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 As e1 is identity of G (usage 1),As e2 is identity of G (usage 1),2a. 2b. e2 is identity of G (usage 3),As e1 is identity of G (usage 3), 3a. 3b. By 2a. and 3a.,By 2b. and 3b., 4a. 4b. By 4a. and 4b., 5. , 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 frame|1. Assume a group has two identities. frame| 2. e1 * e2 = e1 as e2 is identity of G, and e1 is in G. frame| 3. e1 * e2 = e2 as e1 is identity of G, and e2 is in G frame| 4. The two identities are the same. frame| 5. a group only has one identity