Definition

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

  • G has an identity element, usually denoted or just 1, 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.