Abstract Algebra/Group Theory/Group/Inverse is Unique

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Theorem[edit]

In a group, each element only has one inverse.

Proof[edit]

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)