Abstract:T.~Kepka, M.K.~Kinyon and J.D.~Phillips [{\it The structure of F-quasigroups\/}, J. Algebra {\bf 317} (2007), no.~2, 435--461] developed a connection between F-quasigroups and NK-loops. Since NK-loops are contained in the variety generated by groups and commutative Moufang loops, a question that arises is whether or not there exists a nonsplit NK-loop and likewise a nonsplit F-quasigroup. Here we prove that there do indeed exist nonsplit F-quasigroups and show that there are exactly four corresponding nonsplit NK-loops of minimal order~$3^6$.
Keywords: F-quasigroup, NK-loop, Moufang loop
AMS Subject Classification: 20E10 20N05