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Co-maximal Graphs of Subgroups of Groups

Published online by Cambridge University Press:  20 November 2018

Saieed Akbari ,
Babak Miraftab  and
Reza Nikandish
Saieed Akbari
Affiliation:
Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran e-mail: s_akbari@sharif.edu e-mail: babak.math@gmail.com
Babak Miraftab
Affiliation:
Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran e-mail: s_akbari@sharif.edu e-mail: babak.math@gmail.com
Reza Nikandish
Affiliation:
Department of Basic Sciences, Jundi-Shapur University of Technology, Dezful, Iran e-mail: r.nikandish@ipm.ir
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Abstract

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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.

Keywords

05C2505E1520D1020D15co-maximal graphs of subgroups of groupsdiameternilpotent groupsolvable group
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 1 , 01 March 2017 , pp. 12 - 25
Copyright
Copyright © Canadian Mathematical Society 2017

References

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