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Co-maximal Graphs of Subgroups of Groups

  Published:2016-11-10
 Printed: Mar 2017
  • Saieed Akbari,
    Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
  • Babak Miraftab,
    Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
  • Reza Nikandish,
    Department of Basic Sciences, Jundi-Shapur University of Technology, Dezful, Iran
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Abstract

Let $H$ be a group. The co-maximal graph of subgroups of $H$, denoted by $\Gamma(H)$, is a graph 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 number and vertex chromatic number of $\Gamma(H)$. For instance, we show that if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$ is connected with diameter at most $3$. Also, we characterize all 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 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.
Keywords: co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable group co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable group
MSC Classifications: 05C25, 05E15, 20D10, 20D15 show english descriptions Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65]
Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]
Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17]
Nilpotent groups, $p$-groups
05C25 - Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65]
05E15 - Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]
20D10 - Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17]
20D15 - Nilpotent groups, $p$-groups
 

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