Abstract Algebra/Group Theory/Group/Inverse is Unique

TheoremEdit

In a group, each element only has one inverse.

ProofEdit

0. Choose  . Then, inverse g1−1 of g is also in G.
1. Assume g has a different inverse g2−1 in G
2.  
  is associative on G
3.  
g1-1 and g2-1 are inverses of g on G (usage 3)
4.  , contradicting 1.
eG is identity of G (usage 3)