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 |
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: The Group Property: Theorem $1 \ \text {(ii)}$