Simple quasigroups with commuting inner permutations are medial.
@article{118575, author = {Tom\'a\v s Kepka and K. K. \v S\v cukin}, title = {Simple quasigroups whose inner permutations commute}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {34}, year = {1993}, pages = {223-227}, zbl = {0784.20035}, mrnumber = {1241731}, language = {en}, url = {http://dml.mathdoc.fr/item/118575} }
Kepka, Tomáš; Ščukin, K. K. Simple quasigroups whose inner permutations commute. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 223-227. http://gdmltest.u-ga.fr/item/118575/
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