![]() To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. Modern group theory-an active mathematical discipline-studies groups in their own right. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. The concept of a group arose in the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term group (French: groupe) for the symmetry group of the roots of an equation, now called a Galois group. Point groups describe symmetry in molecular chemistry. The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity. Lie groups appear in symmetry groups in geometry, and also in the Standard Model of particle physics. In geometry, groups arise naturally in 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 a general group. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way). Many mathematical structures are groups endowed with other properties. ![]() In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. The manipulations of the Rubik's Cube form the Rubik's Cube group. ![]() For a more advanced treatment, see Group theory. This article is about basic notions of groups in mathematics.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |