# 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.

That is, the group product is cancellable.

## Proof 1

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

Suppose $b a = c a$.

Then:

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \left({b a}\right) a^{-1}$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle \left({c a}\right) a^{-1}$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \implies$$ $$\displaystyle$$ $$\displaystyle b \left({a a^{-1} }\right)$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle c \left({a a^{-1} }\right)$$ $$\displaystyle$$ $$\displaystyle$$ by associativity $$\displaystyle$$ $$\displaystyle \implies$$ $$\displaystyle$$ $$\displaystyle b e$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle c e$$ $$\displaystyle$$ $$\displaystyle$$ by the definition of inverse $$\displaystyle$$ $$\displaystyle \implies$$ $$\displaystyle$$ $$\displaystyle b$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle c$$ $$\displaystyle$$ $$\displaystyle$$ by the definition of identity

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

$\blacksquare$

## Proof 2

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.

$\blacksquare$

## Proof 3

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

$\blacksquare$