Cancellation laws

Statement

Let $G$ be a group and let $a, b, c \in G$. If $ab = ac$, then $b = c$. If $ba = ca$, then $b = c$.

Remarks

The first law is left cancellation and the second law is right cancellation. To prove the first law, multiply $ab = ac$ on the left by $a^{-1}$. A consequence is that in the Cayley table of a finite group, no element occurs two times in one row or in one column. The laws can fail in a monoid: in $(\mathbb{Z}, \cdot)$, $0 \cdot 1 = 0 \cdot 2$ but $1 \neq 2$.

Explanations

L1 human

In a group, you can remove the same element from both sides of an equation, if it is on the same side of each product. This works because each element has an inverse that undoes it. Take care with the side: $ab = ca$ does not give $b = c$ in general.