||Co-Maximal Graphs of Subgroups of Groups|
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
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
Categories:05C25, 05E15, 20D10, 20D15