TITLE: Prove that middle cancellation implies that the group is abelian
QUESTION [5 upvotes]: 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).

To be an Abelian group, it is required that $\forall a,b \in G$, $a. b=b .a$
Can I request for a hint to this question? I'm not sure how to get started on this one.

REPLY [5 votes]: Hint: Given $a$ and $b$, find an element $x$ such that $axb=bxa$. (I can think of two $x$s that would work.)