On Left Conjugacy Closed Loops with a Nucleus of Index Two

A loop Q is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. Loops in which the left and right nuclei coincide and are of index 2 are necesarilly LCC, and they are constructed in the paper explicitly. LCC loops Q with the right nucleus G of index 2 offer a larger diversity. A sample of results: if Z(G) = 1, then Q is also right conjugacy closed. For each m ≥ 2 one can construct Q of order 2m in such a way that its left nucleus is trivial. If Q is involutorial, then it is a Bol loop. There are three motivation sources for this paper: questions induced by recent progress in the structural theory of LCC loops [4]; results on involutorial Bol loops with the right nucleus of index 2 [7]; and the classical result of Goodaire and Robinson stating that every loop with nucleus of index 2 is conjugacy closed [5]. By repeating an argument from [5] one can prove easily that any loop Q with Nλ equal to Nμ and of index 2 has to be left conjugacy closed (LCC). Section 1 contains an explicit description of all such loops. Those of them that possess a trivial right nucleus are examined more closely, and it turns out that there is no such finite loop, while one can easily construct an infinite example. Section 2 is auxiliary. It is concerned with a general descriptions of permutations φ : G→ G, G a group, such that φ(x) = φ(x) for all x, y ∈ G. The description is used in Section 3 to characterize LCC loops Q with Q = NλNρ. Section 4 deals with general aspects of the situation |Q : Nρ| = 2, Q an LCC loop, and Section 5 gives details for the case of Nρ abelian. We shall observe that there is a large degree of freeness when constructing the latter loops. On the other hand, if Nρ has (as a group) a trivial centre, then this freeness is far more restricted—we shall see that such loops are conjugacy closed (i.e., both LCC and RCC), which means that all nuclei coincide and one can use the explicit description of Section 1. In Section 5 we shall construct various LCC loops which have a trivial left nucleus and an abelian right nucleus of index 2. We shall also observe that all involutorial LCC loops with the right nucleus of index two are Bol loops. The first papers dealing systematically with LCC loops seem to be [1] and [9]. The relevance of [7] is based on the fact that (left) Bol loops Q with x ∈ Nλ for all x ∈ Q are LCC, by [9]. For general properties of loops consult [2] or [3]. 2000 Mathematics Subject Classification. Primary 20N05; Secondary 08A05.