Cancellation Laws/Proof 3

Corollary to Group has Latin Square Property

Let $G$ be a group.

Let $a, b, c \in G$.

Then the following hold:

Right cancellation law

$b a = c a \implies b = c$

Left cancellation law

$a b = a c \implies b = c$

Proof

Suppose $x = b a = c a$.

By Group has Latin Square Property, there exists exactly one $y \in G$ such that $x = y a$.

That is, $x = b a = c a \implies b = c$.

Similarly, suppose $x = a b = a c$.

Again by Group has Latin Square Property, there exists exactly one $y \in G$ such that $x = a y$.

That is, $a b = a c \implies b = c$.

$\blacksquare$

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.4$: Theorem $1$ Corollary