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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({b a}\right) a^{-1}\)	\(=\)	\(\displaystyle \left({c a}\right) a^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle b \left({a a^{-1} }\right)\)	\(=\)	\(\displaystyle c \left({a a^{-1} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by associativity
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle b e\)	\(=\)	\(\displaystyle c e\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by the definition of inverse
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle b\)	\(=\)	\(\displaystyle c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	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 Elements of Semigroup Also Cancellable, it follows that all its elements are therefore cancellable.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$: Theorem $7.1$ Corollary
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.4$: Theorem $1$ Corollary
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 35.1$