Abstract Algebra/Group Theory/Group/Inverse is Unique

Theorem

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In a group, each element only has one inverse.

Proof

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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)