For a finite group $G$, the Gruenberg-Kegel graph (also known as the prime graph) of $G$ is the graph with vertices the prime divisors of the order of $G$, and for which two vertices $p$ and $q$ are adjacent if $G$ has an element of order $\mathrm{pq}$.

The GruenbergKegelGraph( 'G' ) command returns the Gruenberg-Kegel graph of the finite group G.

Commands in the GraphTheory package can be used to visualize the graph returned by this command, as well as to analyze its properties.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{GKG}≔\mathrm{GruenbergKegelGraph}\left(\mathrm{Monster}\left(\right)\right)$

$\mathrm{GKG}≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 15\; vertices,\; 23\; edges,\; and\; 3\; self-loops}$

$\mathbf{use}\mathrm{GraphTheory}\mathbf{in}\mathrm{HighlightVertex}\left(\mathrm{GKG}\,\mathrm{SelfLoops}\left(\mathrm{GKG}\right)\,\'\mathrm{stylesheet}\'\=\left[\'\mathrm{shape}\'\=''pentagon''\,\'\mathrm{color}\'\=''red''\right]\right)\mathbf{end\; use}\:$

$\mathbf{use}\mathrm{GraphTheory}\mathbf{in}\mathrm{HighlightVertex}\left(\mathrm{GKG}\,\mathrm{map}\left(\mathrm{op}\,\mathrm{select}\left(c\→\mathrm{nops}\left(c\right)\=1\,\mathrm{ConnectedComponents}\left(\mathrm{GKG}\right)\right)\right)\,\'\mathrm{stylesheet}\'\=\left[\'\mathrm{shape}\'\=''7gon''\,\'\mathrm{color}\'\=''green''\right]\right)\mathbf{end\; use}\:$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{GKG}\right)$

$G≔\mathrm{FrobeniusGroup}\left(2238\,1\right)\:$

$\mathrm{GKG}≔\mathrm{GruenbergKegelGraph}\left(G\right)$

$\mathrm{GKG}≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 3\; vertices\; and\; 1\; edge}$

$\mathrm{GraphTheory}:-\mathrm{ConnectedComponents}\left(\mathrm{GKG}\right)$

$\left[\left[2\,3\right]\,\left[373\right]\right]$

The GroupTheory[GruenbergKegelGraph] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.