论文标题
不可溶解的群体的特征度图具有切割vertex。 ii
Non-solvable groups whose character degree graph has a cut-vertex. II
论文作者
论文摘要
令$ g $为有限的组，让$ {\ rm {cd}}（g）$表示$ g $的不可减至的复杂字符的学位集。然后将字符度图$δ（g）$定义为（简单的无向）图，其顶点是$ {\ rm {cd}}}（g）$的数字的主要划分，而两个不同的顶点$ p $，$ q $在且仅当$ pq $均在$ {\ rm {cd cd}的情况下均为$ pq $时，仅当$ pq $时。本文继续进行工作，始于[7]，朝着有限的不可溶解基团的分类，其度图具有切割vertex，即其去除的顶点增加了图形的连接组件的数量。而在[7]中，没有组成因子同构至$ {\ rm {\ rm {psl}} _ 2（t^a）$（对于任何Prime Power $ t^a \ geq 4 $）的组得到了治疗，在这里，我们考虑当$ t $ t $和$ t^a> 5 $ 5 $ t $ t $时，我们考虑了互补情况。然后，将在本系列的第三篇和最后一篇论文（[8]）中完成此分类的证明，该论文涉及案例$ t = 2 $。
Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. Define then the character degree graph $Δ(G)$ as the (simple undirected) graph whose vertices are the prime divisors of the numbers in ${\rm{cd}}(G)$, and two distinct vertices $p$, $q$ are adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. This paper continues the work, started in [7], toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e., a vertex whose removal increases the number of connected components of the graph. While, in [7], groups with no composition factors isomorphic to ${\rm{PSL}}_2(t^a)$ (for any prime power $t^a\geq 4$) were treated, here we consider the complementary situation in the case when $t$ is odd and $t^a> 5$. The proof of this classification will be then completed in the third and last paper of this series ([8]), that deals with the case $t=2$.
