Finite FRUTE loops

Abstract

A loop [Formula: see text] is called a FRUTE loop if it obeys the identity [Formula: see text]. Interestingly, a FRUTE loop is a Moufang loop but not necessarily an extra loop or a group (and vice versa). In this paper, algebraic properties of the left (right) regular representation set of a FRUTE loop are deduced. A FRUTE loop is shown to be universal and an [Formula: see text]-loop for all [Formula: see text]. A Moufang loop is shown to be a FRUTE loop if and only if it is nuclear cube if and only if it is an [Formula: see text]-loop. It is established that: the smallest, associative, non-commutative FRUTE loop is of order [Formula: see text] (the quaternion group [Formula: see text]); for any [Formula: see text], there exists at least a non-commutative group of order [Formula: see text] that is a FRUTE loop; there exists [Formula: see text]-groups of orders [Formula: see text] that are non-commutative FRUTE loops; there are no non-commutative groups that are FRUTE loops of the following range of orders [Formula: see text]; there are two non-associative FRUTE loops of order [Formula: see text] up to isomorphism and there are six non-isomorphic, non-associative FRUTE loops of order [Formula: see text]. It is noted that there exists a non-associative and non-commutative FRUTE loop of order [Formula: see text]. The study is concluded with some questions, conjectures and problem.