Buchsteiner Loops

Buchsteiner loops are those which satisfy the identity x\(xy · z) = (y · zx)/x. We show that a Buchsteiner loop modulo its nucleus is an abelian group of exponent four, and construct an example where the factor achieves this exponent. A loop (Q, ·) is a set Q together with a binary operation · such that for each a, b ∈ Q, the equations a · x = b and y · a = b have unique solutions x, y ∈ Q, and such that there is a neutral element 1 ∈ Q satisfying 1 · x = x · 1 = x for every x ∈ Q. Standard references in loop theory are [4, 6, 32]. The variety (i.e., equational class) of all loops being too broad for a detailed structure theory, most investigations focus on particular classes of loops. In this paper we investigate a variety of loops which has not hitherto received much attention, despite the fact that it is remarkably rich in structure, namely the variety defined by the identity (B) x\(xy · z) = (y · zx)/x. Here a\b denotes the unique solution x to a · x = b, while b/a denotes the unique solution y to y · a = b. We call (B) the Buchsteiner law and a loop satisfying it a Buchsteiner loop since Hans-Hennig Buchsteiner seems to have been the first to notice their importance [7]. The Buchsteiner law (B) is easily seen to be equivalent to each of the following: xy · z = xu ⇐⇒ y · zx = ux for all x, y, z, u , and (B’) xy · z = xu · v ⇐⇒ y · zx = u · vx for all x, y, z, u, v . (B”) Both the identity (B) and the implications (B’), (B”) will prove useful in what follows. Buchsteiner loops can be understood in terms of coinciding left and right principal isotopes or in terms of autotopisms that identify the left and the right nucleus. Among the equalities that can be obtained by nuclear identification this is the only one which has not been subjected to a systematic study. (The other equalities are the left and right Bol laws, the Moufang laws, the extra laws, and the laws of left and right conjugacy closedness [18].) All Buchsteiner loops are G-loops (a loop Q is said to be a G-loop if every loop isotope of Q is isomorphic to Q). Groups are G-loops, and it is well-known and 2000 Mathematics Subject Classification. Primary 20N05; Secondary 08A05.