Group theory

Group theory is the study of a particular algebraic structure called a group. A group is a set that, in an abstract sense, has a special kind of "structure" with some very "nice" properties. Many of the sets commonly used in mathematics, like the integers and the complex numbers, are groups.

Group theory provides a basic foundation to study other algebraic structures that have even more structure, like rings and fields.

Concepts from group theory

A group

A group is a set G and a binary operator * that has the following properties:

• The group has an identity element: There is an element e, such that x * e=x and e * x=x for all x in the group.
• Every element has an inverse: For each element x in the group, there is another element y, such that x * y=e and y * x=e. (e is the identity element)
• The operation is associative: For all elements x, y, and z in the group, (x * y) * z = x * (y * z).

Normal subgroups

A subgroup is a subset of a group that is itself a group. Not every subset of a group is a subgroup (for example, a subset that does not contain the identity element e cannot be a group). A normal subgroup is a very important kind of subgroup and is defined by a few different equivalent definitions. The role of normal subgroups will be shown in the next few sections.