Abstract Algebra/Group Theory/Group/Cancellation

< Abstract Algebra < Group Theory < Group

Theorem

Let G be a Group.

\forall \;g,a,b\in G:(g\ast a=g\ast b)\rightarrow (a=b)

\forall \;g,a,b\in G:(a\ast g=b\ast g)\rightarrow (a=b)

0. Choose ${\color {OliveGreen}g},a,b\in G$ such that ${\color {OliveGreen}g}\ast a={\color {OliveGreen}g}\ast b$
1. ${\color {BrickRed}g^{-1}}\in G$	definition of inverse of g in G (usage 1)
2. ${\color {BrickRed}g^{-1}}\ast ({\color {OliveGreen}g}\ast a)={\color {BrickRed}g^{-1}}\ast ({\color {OliveGreen}g}\ast b)$	0.
3. $({\color {BrickRed}g^{-1}}\ast {\color {OliveGreen}g})\ast a=({\color {BrickRed}g^{-1}}\ast {\color {OliveGreen}g})\ast b$	$\ast$ is associative in G
4. $e_{G}\ast a=e_{G}\ast b$	g^-1 is inverse of g (usage 3)
5. $a=b\,$	e_G is identity of G(usage 3)

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