Cancellation laws

Proposition (Cancellation laws). Let $G$ be a group and let $a,b,c\in G$.

• (Left cancellation) If $ab=ac$, then $b=c$.
• (Right cancellation) If $ba=ca$, then $b=c$.

Equivalently, for each fixed $a\in G$, the left-translation map $L_a:G\to G$, $L_a(x)=ax$, and the right-translation map $R_a:G\to G$, $R_a(x)=xa$, are injective.

Context. Cancellation is the algebraic shadow of invertibility: you “cancel” by multiplying by $a^{-1}$ on the appropriate side. Uniqueness of inverses (see uniqueness of inverses ) ensures $a^{-1}$ is well-defined.

Proof sketch. If $ab=ac$, multiply on the left by $a^{-1}$ to get $b=c$. The right cancellation law is analogous.