Conclusion
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\a 3 =b 8 =e, ab =ba 2> 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 allnon-Abelian groups have anti-symmetric mappings.
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Gallian, J.A., Mullin, M.D. Groups with anti-symmetric mappings. Arch. Math 65, 273–280 (1995). https://doi.org/10.1007/BF01195537
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DOI: https://doi.org/10.1007/BF01195537