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

(1)

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:

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

Bibliography

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