Cancellation Laws/Corollary 2

Corollary to Cancellation Laws

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

Then:

$h g = g \implies h = e$

$\ds h g$

$=$

$\ds g$

$\ds \leadsto \ \ $

$\ds h g$

$=$

$\ds e g$

Group Axiom $\text G 2$: Existence of Identity Element

$\ds \leadsto \ \ $

$\ds h$

$=$

$\ds e$

$\blacksquare$

$\ds h g$

$=$

$\ds g$

$\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

$\blacksquare$

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