Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian. Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian. (Middle cancellation implies commutativity).
I am having trouble with this homework problem. The way I started was:
Suppose $G$ is a group for which middle cancellation holds. Multiply $x$ by its inverse, $x^{-1}$ using middle cancellation so you have $ab=cd$. Thus $G$ is Abelian. 
I am unsure if this is the way to approach it.

Beginning with $b = b$, which is equivalent to $eb = be$, observe that we have $aa^{-1}b = ba^{-1}a$.
Thus, by hypothesis, $ab = ba$. Hence the group is abelian.
