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.

Bibliography

See R. P. Burn, Groups (1987); J. A. Green, Sets and Groups (1988).

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group

1. a small band of players or singers, esp of pop music

2. a number of animals or plants considered as a unit because of common characteristics, habits, etc.

3. two or more figures or objects forming a design or unit in a design, in a painting or sculpture

4. Chem two or more atoms that are bound together in a molecule and behave as a single unit

5. a vertical column of elements in the periodic table that all have similar electronic structures, properties, and valencies

6. Geology any stratigraphical unit, esp the unit for two or more formations

7. Maths a set that has an associated operation that combines any two members of the set to give another member and that also contains an identity element and an inverse for each element

8. See blood group

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group - 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.