$\mathbf{2}$-Closure of $\mathbf{\frac{3}{2}}$-transitive group in polynomial time.

Abstract

Let $G$ be a permutation group on a finite set $\Omega$. The $k$-closure $G^{(k)}$ of the group $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ having the same orbits as $G$ on the $k$-th Cartesian power $\Omega^k$ of $\Omega$. A group $G$ is called $\frac{3}{2}$-transitive if its transitive and the orbits of a point stabilizer $G_\alpha$ on the set $\Omega\setminus\{\alpha\}$ are of the same size greater than one. We prove that the $2$-closure $G^{(2)}$ of a $\frac{3}{2}$-transitive permutation group $G$ can be found in polynomial time in size of $\Omega$. In addition, if the group $G$ is not $2$-transitive, then for every positive integer $k$ its $k$-closure can be found within the same time. Applying the result, we prove the existence of a polynomial-time algorithm for solving the isomorphism problem for schurian $\frac{3}{2}$-homogeneous coherent configurations, that is the configurations naturally associated with $\frac{3}{2}$-transitive groups.