Principal and doubly homogeneous quandles

Abstract

In the paper we describe the class of principal quandles and we show that connected quandles can be decomposed as a disjoint union of principal quandles. We also prove that simple affine quandles are finite and they can be characterized among finite simple quandles by several different equivalent properties, such as for instance being doubly homogeneous (i.e. having a doubly transitive automorphism group). A complete description of finite doubly-homogeneous quandles is provided extending the result of [34] and solving [3, Problem 6.7]. We also provide a classification of connected cyclic quandles with several fixed points independently from [23].

Superconnected left quasigroups and involutory quandles

“…The orbits of LMlt(Q) and of Dis(Q) coincide, because of the structure of LMlt(Q) given in Lemma 1.1. We extend [8,Proposition 1.4] to the setting of idempotent left quasigroups.…”

“…Hence [a] σ Q is contained in the orbit of a with respect to Dis(Q). Hence, according to [8,Theorem 3.4], [a] σ is principal over Dis(S) ≅ N Dis(Q) (Dis(Q) a ) Dis(Q) a that is solvable (the diplacement group of a finite latin quandle is solvable [27]).…”

“…Q is an abelian cover of Q ip Q . Some of the contents of Section 2.4 of [8] on principal latin quandles extend to principal superconnected quandles. Proposition 2.12.…”

“…Proof. If Q is abelian, the statement is true because subquandles correspond to submodules with respect to the structure given by the affine representation [8,Proposition 2.18]. Let Q be solvable of length n + 1 i.e.…”

“…If C is infinite then Q is not connected. Hence Q is a finite connected affine quandle, and then Q is latin (see [8,18]). In particular, R 0 ∶ x ↦ 2x is bijective and so C is odd.…”