论文标题
置换群体的erdős-ko-rado问题
Erdős-Ko-Rado problems for permutation groups
论文作者
论文摘要
在本文中，我们研究了原始和准脱落置换组中的相交集。令$ g \ leqslant \ mathrm {sym}（ω）$为及时置换组，而$ {s} $相交集。先前的结果表明，如果$ g $是2个传输或Frobenius组，则$ | {s} | \ leqslant |g_Ω| $（对于ω$中的某些$ω\）。此外，对于某些2个传递组，$ | {s} | = |g_Ω| $ if，仅当$ {s} $是稳定器的固定器。在本文中，我们证明了这些陈述远非一般瞬态群体的真相。特别是，我们表明，在原始组的情况下，甚至没有绝对常数$ c $，因此$ | {s} | \ leqslant c |g_Ω| $。在情况下，$ g $是一个原始置换组同构至$ \ mathrm {psl（2，p）} $，我们表征了$ g $的子组，这些子组是相交的集合。我们还表明，如果$ g \ leqslant \ mathrm {sym}（ω）$是Prime Power度的排列组，那么对于任何相交的集合集$ s $，我们都有$ | s | s | \ leq |g_Ω| $（对于某些$ω\ inω$）。这证明了\ cite {mrs}中猜想的一部分。
In this paper, we study intersecting sets in primitive and quasiprimitive permutation groups. Let $G \leqslant \mathrm{Sym}(Ω)$ be a transitive permutation group, and ${S}$ an intersecting set. Previous results show that if $G$ is either 2-transitive or a Frobenius group, then $|{S}|\leqslant|G_ω|$ (for some $ω\in Ω$). Furthermore, for some 2-transitive groups, $|{S}|=|G_ω|$ if and only if ${S}$ is a coset of a stabilizer. In this paper, we prove that these statements are far from the truth for general transitive groups. In particular, we show that in the case of primitive groups, there is even no absolute constant $c$ such that $|{S}|\leqslant c|G_ω|$. In the case $G$ is a primitive permutation group isomorphic to $\mathrm{PSL(2,p)}$, we characterize the subgroups of $G$ which are intersecting sets. We also show that if $G \leqslant \mathrm{Sym}(Ω)$ is a permutation group of prime power degree, then for any intersecting set $S$, we have $|S|\leq |G_ω|$ (for some $ω\in Ω$). This proves a part of a conjecture in \cite{MRS}.
