I am looking for the paper
Graham Higman and Bernhard Hermann Neumann, Groups as groupoids with one law, Publicationes Mathematicae Debrecen 2 (1952), 215–221.
In it, the authors prove (among other things) that one can axiomatize a group as a set $X$ together with a single binary operation $\rho\colon X\times X\to X$ (which is morally right division), where we use reverse Polish notation to write $xy\rho$ for $\rho(x,y)$, and a single axiom: for all $x$, $y$, $z\in X$, we have $$xxx\rho y\rho z\rho xx\rho x\rho z\rho\rho\rho=y.$$ Written with $/$, instead of $\rho$, we have $$\frac{x}{\frac{\frac{x/x}{y}}{z}\Big/\frac{\frac{x/x}{x}}{z}}=y,$$ or $$x \Biggl/ \Biggl(\Bigl(\bigl((x / x) / y\bigr) / z\Bigr) \Big/ \Bigl(\bigl((x / x) / x\bigr) / z\Bigr)\Biggr) = y.$$
It is MR57866 on MathSciNet. The journal webpage archives only go back to volume 40. Google Books and Google Scholar do not reveal any online copies. I did not find any used copies on Abebooks. I was unable to find any useful information on WorldCat.
Thanks.
I believe I've answered the question as declared in the comments. Here is a prank.
Let us parse the Higman-Neumann formula and turn it into a diagram in the sense of category theory. This will allow us to reformulate the Higman-Neumann formula without any variables. We have:
$$\boxed{x \,\boxed{\boxed{\boxed{\boxed{xx\rho\,} \,y\rho} \,z\rho}\;\; \boxed{\boxed{\boxed{xx\rho} \, x\rho} \, z\rho}\, \rho}\, \rho}=y.$$
Let's set up the notation: denote by $G$ the underlying set, by $G^k$ the $k$-th cartesian power of $G$, $\pi_2: G^3\to G, (x,y,z)\mapsto y$, $\Delta_4:G\to G^4, x\mapsto (x,x,x,x)$, $\operatorname{id}=\Delta_1$. Any permutation in $S_k$ acts on $G^k$ by permuting the coordinates; put $ \sigma=(4,8,5,7)\in S_9$. Then we have that the following diagram defines a unique group structure on $G$:
Written as a formula, $(G,\rho)$ is a group iff
\begin{align*} \rho &\circ (\Delta_1\times\rho) \circ (\Delta_1\times \rho\times\rho) \circ (\Delta_1\times\rho\times \Delta_1\times\rho\times \Delta_1) \\ &\circ (\Delta_1\times\rho\times\Delta_1\times\Delta_1\times\rho\times\Delta_1\times\Delta_1) \circ \sigma\circ(\Delta_6\times\Delta_1\times\Delta_2) = \pi_2. \end{align*}
(Jokes aside, this prank points to something potentially interesting: It is known that many algebraic structures can be defined purely in terms of diagrams. Certainly groups can be defined via diagrams (see e.g. http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/group+object), whose homotopy type we can consider, and in this sense the Higman-Neumann formula is simpler than the standard formulation. Thus one can say that the fundamental group of the group object functor is $\mathbb{Z}$. In general given an algebraic object functor one can define its fundamental group as the group of "minimal rank", where the group varies among the fundamental groups of all "geometric realizations" of the functor. As a disclaimer, it is not clear to me if this is indeed a non-trivial line of inquiry, or if it's already been completed.)