History of group theory

The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry.Lagrange, Abel and Galois were early researchers in the field of group theory.

The earliest study of groups as such probably goes back to the work of Lagrange in the late 18th century. However, this work was somewhat isolated, and 1846 publications of Cauchy and Galois are more commonly referred to as the beginning of group theory. The theory did not develop in a vacuum, and so three important threads in its pre-history are developed here.

One foundational root of group theory was the quest of solutions of polynomial equations of degree higher than 4.

An early source occurs in the problem of forming an equation of degree m having as its roots m of the roots of a given equation of degree n > m. For simple cases the problem goes back to Hudde (1659). Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Le Sœur (1748) and Waring (1762 to 1782) still further elaborated the idea.

A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. He discovered that the roots of all resolvents (résolvantes, réduites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a Calcul des Combinaisons. The contemporary work of Vandermonde (1770) also foreshadowed the coming theory.

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