Question

State and prove cancellation laws on groups.

Answer 1

Let $G$ be a group.

Then for all $a,b,c\in G$
(i) $a\ast b=a\ast c\Rightarrow b=c$ (Left cancellation law)
(ii) $b\ast a=c\ast a\Rightarrow b=c$ (Right cancellation law)
Proof:
$a\ast b=a\ast c$
Pre multiplying by $a^{-1}$, we get
$a^{-1}\ast (a\ast b)=a^{-1}\ast (a\ast c)$
$\Rightarrow (a^{-1}\ast a)\ast b=(a^{-1}\ast a)\ast c\Rightarrow e\ast b=e\ast c$
(i.e)$b=c$
(ii) $b\ast a=c\ast a$
Post multiplying by $a^{-1}$, we get
$(b\ast a)\ast a^{-1}=(c\ast a)\ast a^{-1}$
$b\ast (a\ast a^{-1})=c\ast (a\ast a^{-1})$
(i.e.,) $b\ast e=c\ast e\Rightarrow b=c$