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This question already has an answer here:

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.

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marked as duplicate by Martin Sleziak, user147263, Silvia Ghinassi, Jeel Shah, colormegone Feb 22 '16 at 16:14

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    $\begingroup$ Every element in a group has an inverse. In particular, $a$ has an inverse $a^{-1}$ and $a^{-1}ac=c$, $a^{-1}ab=b$. $\endgroup$ – Pedro Tamaroff Feb 12 '14 at 7:57
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If $ab=ac$, then $b=a^{-1}ab=a^{-1}ac=c$.

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

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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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