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Problem 49 (Cancellation Laws For Groups)
Suppose $G$ is a group. Let $a,b,c\in G$. Prove that if $ca=cb$, then $a=b$. A similar proof will show that if $ac=bc$, then $a=b$.
Solution
Let $ac = bc$. Since $G$ is a group, we know that $c^{-1} \in G$. Multiplying both sides by $c^{-1}$ on the right, we find that $(ac)c^{-1} = (bc)c^{-1}$. Because $G$ is a group, we know that these operations are associative. Thus, we know that $a(cc^{-1}) = b(cc^{-1})$. By the definition of the identity, it then follows that $ae = be$. It also follows from the definition of the identity that $a = b$.
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