Cancellation Laws/Corollary 1/Proof 2

Corollary to Cancellation Laws

$g h = g \implies h = e$

Proof

\(\ds g h\)

\(=\)

\(\ds g\)

\(\ds \leadsto \ \ \)

\(\ds g^{-1} \paren {g h}\)

\(=\)

\(\ds g^{-1} g\)

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

$\blacksquare$

Sources

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