Co-maximal Graphs of Subgroups of Groups

Canadian mathematical bulletin

Babak Miraftab¹  Saieed Akbari¹  Reza Nikandish² 

[1] Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran;Department of Basic Sciences, Jundi-Shapur University of Technology, Dezful, Iran

关键词: co-maximal graphs of subgroups of groups;    diameter;    nilpotent group;    solvable group;   

DOI  :  10.4153/CMB-2016-026-0

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $H$ be a group. The co-maximal graph of subgroupsof $H$, denoted by $Gamma(H)$, is agraph whose vertices are non-trivial and proper subgroups of$H$ and two distinct vertices $L$and $K$ are adjacent in $Gamma(H)$ if and only if $H=LK$. In this paper, we study the connectivity, diameter, clique numberand vertexchromatic number of $Gamma(H)$. For instance, we show thatif $Gamma(H)$ has no isolated vertex, then $Gamma(H)$ is connected with diameter at most $3$. Also, we characterizeall finite groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $Gamma(H)$ is connected and moreover thedegree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraicallyclosed field is zero or infinite.

【 授权许可】

Unknown