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).

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.