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.

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    $\begingroup$ It's not difficult to imagine an axiom system based on division actually... IMHO in the narration of "Indiscrete thoughts" it makes more sense this other example: Wiener axiomatized the fields laws by taking $x@y:=1-x/y$ as the only basic operation, and his axiomatization is quite different from the usual axiom systems for fields (but has analogies with Sheffer's axiomatization of Boolean algebras). See "A set of postulates for fields, Trans. Amer. Math. Soc. 21 (1920), 237-246" $\endgroup$ – Luca Ghidelli Aug 4 '17 at 13:51
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    $\begingroup$ By "it's not difficult to imagine" I mean that if one defines $x@y:=xy^{-1}$, then one can recover the identity by $e:=x@x$ for any $x$, the inverse by $x^{-1}:=(x@x)@x$ and the multiplication by $x*y:=x@((y@y)@y)$. From these observations, it's now straightforward to axiomatize. $\endgroup$ – Luca Ghidelli Aug 4 '17 at 14:00
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    $\begingroup$ And as a third comment, there is the article "G. Higman and B. H. Neumann. Groups as groupoids with one law. Publicationes Mathematicae Debrecen, 2:215--227, 1952" on the axiomatization of group theory with only one law $x/y:=xy^{-1}$ and only one axiom: $x / ((((x / x) / y) / z) / (((x / x) / x) / z)) = y$. See also works of McCune, Kinyon... but unfortunately I know nothing of Wiener, nor I see anything from bibliography: "Bibliography of Norbert Wiener. Bull. Amer. Math. Soc. 72 (1966), no. Number 1, Part 2, 135--145". $\endgroup$ – Luca Ghidelli Aug 4 '17 at 14:27
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    $\begingroup$ I agree it's not hard to come up with axioms, but I was wondering specifically what axiomatization Rota was referring to. But that's interesting if he actually had in mind Wiener's 1920 paper on fields that you cited. $\endgroup$ – Noam Zeilberger Aug 4 '17 at 14:29

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}

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    $\begingroup$ Indeed, this is the axiomatization I had in mind! Do you know of a reference? These axioms are closely related to (actually, an instance of) the axioms for a skew-closed category, and I am interested in where they first appeared in the literature on groups. (I was wondering whether this is what Rota was referring to, though it seems from Luca Ghidelli's comment that he was likely referring to something else.) $\endgroup$ – Noam Zeilberger Aug 20 '17 at 15:30
  • $\begingroup$ I have no reference for this. $\endgroup$ – Matt F. Aug 20 '17 at 15:42
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    $\begingroup$ Why not denote the operator by its usual symbol $x/y$? $\endgroup$ – Emil Jeřábek Aug 20 '17 at 15:45

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