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A Class of Loops Which are Isomorphic to all Loop Isotopes

It is convenient and not without precedent (see [2], [1], and also [6]) to call a loop which is isomorphic to all of its loop isotopes a G-loop. Since all groups are readily seen to be G-loops, the only interest in such loops, from a loop-theoretic standpoint, resides with those which are not associative. Examples and ad hoc constructions of such loops have appeared sporadically in the literature (see, for instance, [1], [2], [4], [6], [8], [9], and [13]). Any finite loop of order n < 5 is a group and, hence, must also be a G-loop. R. L. Wilson [11, 12, 13] proved that a finite G-loop of prime order is necessarily a group; he also exhibited for each even integer n > 5 a G-loop of order n which is not associative and then raised questions concerning the existence of finite G-loops which are not groups for every possible composite order n > 5.