Cancellation Laws/Corollary 1

From ProofWiki
Jump to navigation Jump to search

Corollary to Cancellation Laws

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

Then:

$g h = g \implies h = e$


Proof 1

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

$\blacksquare$


Proof 2

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

$\blacksquare$