Abstract view
Comaximal Graphs of Subgroups of Groups


Published:20161110
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, JundiShapur University of Technology, Dezful, Iran
Abstract
Let $H$ be a group. The comaximal graph of subgroups
of $H$, denoted by $\Gamma(H)$, is a
graph whose vertices are nontrivial 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 comaximal 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
comaximal graph of a general linear group over an algebraically
closed field is zero or infinite.
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
