proof of example of medial quasigroup
We shall proceed by first showing that the algebraic systems defined in the parent entry (http://planetmath.org/MedialQuasigroup) are quasigroups and then showing that the medial property is satisfied.
This is equivalent to
Since is an automorphism, there will exist a unique solution to this equation.
Likewise, the equation is equivalent to
which, in turn is equivalent to
so we may also find a unique such that . Hence, is a quasigroup.
To check the medial property, we use the definition of to conclude that
Since and are automorphisms and the group is commutative, this equals
Since and commute this, in turn, equals
Because and are automorphisms, this equals
By defintion of , this equals
which equals , so we have
Thus, the medial property is satisfied, so we have a medial quasigroup.
|Title||proof of example of medial quasigroup|
|Date of creation||2013-03-22 16:27:35|
|Last modified on||2013-03-22 16:27:35|
|Last modified by||rspuzio (6075)|