Two-point homogeneous quandles with prime cardinality. (English) Zbl 1295.57011

Tamaru, Hiroshi

J. Math. Soc. Japan 65, No. 4, 1117-1134 (2013).

A quandle is a set \(X\) endowed with a binary operation \(\ast :X \times X \to X\) such that \(\ast \) is idempotent, right distributive and for all \(a, b \in X\) the equation \(x \ast a=b\) has precisely one solution. For every \(x \in X\) the map \(s_x:X \to X\) is defined by \(y \ast x:=s_x(y)\). Analogously to two-point homogeneous Riemannian manifolds in this paper the author defines and characterizes in terms of the isotropy representations two-point homogeneous quandles, i.e. quandles \((X,s)\) such that for all \((x_1,x_2)\), \((y_1,y_2) \in X \times X\) with \(x_1 \neq x_2\), \(y_1 \neq y_2\) there exists an inner automorphism \(f\) of \((X,s)\) such that \(f(x_i)=y_i\), \(i=1,2\). After this he gives a sufficient condition, called the cyclic type property, for finite quandles \((X,s)\) with cardinality \(\geq 3\) to be two-point homogeneous and studies quandles of cyclic type. In the paper he classifies the two-point quandles with prime cardinality \(p \geq 3\). To obtain a classification of these quandles he determines the inner automorphism groups of linear Alexander quandles.

Reviewer: Ágota Figula (Debrecen)

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
53C35	Differential geometry of symmetric spaces

Keywords:

quandles; symmetric spaces; two-point homogeneous Riemannian manifolds

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