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. Joseph Louis Lagrange, Niels Henrik Abel and Évariste Galois were early researchers in the field of group theory.
Early 19th century
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 Augustin Louis 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.
Development of permutation groups
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 . For simple cases, the problem goes back to Johann van Waveren Hudde (1659). Nicholas Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Thomas Le Seur (1703–1770) (1748) and Edward Waring (1762 to 1782) still further elaborated the idea. Waring proved the fundamental theorem of symmetric polynomials, and specially considered the relation between the roots of a quartic equation and its resolvent cubic.
Lagrange's goal (1770, 1771) was to understand why equations of third and fourth degree admit formulas for solutions, and a key object was the group of permutations of the roots. On this was built the theory of substitutions. He discovered that the roots of all Lagrange 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 Alexandre-Théophile Vandermonde (1770) developed the theory of symmetric functions and solution of cyclotomic polynomials. Leopold Kronecker has been quoted as saying that a new boom in algebra began with Vandermonde's first paper. Similarly Cauchy gave credit to both Lagrange and Vandermonde for studying symmetric functions and permutations of variables.
Paolo Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini was the first person to explore ideas in the theory of permutation groups such as the order of an element of a group, conjugacy, and the cycle decomposition of elements of permutation groups. Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni. He also published a letter from Pietro Abbati to himself, in which the group idea is prominent. However, he never formalized the concept of a group, or even of a permutation group.
Évariste Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the posthumous publication of his collected papers in 1846 (Liouville, Vol. XI). He considered for the first time what is now called the closure property of a group of permutations, which he expressed as
Galois found that if are the n roots of an equation, there is always a group of permutations of the r'''s such that every function of the roots invariable by the substitutions of the group is rationally known, and conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. In modern terms, the solvability of the Galois group attached to the equation determines the solvability of the equation with radicals.
Galois was the first to use the words group (groupe in French) and primitive in their modern meanings. He did not use primitive group but called equation primitive an equation whose Galois group is primitive. He discovered the notion of normal subgroups and found that a solvable primitive group may be identified to a subgroup of the affine group of an affine space over a finite field of prime order.