For which groups does $ab=cd\implies ba=dc$? I just started studying group theory and I thought of this question which I couldn't solve. I tried making some progress by working on the group of functions of the form $f(x)=ax+b$, $a,b\in \mathbb{R}$ where the operation $*$ denotes composition of functions, but to no avail.
Here is the problem: Let the set $X$ be a non-abelian group with respect to some operation $*$. Find all $X$ such that $\forall a,b,c,d\in \mathbb{X}$, $a*b=c*d\implies b*a=d*c$.

As it's written, there is no such group. You can take $a$ to be the group identity. Then you have
$$ b = cd = dc$$
for all $c$ and $d$ in X. This implies the group is abelian, which is a contradiction.
