411 REFERENCE · Wikibooks
Abstract Algebra/Group Theory/Group/Cancellation
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Theorem
Let G be a Group.
1.
2.
Proof
{| style="width: 60%"
|-
| 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
frame|if a*g = b*g...frame|a = a*g*g-1frame|b*g*g-1 = bframe|then a = b.
Usage
Usage1if a, b, x are in the same group, and x*a = x*b, then a = b
Notice
a, b, and g have to be all in the same group.
has to be the binary operator of the group.
G has to be a group.