弱EKR群的刻画及具有指定点稳定子的交集密度
Characterization of Weak EKR Groups and Intersection Densities with Prescribed Point Stabilizers
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中文总结 AI 辅助
本文刻画弱EKR群的性质，建立相关准则，还解决公开问题，证明对指定有限群H和整数n≥m，存在对应传递置换群的交集密度为n/m。
中文摘要 AI 辅助
若有限群的所有传递置换作用都具有EKR性质，则称该群具有弱Erdős-Ko-Rado性质。我们通过正规子群和主因子来刻画该性质，更准确地说，我们引入局部交集密度并建立正规扩张准则，将弱EKR性质简化为初等阿贝尔主因子及其上诱导线性群的差集条件。对于秩1的主因子，该条件自动满足；对于秩2的主因子，该条件等价于诱导线性群在一维子空间上非传递。在本文第二部分，我们通过确定具有指定点稳定子的传递置换群的可能交集密度，解决了一个公开问题：证明对于每个阶为m≥4的有限群H和每个整数n≥m，存在一个忠实传递置换群，其点稳定子同构于H，且交集密度为n/m。
英文摘要
A finite group has the weak Erdos-Ko-Rado property if all of its transitive permutation actions have the EKR property. We characterize this property in terms of normal subgroups and chief factors. More precisely, we introduce a local intersection density and establish a normal-extension criterion which reduces the weak EKR property to difference-set conditions on the elementary abelian chief factors and the linear groups induced on them. For chief factors of rank one the condition is automatically satisfied, and for chief factors of rank two, this condition is equivalent to the induced linear group being intransitive on the one-dimensional subspaces. In the second part of the paper, we solve an open problem by determining the possible intersection densities of transitive permutation groups with a prescribed point stabilizer. We prove that, for every finite group $H$ of order $m\geq 4$ and every integer $n\geq m$, there exists a faithful transitive permutation group with point stabilizer isomorphic to $H$ and intersection density $n/m$.
