A group, usually referred to as *G*, is a finite or infinite set of elements together with a binary operation (often called the group operation) that together satisfy the four fundamental properties—closure, associativity, and the identity and inverse properties (for more information about these properties, see elsewhere in this chapter). A great many of the objects investigated in mathematics turn out to be groups, including familiar number systems—such as the integers, rational, real, and complex numbers under addition; non-zero rational, real, and complex numbers under multiplication; non-singular matrices under multiplication; and so on. The branch of mathematics that studies groups is called group theory, an important area of mathematics that has many applications to mathematical physics (such as particle theory).