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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
 MSC Classifications: 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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