Abstract Algebra/Group Theory/Group/Identity is Unique

= Theorem =


 * Each group only has one identity

= Proof =


 * 0. Let G be any group. Then G has an identity, say e1.
 * 1. Assume G has a different identity e2

By 4a. and 4b.,
 * 5. $$ {\color{blue}e_1} = {\color{OliveGreen}e_2} $$, contradicting 1.

Since a right assumption can't lead to a wrong or contradicting conclusion, our assumption (1.) is false and identity of a group is unique.

= Diagrams =