A **group** is a set paired with an operation on the set. As such, a group can be conceptualized as an ordered pair , where is a set, and is an operation.

A set and operation is a group if and only if it satisfies the following properties:

- Identity element — There exists an , called an identity element, such that , for all
- Inverses — For each , there exists an , called an inverse of , such that
- Associativity — For all
- Closure — For all

Whenever the group operation is , the operation of group elements , , is often abbreviated as simply a juxtaposition of the group elements, .

## Important Results

From the given criterion for a group, the following properties can be shown for any group :

- There exists exactly one identity element;
- For each , there exists exactly one inverse of , and henceforth is referred to as (proof)
- For each
- Groups have the cancellation property: For all implies , and implies .

## Optional Properties

A group is:

- An abelian group if the operation is commutative, i.e. for all
- A cyclic group if there exists a such that , where is copies of being operated together (see exponent)
- A subgroup of a group if (see subset), where the group operation on is a domain restriction on the group operation on

A group with a partial order on it is a partially ordered group if for all , if , then and (translation invariance). It is a totally ordered group if in addition is a total order.