Buchsteiner Loops: Associators and Constructions

Let Q be a Buchsteiner loop. We describe the associator calculus in three variables, and show that |Q| ≥ 32 if Q is not conjugacy closed. We also show that |Q| ≥ 64 if there exists x ∈ Q such that x is not in the nucleus of Q. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that is nilpotency class 3 and possesses an abelian inner mapping group. Buchsteiner loops are those loops that satisfy the Buchsteiner law x\(xy · z) = (y · zx)/x. Their study was initiated by Hans Hoenig Buchsteiner [2]. His paper left many problems open, some of which were recently solved [8]. In particular we know now that the nucleus N = N(Q) is a normal subloop of every Buchsteiner loop Q and that Q/N is an abelian group of exponent four. Buchsteiner loops are closely connected to conjugacy closed loops (CC loops). A CC loop is conjugacy closed if and only if Q/N is a boolean group (i.e. a group of exponent two), by [9]. Not every Buchsteiner loop with Q/N boolean needs to be conjugacy closed (there are plenty of examples now. Some of them appear in this paper, and many other can be derived from the ring construction of [7].) In every Buchsteiner loop Q the mappings L xy LxLy and R −1 yxRxRy are automorphisms of Q, by [8], and this fact effects the behaviour of the associators [x, y, z] = (x · yz)\(xy · z). The group Q/A(Q) acts upon N = N(Q) (that always holds when Q/N is a group since then A(Q) ≤ Z(N(Q)), by [10]. Here A(Q) denotes the least normal subloop A Q such that Q/A is a group. If Q/N is a group, then A(Q) coincides with the subgroup generated by all associators [x, y, z], by [11]). By translating the automorphism behaviour of L xyLxLy into relations between associators we get that [x, y, uv] = [x, y, u][x, y, v] for all x, y, u, v ∈ Q. Note that n is defined as v\(nv), for all v ∈ Q and n ∈ N . If Q is a loop such that Q/N is a group, then one can code the Buchsteiner identity as [x, y, z] = [y, z, x] for all x, y, z ∈ Q. The cyclic shift expressed by this action implies that in every Buchsteiner loop we have [x, y, uv] = [x, y, u][x, y, v], [x, uv, y] = [x, u, y][x, v, y], [uv, x, y] = [u, x, y][v, x, y] 2000 Mathematics Subject Classification. Primary 20N05; Secondary 08A05.