division in group

In any group (G,) one can introduce a division operationMathworldPlanetmath ‘‘:’’ by setting


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

xy=x:((y:y):y),x-1=(x:x):x. (1)

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

  1. 1.


  2. 2.


  3. 3.


The above result may be conversed:


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

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


and using the property 3,


Then we get:



  • 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 TheoremMathworldPlanetmath
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