Structural Interactions of Conjugacy Closed Loops

We study conjugacy closed loops by means of their multiplication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication groups, respectively. Put M = {a ∈ Q; La ∈ R}. We prove that the cosets of A agree with orbits of [L,R], that Q/M ∼= (InnQ)/L1 and that one can define an abelian group on Q/N × Mlt1. We also explain why the study of finite conjugacy closed loops can be restricted to the case of N/A nilpotent. Group [L,R] is shown to be a subgroup of a power of A (which is abelian), and we prove that Q/N can be embedded into Aut([L,R]). Finally, we describe all conjugacy closed loops of order pq. Conjugacy closed loops have been defined independently by Soikis [15] and by Goodaire and Robinson [7], [8]. More recent results concerning their structure have been obtained by Basarab [1], Kunen [10], Drápal [4] and by Kinyon, Kunen and Phillips [11]. This paper can be regarded as a sequel to [4] since its main concern rests in exploring how various features of conjugacy closed loops are manifested in their multiplication groups. We shall use several results of [11] and end by setting down the structure of conjugacy closed loops of order pq. Their study was initiated by Goodaire and Robinson in [7] and resumed by Kunen [10], who proved that if a nonassociative conjugacy closed loop of such an order exists, then q < p has to divide p − 1. Kunen also fully described the case q = 2. Here we shall prove that whenever q divides p−1, then there can be constructed, up to isomorphism, exactly one nonassociative conjugacy closed loop of order pq. Its operation can be given by the formula (i, r) · (j, s) = (i+ jγ + (1− γ)(1− γ), r + s), where i and j are integers modulo p, r and s are integers modulo q, and γ is a fixed integer whose multiplicative order modulo p is equal to q. Unlike in [10], we shall not construct the loop by considering equalities of loop terms, but we shall derive it from knowledge of the loop’s multiplication group. To get the structure of the multiplication group we shall use various facts, some of which have appeared in [11] and [4], and some of which will be proved in this paper. In fact, we shall obtain quite a few structural results, and not all of them will be needed for the case pq. Before turning to their short overview, we shall list those properties of conjugacy closed loops that will be used throughout this paper Received by the editors June 3, 2003 and, in revised form, August 29, 2005. 2000 Mathematics Subject Classification. Primary 20N05; Secondary 08A05.