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Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms.
A p (G) is the intersection of all normal subgroups K such that G/K is an abelian p-group (i.e., K is an index normal subgroup that contains the derived group [,]): G/A p (G) is the largest abelian p-group (not necessarily elementary) onto which G surjects.
Group theory as an increasingly independent subject was popularized by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des substitutions et des équations algébriques (1870) is a classic; and to Eugen Netto (1882), whose Theory of Substitutions and its Applications to Algebra was translated into ...
In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive integer with x p = e, where e is the identity element of G.
The group consists of the finite strings (words) that can be composed by elements from A, together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance (abb) • (bca) = abbbca. Every group (G, •) is basically a factor group of a free group generated by G.
The ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway, Robert Turner Curtis, Simon Phillips Norton, Richard Alan Parker and Robert Arnott Wilson (with computational assistance from J. G. Thackray), published in December 1985 by Oxford University Press and reprinted with corrections in 2003 (ISBN 978-0-19-853199-9).
For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved. [53] Group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form.