group

group in mathematics, system consisting of a set of elements and a binary operation a [symbol] 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 [symbol]( b [symbol] c )=( a [symbol] b )[symbol] c, where [symbol] 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 [symbol] 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 [symbol] a^-1 = I. If, in addition to satisfying these four axioms, the group also satisfies the commutative law for the operation, i.e., a [symbol] b = b [symbol] 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).