论有限群的无势图

Jaime Torres, Ismael Gutierrez, E. J. Garcia-Claro
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引用次数: 0

摘要

如果 G 是一个非零potent 群,并且 nil(G) = {g \in G : is nilpotent forall h\in G},那么 G 的零potent 图就是具有顶点集 G-nil(G) 的图,其中两个不同的顶点如果生成 G 的一个零potent 子群,那么它们就是相关的。根据 G 的 Fitting 子群的大小和强自中心化子群的数量,分别给出了 G 的无穷图的小群数和连通成分数的下限。证明了当且仅当 n 或 n-1 是素数时,阶数为 n 的对称群的无穷图是断开的,并且没有有限非无穷群具有自补无穷图。对于二面体群 Dn,可以确定当 n 为奇数时,其无穷图的连通成分数比 n 多一个;当 n 为偶数时,其无穷图的连通成分数比 n 的 2'- 部分多一个。此外,还提供了 PSL(2,q)(其中 q 是质数幂)无勢图的连通部分数公式。最后,研究了一个群的特定子集(包含其无穷图的连通成分)包含其一个 Sylow p 子群的必要条件和充分条件;并证明了具有偶数阶 nil(G) 的有限非无穷群 G 的无穷图是非欧拉图。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the Nilpotent Graph of a finite Group
If G is a non-nilpotent group and nil(G) = {g \in G : is nilpotent for all h\in G}, the nilpotent graph of G is the graph with set of vertices G-nil(G) in which two distinct vertices are related if they generate a nilpotent subgroup of G. Several properties of the nilpotent graph associated with a finite non-nilpotent group G are studied in this work. Lower bounds for the clique number and the number of connected components of the nilpotent graph of G are presented in terms of the size of its Fitting subgroup and the number of its strongly self-centralizing subgroups, respectively. It is proved the nilpotent graph of the symmetric group of degree n is disconnected if and only if n or n-1 is a prime number, and no finite non-nilpotent group has a self-complementary nilpotent graph. For the dihedral group Dn, it is determined the number of connected components of its nilpotent graph is one more than n when n is odd; or one more than the 2'-part of n when n is even. In addition, a formula for the number of connected components of the nilpotent graph of PSL(2,q), where q is a prime power, is provided. Finally, necessary and sufficient conditions for specific subsets of a group, containing connected components of its nilpotent graph, to contain one of its Sylow p-subgroups are studied; and it is shown the nilpotent graph of a finite non-nilpotent group G with nil(G) of even order is non-Eulerian.
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