On loop identities that can be obtained by a nuclear identification

We start by describing all the varieties of loops Q that can be defined by autotopisms αx, x ∈ Q, where αx is a composition of two triples, each of which becomes an autotopism when the element x belongs to one of the nuclei. In this way we obtain a unifying approach to Bol, Moufang, extra, Buchsteiner and conjugacy closed loops. We reprove some classical facts in a new way and show how Buchsteiner loops fit into the traditional context. In Section 6 we describe a new class of loops with coincinding left and right nuclei. These loops have remarkable properties and do not belong to any of the classical classes. We start this paper by investigating interactions of loop nuclei and loop identities via loop autotopisms. We shall observe that this is a natural way how to obtain nearly all loop varieties that have been studied in the past. To be more exact, we shall get in this way Moufang loops, left Bol loops, right Bol loops, extra loops, left conjugacy closed (LCC) loops, RCC loops and Buchsteiner loops. No other class of loops can be obtained by the method. In all these varieties the middle nucleus has to coincide with the left or the right nucleus. Furthermore, there exists no finite loop with trivial right nucleus and coinciding left and middle nuclei that would be of index two. However, there exist finite loops with coinciding left and middle nuclei that are of index two, in which the right nucleus is of order two. All such loops have to be left conjugacy closed. In Section 6 we shall construct a class of finite loops in which the left and right nuclei are coinciding and of index two, and the middle nucleus is of order two. Such loops clearly belong to none of the varieties mentioned above. After computing their multiplication groups we shall see that in many cases the inner mapping group is of order pq. Loops of this kind have been studied intensively in the past from the point of solvability of the multiplication group. This paper seems to describe the first known examples of such loops in which the nuclei do not coincide. The construction of Section 6 can be also used to present a new infinite series of finite G-loops. In Section 1 we explicitly describe the method of nuclear identification. This leads to a dozen of identities (cf. Table 1). By investigating them from the standpoint of the weak inverse property (WIP ) we obtain new conceptual proofs for a number of classical results. In Corollary 1.7 we point out that many of the investigated varieties can be described by normalizing properties involving the left 2000 Mathematics Subject Classification. Primary 20N05; Secondary 08A05.