Cancellation Laws/Proof 1

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

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

Suppose $b a = c a$.

Then:

 $\displaystyle \paren {b a} a^{-1}$ $=$ $\displaystyle \paren {c a} a^{-1}$ $\displaystyle \leadsto \ \$ $\displaystyle b \paren {a a^{-1} }$ $=$ $\displaystyle c \paren {a a^{-1} }$ Definition of Associative Operation $\displaystyle \leadsto \ \$ $\displaystyle b e$ $=$ $\displaystyle c e$ Definition of Inverse Element $\displaystyle \leadsto \ \$ $\displaystyle b$ $=$ $\displaystyle c$ Definition of Identity Element

Thus, the right cancellation law holds.

The proof of the left cancellation law is analogous.

$\blacksquare$