Cancellation Laws/Proof 1

Theorem

Let $G$ be a group.

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

Then the following hold:

$b a = c a \implies b = c$

$a b = a c \implies b = c$

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

Suppose $b a = c a$.

Then:

$\ds \paren {b a} a^{-1}$

$=$

$\ds \paren {c a} a^{-1}$

$\ds \leadsto \ \ $

$\ds b \paren {a a^{-1} }$

$=$

$\ds c \paren {a a^{-1} }$

$\ds \leadsto \ \ $

$\ds b e$

$=$

$\ds c e$

$\ds \leadsto \ \ $

$\ds b$

$=$

$\ds c$

Thus, the right cancellation law holds.

The proof of the left cancellation law is analogous.

$\blacksquare$

1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 35.1$: Elementary consequences of the group axioms