Group theory

Group theory is that branch of mathematics concerned with the study of groups.

History

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.

Related Topics:
Algebraic equation - Number theory - Geometry - Euler - Gauss - Lagrange - Abel - Galois - Field theory - Galois theory

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An early source occurs in the problem of forming an mth-degree equation having as its roots m of the roots of a given

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nth-degree equation (m < n). 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.

Related Topics:
Hudde - Saunderson - Le Sœur - Waring

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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.

Related Topics:
Permutations - Lagrange - Vandermonde

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Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini distinguished what are now called

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intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation

Related Topics:
Transitive - Imprimitive

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under the name l'assieme della permutazioni. He also published a letter from Abbati to himself, in which the group idea is prominent.

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Galois found that if r_1, r_2, ldots r_n are the n roots of an equation, there is always a group of permutations of the r'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 equations and to that of elliptic functions. 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).

Related Topics:
Galois - Modular equation - Elliptic function

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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 Emile Mathieu.

Related Topics:
Arthur Cayley - Augustin Louis Cauchy - Serret - Camille Jordan - Netto - Nineteenth century - Bertrand - Charles Hermite - Frobenius - Leopold Kronecker - Emile Mathieu

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It was Walther von Dyck who, in 1882, gave the modern definition of a group.

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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, Maurer, and Cartan. The discontinuous (discrete group) theory was built up by Felix Klein, Lie, Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy.

Related Topics:
Lie group - Discrete subgroup - Transformation group - Sophus Lie - Killing - Study - Schur - Maurer - Cartan - Discrete group - Felix Klein - Poincaré - Charles Émile Picard - Modular form - Monodromy

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Other important mathematicians in this subject area include Emil Artin, Emmy Noether, Sylow, and many others.

Related Topics:
Emil Artin - Emmy Noether - Sylow

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