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:

1. Identity elementThere exists an , called an identity element, such that , for all
2. InversesFor each , there exists an , called an inverse of , such that
3. Associativity — For all
4. 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.