# 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 |

$\blacksquare$

## Sources

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