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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 $G$ be a group. Pick $a,b,c\in G$ such that $ca=cb$. Pick $c^{-1}\in G$ such that $c^{-1}$ is the inverse of $c$ in $G$. Thus we have $$\begin{align} a&=ea &\qquad\text{identity}\\ &=(c^{-1}c)a &\qquad\text{inverses}\\ &=c^{-1}(ca) &\qquad\text{associativity}\\ &=c^{-1}(cb) &\qquad\text{assumption from above}\\ &=(c^{-1}c)b &\qquad\text{associativity}\\ &=eb &\qquad\text{inverses}\\ &=b. &\qquad\text{identity} \end{align}$$

Similarly, pick $a,b,c\in G$ such that $ac=ab$ and $c^{-1}$ is the inverse of $c$ in $G$. Thus we have $$\begin{align} a&=ae &\qquad\text{identity}\\ &=a(cc^{-1}) &\qquad\text{inverses}\\ &=(ac)c^{-1} &\qquad\text{associativity}\\ &=(bc)c^{-1} &\qquad\text{assumption from above}\\ &=b(cc^{-1}) &\qquad\text{associativity}\\ &=be &\qquad\text{inverses}\\ &=b. &\qquad\text{identity} \end{align}$$

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