Cancellation property in groups

Question

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.

Every element in a group has an inverse. In particular, $a$ has an inverse $a^{-1}$ and $a^{-1}ac=c$, $a^{-1}ab=b$. — Pedro, Feb 12 '14 at 07:57

score 3 · Answer 1 · answered Feb 12 '14 at 07:57

3

If $ab=ac$, then $b=a^{-1}ab=a^{-1}ac=c$.

If $ba=ca$, then $b=baa^{-1}=caa^{-1}=c$.

answered Feb 12 '14 at 07:57

score 2 · Answer 2 · answered Feb 12 '14 at 08:43

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.

2 Answers2