Groups

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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Noun
1. any number of entities (members) considered as a unit
(synonym) grouping
(hyponym) arrangement
(derivation) aggroup
2. (chemistry) two or more atoms bound together as a single unit and forming part of a molecule
(synonym) radical, chemical group
(hypernym) unit, building block
(hyponym) acyl, acyl group
(part-holonym) molecule
(classification) chemistry, chemical science
3. a set that is closed, associative, has an identity element and every element has an inverse
(synonym) mathematical group
(hypernym) set
(hyponym) subgroup
Verb
1. arrange into a group or groups; "Can you group these shapes together?"
(hypernym) classify, class, sort, assort, sort out, separate
(hyponym) regroup
(derivation) grouping
2. form a group or group together
(synonym) aggroup
(hypernym) meet, gather, assemble, forgather, foregather
(hyponym) team, team up
(derivation) grouping

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)

GRUPPI. CROCCHI. GRUPPI FINANZIARI. TRUST. RAGGRUPPA. RADUNA