group

In mathematics, a finite or infinite set of elements that can be combined by an operation; formally, a group must satisfy certain conditions. For example, the set of all integers (positive or negative whole numbers) forms a group with regard to addition because: (1) addition is associative, that is, the sum of two or more integers is the same regardless of the order in which the integers are added; (2) adding two integers gives another integer; (3) the set includes an identity element 0, which has no effect on any integer to which it is added (for example, 0 + 3 = 3); and (4) each integer has an inverse (for instance, 7 has the inverse −7), such that the sum of an integer and its inverse is 0. Group theory is the study of the properties of groups.