group

Definition

A group is a pair consisting a set and a binary operation of such that

• G has an identity element , with the property that
• The operation is associative, a.k.a
• Each elements of has an inverse s.t.

Some authors also add a closure property to explicitly state that . Though this property is implied by the fact that is a binary operator of .

Examples

• The pair is not a group since does not have an inverse. By contrast, if we change to non-zero rationals, then this pair becomes a group.

Relation to Other Structures

Specializations

• An abelian group, also called a commutative group, is a group in which the operation is commutative.
• An additive group is a specific type of abelian group where the operation is addition.

Generalizations

• A monoid is a more general structure than a group, lacking the requirement for inverses.
• A semigroup further generalizes a monoid by omitting the identity element.