Let $(G,*)$ be a group with
the following cancellation property: $x*a*y=b*a*c$ implies
$x*y=b*c$ for all $x$, $y$, $a$, $b$, $c$ in $G$. Prove that $G$
is abelian (that is, that the operation $*$ is commutative.)
Solution. Let
$x$ and $y$ denote any two members of the group. Then
x*x^{-1}*y = e*y = y = y*e=y*x^{-1}*x \Longrightarrow
x*y = y*x,
by the given cancellation rule.