How to prove $b=c$ if $ab=ac$ (cancellation law in groups)?

Solution 1:

Suppose $$a\cdot b = a\cdot c$$ Let $a^{-1}$ be the inverse element of $a$ in $G$ (s.t. $a^{-1}\cdot a = a\cdot a^{-1} = e$ where $e$ is the identity element), which must exist by the axioms of groups. Now consider

$$a^{-1}\cdot(a \cdot b) =a^{-1}\cdot(a\cdot c)$$

By associativity, we have

$$(a^{-1}\cdot a)\cdot b = (a^{-1}\cdot a)\cdot c$$

By the definition of inverse, we have

$$e\cdot b = e\cdot c$$

where $e$ is the identity element (s.t. $e\cdot x = x\cdot e = x$ for all $x \in G$). By the definition of the identity element,

$$b = c$$

Solution 2:

Hint:

If you know that $4\cdot x = 4\cdot y$, how do you prove that $x=y$?

Hint 2:

Think about inverses

Solution 3:

$G$ is a group. One of the axioms of a group is that every element has an inverse. This means that $a\in G$ has an inverse $a^{-1} \in G$. This will help a lot.