an associative quasigroup is a group

Proposition 1.

Let G be a set and a binary operationMathworldPlanetmath on G. Write ab for ab. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    (G,) is an associative quasigroupPlanetmathPlanetmath.

  2. 2.

    (G,) is an associative loop.

  3. 3.

    (G,) is a group.


We will prove this in the following direction (1)(2)(3)(1).


Let xG, and e1,e2G such that xe1=x=e2x. So xe12=xe1=x, which shows that e12=e1. Let aG be such that e1a=x. Then e2e1a=e2x=x=e1a, so that e2e1=e1=e12, or e2=e1. Set e=e1. For any yG, we have ey=e2y, so y=ey. Similarly, ye=ye2 implies y=ye. This shows that e is an identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of G.


First note that all of the group axioms are automatically satisfied in G under , except the existence of an (two-sided) inverse element, which we are going to verify presently. For every xG, there are unique elements y and z such that xy=zx=e. Then y=ey=(zx)y=z(xy)=ze=z. This shows that x has a unique two-sided inverseMathworldPlanetmathPlanetmathPlanetmath x-1:=y=z. Therefore, G is a group under .


Every group is clearly a quasigroup, and the binary operation is associative.

This completesPlanetmathPlanetmathPlanetmathPlanetmath the proof. ∎

Remark. In fact, if on G is flexible, then every element in G has a unique inverse: for z(xz)=(zx)z=ez=z=ze, so by left division (by z), we get xz=e=xy, and therefore z=y, again by left division (by x). However, G may no longer be a group, because associativity may longer hold.

Title an associative quasigroup is a group
Canonical name AnAssociativeQuasigroupIsAGroup
Date of creation 2013-03-22 18:28:50
Last modified on 2013-03-22 18:28:50
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Derivation
Classification msc 20N05
Related topic Group