Abstract Algebra/Group Theory/Group/Identity is Unique
< Abstract Algebra  Group Theory  Group
TheoremEdit
 Each group only has one identity
ProofEdit
 0. Let G be any group. Then G has an identity, say e_{1}.
 1. Assume G has a different identity e_{2}
As e_{1} is identity of G (usage 1), 
As e_{2} is identity of G (usage 1), 


e_{2} is identity of G (usage 3), 
As e_{1} is identity of G (usage 3), 


By 2a. and 3a., 
By 2b. and 3b., 


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.