group
group, in mathematics, system consisting of a set of elements and a binary operation a∘b defined for combining two elements such that the following requirements are satisfied: (1) The set is closed under the operation; i.e., if a and b are elements of the set, then the element that results from combining a and b under the operation is also an element of the set; (2) the operation satisfies the associative law; i.e., a∘(b∘c)=(a∘b)∘c, where ∘ represents the operation and a, b, and c are any three elements; (3) there exists an identity element I in the set such that a∘I=a for any element a in the set; (4) there exists an inverse a^{1} in the set for every a such that a∘a^{1}=I. If, in addition to satisfying these four axioms, the group also satisfies the commutative law for the operation, i.e., a∘b=b∘a, then it is called a commutative, or Abelian, group. The real numbers (see number) form a commutative group both under addition, with 0 as identity element and a as inverse, and, excluding 0, under multiplication, with 1 as identity element and 1/a as inverse. The elements of a group need not be numbers; they may often be transformations, or mappings, of one set of objects into another. For example, the set of all permutations of a finite collection of objects constitutes a group. Group theory has wide applications in mathematics, including number theory, geometry, and statistics, and is also important in other branches of science, e.g., elementary particle theory and crystallography.
See R. P. Burn, Groups (1987); J. A. Green, Sets and Groups (1988).
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group
group / groōp/ • n. [treated as sing. or pl.] a number of people or things that are located close together or are considered or classed together: these bodies fall into four distinct groups. ∎ a number of people who work together or share certain beliefs: I now belong to my local drama group. ∎ a commercial organization consisting of several companies under common ownership. ∎ a number of musicians who play popular music together. ∎ Mil. a unit of the U.S. Air Force, consisting of two or more squadrons. ∎ Mil. a unit of the U.S. Army, consisting of two or more battalions. ∎ Art two or more figures or objects forming a design. ∎ Chem. a set of elements occupying a column in the periodic table and having broadly similar properties arising from their similar electronic structure. ∎ Chem. a combination of atoms having a recognizable identity in a number of compounds. ∎ Math. a set of elements, together with an associative binary operation, that contains an inverse for each element and an identity element. ∎ Geol. a stratigraphic division consisting of two or more formations. • v. [tr.] (often be grouped) put together or place in a group or groups: three wooden chairs were grouped around a dining table. ∎ put into categories; classify: we group them into species merely as a convenience. ∎ [intr.] form a group or groups: many growers began to group together to form cooperatives.
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"group." The Oxford Pocket Dictionary of Current English. . Encyclopedia.com. 13 Dec. 2017 <http://www.encyclopedia.com>.
"group." The Oxford Pocket Dictionary of Current English. . Encyclopedia.com. (December 13, 2017). http://www.encyclopedia.com/humanities/dictionariesthesaurusespicturesandpressreleases/group0
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group
group A set G on which there is defined a dyadic operation ◦ (mapping G × G into G) that satisfies the following properties:
(a) ◦ is associative;
(b) ◦ has an identity, i.e. there is a unique element e in G with the property that x ◦ e = e ◦ x = x
for all x in G; e is called the identity of the group;
(c) inverses exist in G, i.e. for each x in G there is an inverse, denoted by x^{–1}, with the property that x ◦ x^{–1} = x^{–1} ◦ x = e
These are the group axioms.
Certain kinds of groups are of particular interest. If the dyadic operation ◦ is commutative, the group is said to be a commutative group or an abelian group (named for the Norwegian mathematician Niels Abel).
If there is only a finite number of elements n in the group, the group is said to be finite; n is then the order of the group. Finite groups can be represented or depicted by means of a Cayley table.
If the group has a generator then it is said to be cyclic; a cyclic group must be abelian.
The group is a very important algebraic structure that underlies many other algebraic structures such as rings and fields. There are direct applications of groups in the study of symmetry, in the study of transformations and in particular permutations, and also in error detecting and error correcting as well as in the design of fast adders.
Groups were originally introduced for solving an algebraic problem. By group theory it can be shown that algorithmic methods of a particular kind cannot exist for finding the roots of a general polynomial of degree greater than four. See also semigroup.
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"group." A Dictionary of Computing. . Encyclopedia.com. 13 Dec. 2017 <http://www.encyclopedia.com>.
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group
group, social group A number of individuals, defined by formal or informal criteria of membership, who share a feeling of unity or are bound together in relatively stable patterns of interaction. The latter criterion is necessary in order to distinguish social groups from other aggregates dealt with by sociologists which are grouped only in the statistical sense that they share some socially relevant characteristic (including, for example, social categories such as suburban residents or junior managers). However, the term is one of the most widely used in sociology, and will often be found applied to combinations of people who may or may not share a feeling of unity (as in social class groups) and may or may not be involved in regular social interaction (as in the case of members of certain ethnic groups). See also COOLEY, CHARLES HORTON; DESCENT GROUPS; DYAD; GROUP DYNAMICS; OUTGROUP; PARIAH GROUP; PEER GROUP; PRESSURE GROUPS; REFERENCE GROUP; STATUS GROUP; SUMNER, WILLIAM GRAHAM; TRIAD.
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"group." A Dictionary of Sociology. . Encyclopedia.com. 13 Dec. 2017 <http://www.encyclopedia.com>.
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Group
Group
a set of things collected as a unit. See also gathering.
Examples: group of columns [three or four columns joined together on the same pedestal], 1731; of company, 1748; of crystals, 1830; of islands; of musicians; of partisans, 1809; of rocks, 1859; of singers; of trees; of woes, 1729; of words, 1748.
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"Group." Dictionary of Collective Nouns and Group Terms. . Encyclopedia.com. 13 Dec. 2017 <http://www.encyclopedia.com>.
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group
group
1. A number of geophones whose output is summed to feed one seismic channel. A particularly large number of geophones used per channel may be referred to as a ‘patch’. See also ARRAY.
2. See FORMATION.
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"group." A Dictionary of Earth Sciences. . Encyclopedia.com. 13 Dec. 2017 <http://www.encyclopedia.com>.
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group
group assemblage of figures or objects in an artistic design XVII; assemblage of persons or things (gen.) forming a unity XVIII. — F. groupe — It. gruppo, of Gmc. orig.
Hence, or — F., group vb. XVIII.
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"group." The Concise Oxford Dictionary of English Etymology. . Encyclopedia.com. 13 Dec. 2017 <http://www.encyclopedia.com>.
"group." The Concise Oxford Dictionary of English Etymology. . Encyclopedia.com. (December 13, 2017). http://www.encyclopedia.com/humanities/dictionariesthesaurusespicturesandpressreleases/group2
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group
group
•bloop, cockahoop, coop, croup, droop, drupe, dupe, goop, group, Guadeloupe, hoop, loop, poop, recoup, roup, scoop, sloop, snoop, soup, stoep, stoop, stoup, stupe, swoop, troop, troupe, whoop
•hulahoop • cantaloupe • nincompoop
•playgroup • subgroup • peer group
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