Let m be a positive integer and let Ω be a finite set. The m-closure of $G\le \mathrm{Sym}(\mathrm{\Omega })$ is the largest permutation group on Ω having the same orbits as G in its induced action on the Cartesian product ${\mathrm{\Omega }}^m$. 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 $m\ge 3$.

中文翻译：

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可解置换群的 3-闭包是可解的

令m为正整数，Ω 为有限集。的m -闭包$G\le \mathrm{符号}(\mathrm{\Omega })$是 Ω 上最大的置换群，其轨道与G对笛卡尔积的诱导作用相同${\mathrm{\Omega }}^{米}$. 可解置换群的 1-闭包和 2-闭包不一定是可解的。我们证明可解置换群的m -闭包总是可解的$米\ge 3$.