Abstract Algebra/Group Theory/Group/Double Inverse

TheoremEdit

Let G be any group with operation  .

 
In Group G, inverse of inverse of any element g is g.

ProofEdit

0. Choose  
1.   definition of inverse of g in G (usage 1,3)
2.   let a = g−1
3.  
4.   definition of inverse of a in G (usage 2)
5.   as a = g−1

DiagramsEdit

 
1. inverse of filled circle is empty circle.
 
2. inverse of empty circle is filled circle, given 1.