group

v. divide into groups; assemble, gather; classify, sort

n. bunch; ensemble, band; (Computers) window which contains several application icons (in a Windows environment)

Group can refer to:Group (auto racing)Group identifier (Unix)Business group, a collection of legally distinct business entitiesGroup (travel), a set of individuals traveling togetherA musical ensemble or bandBoy bandGirl groupGroup insurance covering people with common characteristicsPlaygroupTask forceCrown groupCultivar groupGroup racesGroup therapy
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In abstract algebra, a group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory.Many of the structures investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Other important examples are the group of non-singular matrices under multiplication and the group of invertible functions under composition. Group theory allows for the properties of such structures to be investigated in a general setting.
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(n.)
To form a group of; to arrange or combine in a group or in groups, often with reference to mutual relation and the best effect; to form an assemblage of.
  

(n.)
An assemblage of objects in a certain order or relation, or having some resemblance or common characteristic; as, groups of strata.
  

(n.)
A variously limited assemblage of animals or plants, having some resemblance, or common characteristics in form or structure. The term has different uses, and may be made to include certain species of a genus, or a whole genus, or certain genera, or even several orders.
  

(n.)
A number of eighth, sixteenth, etc., notes joined at the stems; -- sometimes rather indefinitely applied to any ornament made up of a few short notes.
  

(n.)
A cluster, crowd, or throng; an assemblage, either of persons or things, collected without any regular form or arrangement; as, a group of men or of trees; a group of isles.
  

1. In frequency-division multiplexing, a specific number of associated voice channels, either within a supergroup or as an independent entity. Note 1: In wideband systems, a group usually consists of 12 voice channels and occupies the frequency band from 60 kHz to 108 kHz. Note 2: this is CCITT group B. Note 3 : CCITT Basic Group A, for carrier telephone systems, consists of 12 channels occupying upper sidebands in the 12-kHz to 60-kHz band. Basic Group A is no longer mentioned in CCITT Recommendations. Note 4: A supergroup usually consists of 60 voice channels, i.e., 5 groups of 12 voice channels each, occupying the frequency band from 312 kHz to 552 kHz. (188 ) Note 5: A mastergroup consists of 10 supergroups or 600 voice channels. (188 ) Note 6: The CCITT standard mastergroup consists of 5 supergroups. The U.S. commercial carrier standard mastergroup consists of 10 supergroups. Note 7: The terms "supermaster group" or "jumbo group" are sometimes used to refer to 6 mastergroups. 2. A set of characters forming a unit for transmission or cryptographic treatment. (188 )

A group G is a non-empty set upon which a binary operator * is defined with the following properties for all a,b,c in G:
Closure: G is closed under *, a*b in G Associative: * is associative on G, (a*b)*c = a*(b*c) Identity: There is an identity element e such that a*e = e*a = a. Inverse: Every element has a unique inverse a' such that a * a' = a' * a = e. The inverse is usually written with a superscript -1.
(1998-10-03)