Abstract Algebra/Group Theory/Group/Cancellation

Theorem

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Let G be a Group.
1.  
2.  

Proof

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0. Choose   such that  
1.   definition of inverse of g in G (usage 1)
2.   0.
3.     is associative in G
4.   g-1 is inverse of g (usage 3)
5.   eG is identity of G(usage 3)

Diagrams

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if a*g = b*g...
 
a = a*g*g-1
 
b*g*g-1 = b
 
then a = b.

Usage

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  1. if a, b, x are in the same group, and x*a = x*b, then a = b

Notice

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  1. a, b, and g have to be all in the same group.
  2.   has to be the binary operator of the group.
  3. G has to be a group.