group

In mathematics, a group is a set equipped with a binary operation that is associative, has an identity element, and is such that every element has an inverse. These three conditions, called group axioms, hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The formulation of the axioms is, however, detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many mathematical objects. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizing principle of contemporary mathematics.Groups arise naturally in geometry for the study of symmetries and geometric transformations: the symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form generally a group. These examples were at the origin of the concept of group (together with Galois groups). Lie groups arise as symmetry groups in geometry but appear also in the Standard Model of particle physics. The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity. Point groups describe symmetry in molecular chemistry.
The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term group (groupe, in French) for the symmetry group of the roots of an equation, now called a Galois group. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.

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