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Question

State and prove cancellation laws on groups.

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Solution

Let G be a group.
Then for all a,b,cG
(i) ab=acb=c (Left cancellation law)
(ii) ba=cab=c (Right cancellation law)
Proof:
ab=ac
Pre multiplying by a1, we get
a1(ab)=a1(ac)
(a1a)b=(a1a)ceb=ec
(i.e)b=c
(ii) ba=ca
Post multiplying by a1, we get
(ba)a1=(ca)a1
b(aa1)=c(aa1)
(i.e.,) be=ceb=c

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