Doubly transitive groups and cyclic quandles

Abstract

We prove that for $n \gt 2$ there exists a quandle of cyclic type of size $n$ if and only if $n$ is a power of a prime number. This establishes a conjecture of S. Kamada, H. Tamaru and K. Wada. As a corollary, every finite quandle of cyclic type is an Alexander quandle. We also prove that finite doubly transitive quandles are of cyclic type. This establishes a conjecture of H. Tamaru.