# division in group

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

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

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

1. 1.

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

2. 2.

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

3. 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:

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

## References

• 1 А. И. Мальцев: Алгебраические  системы.  Издательство  ‘‘Наука’’. Москва (1970).
 Title division in group Canonical name DivisionInGroup Date of creation 2013-03-22 15:08:01 Last modified on 2013-03-22 15:08:01 Owner pahio (2872) Last modified by pahio (2872) Numerical id 13 Author pahio (2872) Entry type Theorem Classification msc 08A99 Classification msc 20A05 Classification msc 20-00 Related topic Group Related topic Division Related topic Groupoid Related topic AlternativeDefinitionOfGroup Defines division groupoid