a "group" consists of a set of elements and a single operation which combines a first and a second element and returns a third. the operation must satisfy certain constraints for it to determine a group: it must be closed (when applied to any pair of elements of the associated set, the generated element must also be a member of that set), it must be associative, there must be an identity element (an element which, when combined with another element using the operation, results in the original element, such as adding zero to a number or multiplying it by one), and for every element there must be an inverse element.