Prove that the following ‘cancellation property’ holds in any group: ab = ac implies b = c, and ba = ca implies b = c.
Need help with this, don't know how to prove it for a group.
$\endgroup$ 12 Answers
$\begingroup$If $ab=ac$, then $b=a^{-1}ab=a^{-1}ac=c$.
If $ba=ca$, then $b=baa^{-1}=caa^{-1}=c$.
$\endgroup$ $\begingroup$Denote the unit of the group by $e$ then:
$ab=ac$ implies: $b=eb=(a^{-1}a)b=a^{-1}(ab)=a^{-1}(ac)=(a^{-1}a)c=ec=c$
$ba=ca$ implies: $b=be=b(aa^{-1})=(ba)a^{-1}=(ca)a^{-1}=c(aa^{-1})=ce=c$
Note that unit, inverse and associativity (characteristics of group) all play a part.
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