Question

If$ab=ac$ in a group$G$, prove that$b=c$ .

Short answer

It is proved that$b=c$ .

Step-by-step solution

Inverse of an element in a group

Inverse of an element $a\in G$in a group$G$ is defined as another element $a^{-1}\in G$such that $a\ast a^{-1}=a^{-1}\ast a=e$where,$e\in G$is the identity element in $(G,\ast )$.

Proof

It is given that$ab=ac$ in some group$G$

Now, multiplying both sides from the left of the above equation by, $a^{-1}$we get,

$\begin{array}{l}{a}^{-1}ab=a^{-1}ac\\ \Rightarrow eb=ec\\ \Rightarrow b=c\end{array}$

Hence, it is proved that .$b=c$