What is a simple axiomatisation of groups using division?

Question

I recall from an old exercise I did as an undergrad that groups can be axiomatised using division rather than multiplication:

A group is a non-empty set equipped with a binary division operator / satisfying some equational axioms.

I don't remember the axioms, but I do remember that $x/y = x\cdot y^{-1}$. Because $x/x = 1$ and $1/x = x^{-1}$ we can recover the group structure by setting $x\cdot y = x/(1/y)$ and so such an axiomatisation should exist.

However, I am interested in a simple, preferably natural, axiomatisation. Does anyone know one?

Also, if you know a classic reference for this result/exercise, I will be grateful.

Answer 1

Set $x/x=1$ for all $x$ in $G$ and $x/1=x$, so there is a unique element $1$ and for all $x/x$ is equal to that. $(x/(1/y))/(1/z)= x/(1/(y/(1/z))$ is clearly equivalent to the associative group law where $x/(1/y)=x*y$ . So we have associative law and the $1$ defined is certainly an identity. For inverses, you can have define $1/(1/x)=x$ so $x*1/x=1$ as desired. Perhaps there is a cleaner formulation of the associative rule though the others seem natural enough.