ABSTRACT
Central quasigroups are the quasigroup analogues of abelian groups. For a quasigroup Q, the diagonal quasigroup is defined as
Q̂ = {(x, x) | x ∈ Q}. (3.1)
Now a group Q is abelian if and only if the diagonal is a normal subgroup of Q2. Certainly, if Q is abelian, then so is Q2, whence each subgroup of Q2 is normal. Conversely, suppose that Q̂ C Q2. Then for all x, y ∈ Q, one has
(x, y)−1(x, x)(x, y) = (x, y−1xy) ∈ Q̂,
so that x = y−1xy. Thus Q is abelian. A subquasigroup N of a quasigroup P is said to be a normal subquasigroup, written as N C P , if there is a congruence W on P having N as a congruence class.1 Note that the congruence W is specified uniquely by the fact that for any e ∈ N,x ∈ P , the map ρ(e, x) : N → xW of (2.26) provides a bijection between N and the congruence class xW of the element x of P . One may then use the usual division notation P/N to denote the unambiguously specified quotient quasigroup PW .2 By analogy with the group case, a quasigroup Q is defined to be central , or in the class Z, if the diagonal is a normal subquasigroup of Q2. One also says that the universal congruence Q2 on Q is central. More generally, a congruence V on a quasigroup Q is defined to be central if Q̂ C V . Following a brief discussion of general quasigroup congruences in Section 3.1,
Section 3.2 examines the central congruences of a quasigroup, in the slightly broader context of centrality that will be useful for the treatment of quasigroup modules in Chapter 10. Now abelian groups are the nilpotent groups of class 1. Section 3.3 uses central congruences to give a definition of nilpotence for quasigroups that specializes appropriately to groups. Central quasigroups then become the nilpotent quasigroups of class 1. Section 3.4 discusses the relation of central isotopy, which is weaker than isomorphism, but stronger than general isotopy. In many ways, the relation of central isotopy between quasigroups is more important than the relation of isomorphism. Theorem 3.4
phenomena under the direct product. Theorem 3.8 is another example. Here, central isotopy classes of central quasigroups are shown to correspond exactly to isomorphism classes of central piques. The key Section 3.6, describing the structure of central quasigroups, is thus preceded by Section 3.5 dealing with central piques. Section 3.7 uses centrality to classify quasigroups of prime order. The main Theorem 3.10 may be summarized as saying that for prime order, either a quasigroup is central, or else its multiplication group is almost simple — sandwiched between a simple group and its automorphism group, the simple group being alternating, linear, or one of the Mathieu groups M11 or M23. Section 3.8 examines the stability congruence of a quasigroup. For loops, the stability and center congruences coincide, but they may separate for general quasigroups. There is a corresponding concept of stable nilpotence. Nilpotence of the multiplication group implies stable nilpotence of a finite quasigroup. Conversely, stably nilpotent quasigroups have solvable multiplication groups. Chapter 3 concludes with a brief discussion of some so-called “no-go theorems,” showing that certain groups or group actions cannot be represented as multiplication groups of quasigroups.
