论文标题
有限组，最小基础和交点号
Finite groups, minimal bases and the intersection number
论文作者
论文摘要
令$ g $为有限的组，并回想起Frattini子组$ {\ rm frat}（g）$是$ g $的所有最大亚组的交集。在本文中，我们调查了$ g $的交叉点，表示为$α（g）$，这是其交点与$ {\ rm frat}（g）$相吻合的最大亚组数量的最小数量。在较早的工作中，我们在特殊情况下研究了$α（g）$，在这种情况下，$ g $很简单，在这里我们将分析扩展到几乎简单的组。特别是，我们证明了每个几乎简单的$ g $ $α（g）\ leqslant 4 $，这是最好的。我们还在任意有限群体的交集数量上建立了新的结果，该数量获得了根据该组的主要因素定义的上限。最后，对于几乎简单的组$ g $，我们在相关的不变$β（g）$上提出了最佳的界限，我们称其为$ g $的基本数字。在这种情况下，$β（g）$是$ g $的最小基本大小，因为我们比该组的所有忠实原始动作都差异，并且我们证明了绑定的$β（g）\ leqslant 4 $是最佳的。一路上，我们研究了对称组$ s_ {ab} $原始操作的基础，$ [1，ab] $的分区集中到$ a $ a $ a $ a $ a $ a $ a $ a $ b $，确定$ a \ geqslant b $的确切基本尺寸。这扩展了Benbenishty，Cohen和Niemeyer的早期工作。
Let $G$ be a finite group and recall that the Frattini subgroup ${\rm Frat}(G)$ is the intersection of all the maximal subgroups of $G$. In this paper, we investigate the intersection number of $G$, denoted $α(G)$, which is the minimal number of maximal subgroups whose intersection coincides with ${\rm Frat}(G)$. In earlier work, we studied $α(G)$ in the special case where $G$ is simple and here we extend the analysis to almost simple groups. In particular, we prove that $α(G) \leqslant 4$ for every almost simple group $G$, which is best possible. We also establish new results on the intersection number of arbitrary finite groups, obtaining upper bounds that are defined in terms of the chief factors of the group. Finally, for almost simple groups $G$ we present best possible bounds on a related invariant $β(G)$, which we call the base number of $G$. In this setting, $β(G)$ is the minimal base size of $G$ as we range over all faithful primitive actions of the group and we prove that the bound $β(G) \leqslant 4$ is optimal. Along the way, we study bases for the primitive action of the symmetric group $S_{ab}$ on the set of partitions of $[1,ab]$ into $a$ parts of size $b$, determining the exact base size for $a \geqslant b$. This extends earlier work of Benbenishty, Cohen and Niemeyer.
