On $\operatorname {c}$-3-Transitive Automorphism Groups of Cyclically Ordered Sets

An automorphism group $G$ of a cyclically ordered set $\langle X,C\rangle$ is said to be $\operatorname {c}$-3-transitive if for any elements $x_i,y_i\in X$ ($i=1,2,3$), such that $C(x_1,x_2,x_3)$ and $C(y_1,y_2,y_3)$ there exists an element $g\in G$ satisfying $g(x_i)=y_i$, $i=1,2,3$. We prove that if an automorphism group of a cyclically ordered set is $\operatorname {c}$-3-transitive, then it is simple. A description of $\operatorname {c}$-3-transitive automorphism groups with Abelian two-point stabilizer is given.