论文标题
对称和交替组的nilpotent封面
Nilpotent covers of symmetric and alternating groups
论文作者
论文摘要
我们证明，对称组$ s_n $由Maximal nilpotent子组具有独特的最小封面$ \ Mathcal {M} $，并且我们获得了$ \ Mathcal {M} $的订单的明确且易于计算的公式。此外，我们证明$ \ MATHCAL {M} $的顺序等于$ S_N $的最大非Nilpotent子集的顺序。此封面$ \ Mathcal {M} $具有吸引人的属性；例如，这是一个普通的封面，封面中子组的共轭类别的数量等于在不同的正整数中的分区数量。我们表明，这些结果与交替组$ a_n $的结果形成对比。 In particular, we prove that, for all but finitely many values of $n$, no minimal cover of $A_n$ by maximal nilpotent subgroups is a normal cover and the order of a minimal cover of $A_n$ by maximal nilpotent subgroups is strictly greater than the order of a maximal non-nilpotent subset of $A_n$.
We prove that the symmetric group $S_n$ has a unique minimal cover $\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the order of $\mathcal{M}$. In addition, we prove that the order of $\mathcal{M}$ is equal to the order of a maximal non-nilpotent subset of $S_n$. This cover $\mathcal{M}$ has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of $n$ into distinct positive integers. We show that these results contrast with those for the alternating group $A_n$. In particular, we prove that, for all but finitely many values of $n$, no minimal cover of $A_n$ by maximal nilpotent subgroups is a normal cover and the order of a minimal cover of $A_n$ by maximal nilpotent subgroups is strictly greater than the order of a maximal non-nilpotent subset of $A_n$.
