Finite FRUTE loops

“

It is established that: the smallest, associative, non-commutative FRUTE loop is of order 8 (the quarternion group Q_8); for any n between 1 and 12, there exists at least a non-commutative group of order 8n that is a FRUTE loop; there exists 2-groups of orders 8, 16, 32, 64 that are non-commutative FRUTE loops; there are no non-commutative groups that are FRUTE loops of the following range of orders 1-7, 9-15, 17-23, 25-31, 33-39, 41-47, 49-55, 57-63, 65-71, 73-79, 81-87, 89-95; there are 2 non-associative FRUTE loops of order 81 up to isomorphism and there are 6 non-isomorphic, non-associative FRUTE loops of order 243. It is noted that there exists a non-associative and non-commutative FRUTE loop of order 648. The study is concluded with some questions, conjectures and problem.

”