Cancellation Laws

Theorem
Let $G$ be a group.

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

Then:
 * $b a = c a \implies b = c$
 * $a b = a c \implies b = c$

These are respectively called the right and left cancellation laws.

Proof 1
Let $a, b, c \in G$ and let $a^{-1}$ be the inverse of $a$.

Suppose $b a = c a$.

Then:

Thus, the right cancellation law holds. The proof of the left cancellation law is analogous.

Proof 2
From its definition, a group is a monoid, all of whose elements have inverses and thus are invertible.

From Invertible Elements of Semigroup Also Cancellable, it follows that all its elements are therefore cancellable.