After proving that identity elements and inverse elements are unique, we now prove the cancellation laws in a group. Cancellation is one of the most useful algebraic tools: it allows us to remove a common factor from the left or from the right of an equation. In ordinary arithmetic, students cancel numbers almost automatically, but in abstract algebra cancellation must be justified from the group axioms. The proof uses inverses and associativity. In this lesson, students will learn the left cancellation law, the right cancellation law, and the correct way to apply them in group equations.

Let $(G,\circ)$ be a group. The $\textbf{left cancellation law}$ says that for all $a,b,c\in G$, $$\begin{equation*} a\circ b=a\circ c \implies b=c. \end{equation*}$$ It means that a common left factor may be cancelled from both sides of a group equation.