Cancellation Laws/Proof 2

Theorem

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

From its definition, a group is a monoid, all of whose elements have inverses and thus are invertible.

From Invertible Element of Monoid is Cancellable, it follows that all its elements are therefore cancellable.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Theorem $7.1$ Corollary