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

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

$\blacksquare$


Proof 2

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

$\blacksquare$


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