Cancellation Theorem

Suppose $G$ is a Group. Then we have left cancellation and right cancellation, defined as:

1. If $a,b,c\in G$ and $a\star b=a\star c$ then $b=c$ (left cancellation)
2. If $a,b,c\in G$ and $b\star a=c\star a$ then $b=c$ (right cancellation)

We cannot say that $a\star b=c\star a$ implies $b=c$ (only when $\star$ is commutative).