Buchsteiner Loops

Abstract

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.

The Many Formulae for the Number of Latin Rectangles

“…If (Q, ⊕) possesses an identity element e, that is e satisfies e⊕g = g = g ⊕e for all g ∈ Q, then Q is called a loop. 2007 Hämäläinen and Cavenagh [66] Aut(L) 2007 Csörgő, Drápal and Kinyon [27]…”

Cycle structure of autotopisms of quasigroups and latin squares

“…Thus, Moufang loops can be studied by considering these and other autotopisms , Chapter V, a point of view that culminates in the theory of groups with triality . Other varieties of loops in which the defining identities have autotopic characterizations include conjugacy closed loops , extra loops , and Buchsteiner loops . A new, systematic look at the basic theory of loops defined in this way can be found in .…”

Buchsteiner loops: Associators and constructions

“…By [6], if Q is a Buchsteiner loop, then Q/Z(Q) is a conjugacy closed loop. The associator calculus developed in [8] (which is an earlier paper than [6]) can be formulated within such a theory, and this is also true for results of Section 1. The main results of this paper are independent of Proposition 2.1 since for them it suffices to know the statement only for 3-generated groups.…”

“…Hence |A(Q)| ≥ 8, and so Q/N has to be generated by at most two elements if |Q| ≤ 64. In such a case we can assume that Q is generated by two elements, by Corollary 2.5, and we can also assume that [ The proof is not difficult, and can be found, e. g., in [8].…”

“…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.…”