Abstract Algebra/Group Theory/Group/Inverse is Unique

Theorem

In a group, each element only has one inverse.

Proof

0. Choose $g\in G$. Then, [[../Definition of a Group/Definition of Inverse#Usage1|inverse]] g1−1 of g is also in G.
1. Assume g has a different [[../Definition of a Group/Definition of Inverse|inverse]] g2−1 in G
2. $(g_1^{-1}\ast g)\ast g_2^{-1}=g_1^{-1}\ast (g\ast g_2^{-1})$	[[../Definition of a Group/Definition of Associativity#Usage1|$\ast$ is associative on G]]
3. $e_G\ast g_2^{-1}=g_1^{-1}\ast e_G$	[[../Definition of a Group/Definition of Inverse#Usage3|g1-1 and g2-1 are inverses of g on G (usage 3)]]
4. $g_2^{-1}=g_1^{-1}$, contradicting 1.	[[../Definition of a Group/Definition of Identity#Usage3|eG is identity of G (usage 3)]]