论文标题
交替组和对称组的生成图中的最大Cocliques
Maximal Cocliques in the Generating Graphs of the Alternating and Symmetric Groups
论文作者
论文摘要
有限的组$ g $的生成图$γ（g）$具有$ g $的非身份元素，当它们生成$ g $时，它们的两个元素正好连接。图中的Coclique是一个空的诱导子图，因此$γ（g）$中的Coclique是$ g $的子集，因此没有一对元素会产生$ g $。如果没有较大的可可菌中的Coclique最大。很容易看出，$ g $的最大亚组的非身份元素以$γ（g）$形式形成Coclique，但这种Coclique不必最大。在本文中，我们确定$ \ textrm {s} _n $和$ \ textrm {a} _n $的不及物最大亚组何时是生成图中的最大Cocliques。此外，在$ g = \ textrm {a} _n $和$ \ textrm {s} _n $的情况下，我们证明了Cameron，Lucchini和Roney-Dougal [3]的猜想2 $。也就是说，我们表明$ g $的两个元素在$γ（g）$中具有相同的邻居，并且仅当它们属于完全相同的最大亚组时。
The generating graph $Γ(G)$ of a finite group $G$ has vertex set the non-identity elements of $G$, with two elements connected exactly when they generate $G$. A coclique in a graph is an empty induced subgraph, so a coclique in $Γ(G)$ is a subset of $G$ such that no pair of elements generate $G$. A coclique is maximal if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of $G$ form a coclique in $Γ(G)$, but this coclique need not be maximal. In this paper we determine when the intransitive maximal subgroups of $\textrm{S}_n$ and $\textrm{A}_n$ are maximal cocliques in the generating graph. In addition, we prove a conjecture of Cameron, Lucchini, and Roney-Dougal [3] in the case of $G = \textrm{A}_n$ and $\textrm{S}_n$, when n is prime and $n \neq \frac{(q^d -1)}{(q-1)}$ for all prime powers $q$ and $d \geq 2$. Namely, we show that two elements of $G$ have identical sets of neighbours in $Γ(G)$ if and only if they belong to exactly the same maximal subgroups.
