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Co-maximal graphs of subgroups of groups

Akbari, S ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.4153/CMB-2016-026-0
  3. Abstract:
  4. Let 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 if H = 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 finite groupswhose co-maximal graphs are connected. Among other results, we show that if H is a finitely 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 field is zero or infinite. © 2016 Canadian Mathematical Society
  5. Keywords:
  7. Source: Canadian Mathematical Bulletin ; Volume 60, Issue 1 , 2017 , Pages 12-25 ; 00084395 (ISSN)
  8. URL: https://cms.math.ca/10.4153/CMB-2016-026-0