Abstract Algebra/Group Theory/Group/Identity is Unique

TheoremEdit

Each group only has one identity

ProofEdit

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.

DiagramsEdit

 
1. Assume a group has two identities.
 
2. e1 * e2 = e1
as e2 is identity of G,
and e1 is in G.
 
3. e1 * e2 = e2
as e1 is identity of G,
and e2 is in G
 
4. The two identities are the same.
 
5. a group only has one identity