TITLE: Wiener's axiomatization of the group law based on division
QUESTION [9 upvotes]: Gian-Carlo Rota wrote that [*]:

Wiener axiomatized the group law by taking $xy^{-1}$ as the basic operation, and his axiomatization is quite different from any of the other axiom systems for groups. 

Does anyone know what axiomatization Rota is referring to?
[*] In "The Barber of Seville, or The Useless Precaution", from Indiscrete Thoughts.

REPLY [6 votes]: One surprisingly nice axiomatization for this operator is:
$$x/x=1$$
$$x/1=x$$
$$(x/z)\,/\,(y/z)=x/y$$
The proof is straightforward, and most easily dealt with by writing it out in detail:
We translate sentences from group theory with $(x^{-1})^*=1/x$, $(xy)^*=x/(1/y)$.
The first two axioms of group theory then are
\begin{align}
a1=a&:  a\,/\,(1/1)=a/1=a\\
1a=a&:  1\,/\,(1/a)=(a/a)\,/\,(1/a)=a/1=a\\
aa^{-1}=1&:  a\,/\,(1/(1/a))=a/a=1\\
a^{-1}a=1&:  (1/a)\,/\,(1/a)=1.\\
\end{align}
Associativity is
$$(ab)c=a(bc):  (a/(1/b))\,/\,(1/c)=a\,/\,(1/(b/(1/c)))$$
which is a special case of
\begin{align}
(a/(1/b))\,/\,d
&=(a/(1/b))\,/\,(d/1)\\
&=(a/(1/b))\,/\,((d/b)/(1/b))\\
&=a\,/\,(d/b)\\
&=a\,/\,((d/d)/(b/d))\\
&=a\,/\,(1/(b/d)).
\end{align}