Group (mathematics)

In mathematics, a group is a set equipped with an operation that combines two elements into another element of the same set. The operation must be associative, have an identity element, and give every element an inverse. Groups provide a unified framework for structures including numbers, geometric shapes, and polynomial roots. They arise in the study of symmetries and transformations and are studied in modern group theory through notions such as subgroups, quotient groups, and representations.

Defining conditions

Associativity, an identity element, and an inverse for every element.

Example

The integers under addition form a group.

Historical origin

The concept arose in the study of polynomial equations.