division in group

In any group $(G,\cdot )$ one can introduce a division operation ‘‘:’’ by setting

$$x:y=x\cdot y^{-1}$$

for all elements $x$, $y$ of $G$.  On the contrary, the group operation and the unary inverse forming operation may be expressed via the division by

$x\cdot y=x:((y:y):y),x^{-1}=(x:x):x.$(1)

The division, which of course is not associative, has the properties

1. $(x:z):(y:z)=x:y,$

2. $x:(y:y)=x,$

3. $(x:x):(y:z)=z:y.$

The above result may be conversed:

Theorem.

If the operation ‘‘:’’ of the non-empty groupoid $G$ has the properties 1, 2, and 3, then $G$ equipped with the ‘‘multiplication’’ and inverse forming by (1) is a group.

Proof.  Here we prove only the associativity of ‘‘$\cdot$’’.  First we derive some auxiliary results.  Using definitions and the properties 1 and 2 we obtain

$$(x:y):y^{-1}=(x:y):((y:y):y)=x:(y:y)=x,$$

$$(x:y^{-1}):y=(x:y^{-1}):((y:y):y^{-1})=x:(y:y)=x$$

and using the property 3,

$${(x:y)}^{-1}=((x:y):(x:y)):(x:y)=y:x.$$

Then we get:

$$(xy)z=(x:y^{-1}):z^{-1}=((x:y^{-1}):y):(z^{-1}:y)=x:(z^{-1}:y)=x:{(y:z^{-1})}^{-1}=x(yz)$$