Abstract Algebra/Group Theory/Group/Cancellation

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Theorem[edit | edit source]

Let G be a Group.

Proof[edit | edit source]

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[edit | edit source]

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

Usage[edit | edit source]

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

Notice[edit | edit source]

  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.