an associative quasigroup is a group
We will prove this in the following direction .
Let , and such that . So , which shows that . Let be such that . Then , so that , or . Set . For any , we have , so . Similarly, implies . This shows that is an identity of .
First note that all of the group axioms are automatically satisfied in under , except the existence of an (two-sided) inverse element, which we are going to verify presently. For every , there are unique elements and such that . Then . This shows that has a unique two-sided inverse . Therefore, is a group under .
Every group is clearly a quasigroup, and the binary operation is associative.
This completes the proof. ∎
Remark. In fact, if on is flexible, then every element in has a unique inverse: for , so by left division (by ), we get , and therefore , again by left division (by ). However, may no longer be a group, because associativity may longer hold.
|Title||an associative quasigroup is a group|
|Date of creation||2013-03-22 18:28:50|
|Last modified on||2013-03-22 18:28:50|
|Last modified by||CWoo (3771)|