Cancellation Laws/Corollary 2

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Corollary to Cancellation Laws

Let $g$ and $h$ be elements of a group $G$ whose identity element is $e$.

Then:

$h g = g \implies h = e$


Proof 1

\(\displaystyle h g\) \(=\) \(\displaystyle g\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle h g\) \(=\) \(\displaystyle e g\) Group Axiom $\text G 2$: Existence of Identity Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle h\) \(=\) \(\displaystyle e\) Right Cancellation Law

$\blacksquare$


Proof 2

\(\displaystyle h g\) \(=\) \(\displaystyle g\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {h g} g^{-1}\) \(=\) \(\displaystyle g g^{-1}\) Group Axiom $\text G 2$: Existence of Identity Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle h \paren {g g^{-1} }\) \(=\) \(\displaystyle g g^{-1}\) Group Axiom $\text G 1$: Associativity
\(\displaystyle \leadsto \ \ \) \(\displaystyle h e\) \(=\) \(\displaystyle e\) Group Axiom $\text G 3$: Existence of Inverse Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle h\) \(=\) \(\displaystyle e\) Group Axiom $\text G 2$: Existence of Identity Element

$\blacksquare$


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