Cancellation Laws/Corollary 2/Proof 2

Corollary to Cancellation Laws

$h g = g \implies h = e$

Proof

\(\ds h g\)

\(=\)

\(\ds g\)

\(\ds \leadsto \ \ \)

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

\(=\)

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

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: Exercise $\text{D}$