# Abstract Algebra/Group Theory/Group/Inverse is Unique

< Abstract Algebra | Group Theory | Group

# TheoremEdit

- In a group, each element only has one inverse.

# ProofEdit

0. Choose . Then, inverse *g*_{1}^{−1}of*g*is also in G.1. Assume *g*has a*different*inverse*g*_{2}^{−1}in G- 2.

is associative on G - 3.

*g*_{1}^{-1}and*g*_{2}^{-1}are inverses of*g*on G (usage 3)- 4. , contradicting 1.

*e*_{G}is identity of G (usage 3)