In mathematics, a group is a set equipped with an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists, and every element has an inverse. These three conditions, called group axioms, hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and its definition through the group axioms was elaborated for handling, in a unified way, essential structural properties of entities of very different mathematical nature (such as numbers, geometric shapes and polynomial roots). Because of the ubiquity of groups in numerous areas (both within and outside mathematics), some authors cons