alternative definition of group

The below theorem gives three conditions that form alternative postulatesMathworldPlanetmath.  It is not hard to show that they hold in the group defined ordinarily.


Let the non-empty set G satisfy the following three conditions:
I.     For every two elements a, b of G there is a unique element ab of G.
II.   For every three elements a, b, c of G the equation(ab)c=a(bc)  holds.
III. For every two elements a and b of G there exists at least one such element x and at least one such element y of G that  xa=ay=b.
Then the set G forms a group.

Proof.  If a and b are arbitrary elements, then there are at least one such ea and such eb that  eaa=a  and  beb=b.  There are also such x and y that  xb=ea  and  ay=eb.  Thus we have


i.e. there is a unique neutral elementPlanetmathPlanetmath e in G.  Moreover, for any element a there is at least one couple a, a′′ such that  aa=aa′′=e.  We then see that


i.e. a has a unique neutralizing element a.

Title alternative definition of group
Canonical name AlternativeDefinitionOfGroup
Date of creation 2013-03-22 15:07:58
Last modified on 2013-03-22 15:07:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Theorem
Classification msc 20A05
Classification msc 20-00
Classification msc 08A99
Related topic Characterization
Related topic ACharacterizationOfGroups
Related topic DivisionInGroup
Related topic MoreOnDivisionInGroups
Related topic LoopAndQuasigroup