The $3$-closure of a solvable permutation group is solvable

Abstract

Let $m$ be a positive integer and let $\Omega$ be a finite set. The $m$-closure of $G\le \mathrm{Sym}(\Omega )$ is the largest permutation group on $\Omega$ having the same orbits as $G$ in its induced action on the Cartesian product ${\Omega }^m$. The $1$-closure and $2$-closure of a solvable permutation group need not be solvable. We prove that the $m$-closure of a solvable permutation group is always solvable for $m\ge 3$.

References in corpus (1)

• Two-closure of supersolvable permutation group in polynomial time

Cited by in corpus (2)

• On computing the closures of solvable permutation groups
• Closures of permutation groups with restricted nonabelian composition factors