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Group theory is that branch of mathematics concerned with the study of groups.
Please refer to the Glossary of group theory for the definitions of terms used throughout group theory.
See also list of group theory topics.
There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Euler, Gauss, Lagrange, Abel and Galois were early researchers in the field of group theory. Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory.
An early source occurs in the problem of forming an th-degree equation having as its roots m of the roots of a given th-degree equation (). 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 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.
RuffiniPaolo Ruffini ( Valentano, 1765 Modena, 1822) was an Italian mathematician and philosopher. Among his work was the proof that quintic (and higher-order) equations cannot be solved by radicals and Ruffini's rule, a quick method for polynomial division. (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini distinguished what are now calledintransitive and transitiveIn mathematics, the word transitive admits at least two distinct meanings: A group G acts transitively on a set S if for any x y ∈ S there is some g ∈ G such that gx y''. See group action. A somewhat related meaning is explained at ergodic theor, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme della permutazioni. He also published a letter from Abbati to himself, in which the group idea is prominent.
Galois found that if are the roots of an equation, there is always a group of permutations of the 's such that (1) every function of the roots invariable by the substitutions of the group is rationally known, and (2), conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. Galois also contributed to the theory of modular equationIn mathematics, a modular equation is an algebraic equation satisfied by moduli in the sense of moduli problem. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity fs and to that of elliptic functionIn complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. The elliptic functions can be seen as analogs of the trigonometric functions (which have a single period only).s. His first publication on the group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI). Arthur Cayley and Augustin Louis Cauchy were among the first to appreciate the importance of the theory, and to the latter especially are due a number of important theorems. The subject was popularised by Serret , who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des Substitutions is a classic; and to Netto (1882), whose was translated into English by Cole (1892). Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Mathieu.It was Walther Van Dyck who, in 1882, gave the modern definition of a group.
The study of what are now called Lie groups, and their discrete subgroups, as transformation groups, started systematically in 1884 with Sophus Lie; followed by work of Killing, Study, Schur and Maurer . The discontinuous ( discrete group) theory was built up by Felix Klein, Lie, Poincaré, and Charles Emile Picard, in connection in particular with modular forms and monodromy.
Other important mathematicians in this subject area include Emil Artin, Emmy Noether, Sylow, and many others.