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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
    Commented Aug 4, 2017 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
    Commented Aug 4, 2017 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
    Commented Aug 4, 2017 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
    Commented Aug 4, 2017 at 14:29
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    $\begingroup$ The article doi.org/10.1007/s00407-017-0193-8 gives a whole list of references on axiomatizations of groups in terms of division (footnotes 6 and 62), and also cites a paper by Wiener, but the Wiener paper seems to be on fields, not groups. So Rota might have gotten it mixed up, or it's implicit in the work on fields, or he's referring to unpublished work by Wiener (it's not even clear to me if it's the same Wiener). $\endgroup$
    – Jonas Frey
    Commented Aug 2, 2023 at 2:22

1 Answer 1

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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
    Commented Aug 20, 2017 at 15:30
  • $\begingroup$ I have no reference for this. $\endgroup$
    – user44143
    Commented Aug 20, 2017 at 15:42
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    $\begingroup$ Why not denote the operator by its usual symbol $x/y$? $\endgroup$
    – Emil Jeřábek
    Commented Aug 20, 2017 at 15:45

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