Co-maximal Graphs of Subgroups of Groups

Akbari, Saieed ; Miraftab, Babak ; Nikandish, Reza

Canadian mathematical bulletin, Tome 60 (2017) no. 1, pp. 12-25

Résumé

Let $H$ be a group. The co-maximal graph of subgroups of $H$ , denoted by $\Gamma \left( H \right)$ , is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma \left( H \right)$ if and only if $H\,=\,LK$ . In this paper, we study the connectivity, diameter, clique number, and vertex chromatic number of $\Gamma \left( H \right)$ . For instance, we show that if $\Gamma \left( H \right)$ has no isolated vertex, then $\Gamma \left( H \right)$ is connected with diameter at most 3. Also, we characterize all finitely groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma \left( H \right)$ is connected, and moreover, the degree 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 algebraically closed field is zero or infinite.

Détail

DOI : 10.4153/CMB-2016-026-0

Mots-clés : 05C25, 05E15, 20D10, 20D15, co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable group