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1) an affinity; age; control; discussion; encounter; ethnic, minority; peer; pressure; social; special-interest; splinter group

2) (*BE*) a ginger group ('a group of activists')

3) a blood group

1) (*d*; *intr.*) to group around (the scouts grouped around their leader)

2) (*d*; *tr.*) to group under (to group several types under one heading)

- sometimes rather indefinitely applied to any ornament made up of a few short notes.

¦ noun [treated as sing. or plural]

1. a number of people or things located, gathered, or classed together.

2. a number of musicians who play popular music together.

3. Chemistry a set of elements occupying a column in the periodic table and having broadly similar properties.

4. Chemistry a combination of atoms having a recognizable identity in a number of compounds.

5. Mathematics a set of elements, together with an associative binary operation, which contains an inverse for each element and an identity element.

6. a division of an air force, usually consisting of two or more stations.

¦ verb place in or form a group or groups.

Group (mathematics)

In mathematics, a **group** is a non-empty set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. 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.

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. Lie groups appear in symmetry groups in geometry, and 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 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. 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.

Examples of use of group

1. Banks Mizuho Financial Group, Mitsubishi Tokyo Financial Group and Sumitomo Mitsui Financial Group all finished lower.

2. Ltd., Tamimi Group of Companies and Arabian Services Group.

3. Group leader training: Belkin is the only group leader.

4. At the folk games the Pyongyang City Group placed first in total, the South Phyongan Provincial Group and the North Hwanghae Provincial Group second and the Kangwon Provincial Group and the South Hamgyong Provincial Group third.

5. Any armed group affiliated with a particular ethnic group or religious group should be considered illegal and should be disarmed.