Groups with anti-symmetric mappings

The above theorems are sufficient to demonstrate the existence of antisymmetric mappings for all non-Abelian group of order less than 36, except for the group <a, b\a³ =b⁸ =e, ab =ba²> of order 24, and this group can be shown to have one. On the basis of our results, and the fact that we have no conditions on a non-Abelian group that would eliminate any from having anti-symmetric mappings, we conjecture that all non-Abelian groups have anti-symmetric mappings.