Theorem 4.15
Return to Theorems, Glossary, Homework Problems.
Statement:
If $G$ is a group with binary operation $*$, then the left and right cancellation laws hold in $G$, that is, $a * b = a * c$ implies $b = c$, and $b * a = c * a$ implies $b = c$ for all $a, b, c \in G$.
Proof:
Suppose $a * b = a * c$. Then by $\mathscr{G}_{3}$, there exists $a'$, and
(1)\begin{equation} a' * (a * b) = a' * (a * c). \end{equation}
By the associative law, $\mathscr{G}_{1}$
(2)\begin{equation} (a' * a) * b = (a' * a) * c. \end{equation}
By the definition of $a'$ in $\mathscr{G}_{3}$, $a' * a = e$, so
(3)\begin{equation} e * b = e * c. \end{equation}
By the definition of $e$ in $\mathscr{G}_{2}$,
(4)\begin{equation} b = c. \end{equation}
Similarly, from $b * a = c * a$ one can deduce that $b = c$ upon multiplication on the right by $a'$ and use of the axioms for a group.