation as any. The concept of a group is surely one of the central ideas of mathematics. Certainly, there are a few branches of that science in which groups are not employed implicitly or explicitly. Nor is the use of groups confined to pure mathematics. Quantum theory, molecular and atomic structure, and crystallography are just a few of the areas of science in which the idea of a group as a measure of symmetry has played an important part. The theory of groups is the oldest branch of modern algebra. Its origins are to be found in the work of Joseph Louis Lagrange (1736-1813), Paulo Ruffini (1765-1822), and Evariste Galois (1811-1832) on the theory of algebraic equations. Their groups consisted of permutations of the variables or of the roots of polynomials, and indeed much of the nineteenth century all groups were finite permutation groups. Nevertheless, many of the fundamental ideas of group theory were introduced by these early workers and their successors, Augustin Louis Cauchy (1789-1857), Ludwig Sylow (1832-1918), Camille Jordan (1838-1922), among others.