group is one of the fundamental objects of study in the field of mathematics known as abstract algebra. The branch of algebra that studies groups is called group theory. Group theory has extensive applications in mathematics, science, and engineering. Many algebraic structures such as fields and vector spaces may be defined concisely in terms of groups, and group theory provides an important tool for studying symmetry, since the symmetries of any object form a group. Groups are thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics. Furthermore, their ability to represent geometric transformations finds applications in chemistry, computer graphics, and other fields.
Many structures investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers under multiplication. Other important examples are the group of non-singular matrices under multiplication, and the group of invertible functions under composition. Group theory allows for the properties of such structures to be investigated in a general setting.
This article covers only the basic notions related to groups. More advanced facets, applications and history of group theory are covered in group theory.
Reference:
http://en.wikipedia.org/wiki/Group_%28mathematics%29
