Co-maximal Graphs of Subgroups of Groups

Abstract

AbstractLet H be a group. The co-maximal graph of subgroups of H, denoted by Γ(H), is a graph whose vertices are non-trivial and proper subgroups of H and two distinct vertices L and K are adjacent in Γ(H) if and only ifH = LK. In this paper, we study the connectivity, diameter, clique number, and vertex chromatic number of Γ(H). For instance, we show that if Γ(H) has no isolated vertex, then Γ(H) is connectedwith diameter at most h. Also, we characterize all ûnite groupswhose co-maximal graphs are connected. Among other results, we show that if H is a ûnitely generated solvable group and Γ(H) 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 ûeld is zero or infinite.