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

DOI10.1016/J.JALGEBRA.2021.07.002arXiv2012.14166MaRDI QIDQ6357056

Evgeny Vdovin, E. A. O'Brien, Ilya Ponomarenko, Andrey Vasil'ev

Publication date: 28 December 2020

Abstract: Let $m$ be a positive integer and let $Omega$ be a finite set. The $m$-closure of $GleqoperatornameSym(Omega)$ is the largest permutation group on $Omega$ having the same orbits as $G$ in its induced action on the Cartesian product $Omeg{a}^m$. 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 $mgeq3$.

Mathematics Subject Classification ID

Finite solvable groups, theory of formations, Schunck classes, Fitting classes, (pi)-length, ranks (20D10) General theory for finite permutation groups (20B05)

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