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
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.