Group Definition

The group is the most fundamental object you will study in abstract algebra. Groups generalize a wide variety of mathematical sets: the integers, symmetries of shapes, modular arithmetic, NxM matrices, and much more. After learning about groups in detail, you will then be ready to continue your study of abstract algebra be learning about rings, fields, modules and vector spaces.

The group is an abstract mathematical idea that comes up in a surprising variety of topics, including geometry, topology, number theory, and more. In the 1800s, it became apparent that this same idea was being used to address many different types of problems. So mathematicians began to study the tool itself! Thus the subject Abstract Algebra was born. 

What is this remarkable idea? It’s a way to generalize a set of elements that behaves, mathematically, in a certain way. It may help to see examples of how all groups share these common characteristics (see video below). These include the following properties:

▪ You can perform an operation (e.g. addition) to combine the elements in the set.
▪ The group is closed under that operation (it produces  another element in the set).
▪ A group has an inverse for each element in the set.
▪ When you combine an element and its inverse, you get the identity element.
▪ The elements of a group obey the associative property

You may wonder what motivated the definition of the group? Find this and other lessons below.

Group Example - A Complex Circle
Learn about an interesting group: the set of all complex numbers of magnitude 1. This is both a circle and a group.