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, .
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 .
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