# Cancellation Laws/Corollary 2

## 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

$\blacksquare$