Cancellation laws - Imported Knowlpedia Content

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 $b=c$ .

Equivalent characterizations

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.

Remarks

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.