proof of example of medial quasigroup

We shall proceed by first showing that the algebraic systems defined in the parent entry ( are quasigroupsPlanetmathPlanetmath and then showing that the medial property is satisfied.

To show that the system is a quasigroup, we need to check the solubility of equations. Let x and y be two elements of G. Then, by definition of , the equation xz=y is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to


This is equivalent to


Since g is an automorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, there will exist a unique solution z to this equation.

Likewise, the equation zx=y is equivalent to


which, in turn is equivalent to


so we may also find a unique z such that zx=y. Hence, (G,) is a quasigroup.

To check the medial property, we use the definition of to conclude that

(xy)(zw) = (f(x)+g(y)+c)(f(z)+g(w)+c)
= f(f(x)+g(y)+c)+g(f(z)+g(w)+c)+c

Since f and g are automorphisms and the group is commutativePlanetmathPlanetmathPlanetmathPlanetmath, this equals


Since f and g commute this, in turn, equals


Using the commutative and associative laws, we may regroup this expression as follows:


Because f and g are automorphisms, this equals


By defintion of , this equals


which equals (xz)(yz), so we have


Thus, the medial property is satisfied, so we have a medial quasigroup.

Title proof of example of medial quasigroup
Canonical name ProofOfExampleOfMedialQuasigroup
Date of creation 2013-03-22 16:27:35
Last modified on 2013-03-22 16:27:35
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Proof
Classification msc 20N05