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A group has exactly one identity element.
Proposition (Uniqueness of identity). Let be a group with binary operation written multiplicatively. An element is called an identity element if for every one has and . If are both identity elements, then .
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
A binary operation on a set is a function , where is the Cartesian product of with itself. The value of is usually written .