  We prove that for n>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.
