The General Validity of $ab=cd$ Implies $ba=dc$. I am doing some research of my own and I have a brief question and I don't recall studying this particular thing.  Is it generally true that if a set coupled with a binary operation is closed under inverses and has an identity then $ab=cd$ implies $ba=dc$ for all $a,b,c,d$ in the set?  If not, would this statement hold true for groups?

Consider $GL(2,\mathbb R)$. It is a non empty set with binary associative composition, unit and inverses. 
Let $A$ be the unit matrix, $B=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$, $C=\
\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$ and $D=\
\begin{bmatrix}-1 & 1\\1 & 0\end{bmatrix}$.
Then $AB=B=CD$; but $B=DC$ is not true.
