GruenbergKegelGraph - Maple Help

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GroupTheory

 GruenbergKegelGraph
 construct the Gruenberg-Kegel graph of a group

 Calling Sequence GruenbergKegelGraph( G )

Parameters

 G - a small group

Description

 • 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.

Examples

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

The vertices of the Gruenberg-Kegel graph of the Monster sporadic finite simple group are the so-called supersingular primes.

 > $\mathrm{GKG}≔\mathrm{GruenbergKegelGraph}\left(\mathrm{Monster}\left(\right)\right)$
 ${\mathrm{GKG}}{≔}{\mathrm{Graph 1: an undirected unweighted graph with 15 vertices, 23 edge\left(s\right), and 3 self-loop\left(s\right)}}$ (1)
 > $\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{GraphTheory}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{HighlightVertex}\left(\mathrm{GKG},\mathrm{SelfLoops}\left(\mathrm{GKG}\right),'\mathrm{stylesheet}'=\left['\mathrm{shape}'="pentagon",'\mathrm{color}'="red"\right]\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}:$
 > $\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{GraphTheory}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{HighlightVertex}\left(\mathrm{GKG},\mathrm{map}\left(\mathrm{op},\mathrm{select}\left(c→\mathrm{nops}\left(c\right)=1,\mathrm{ConnectedComponents}\left(\mathrm{GKG}\right)\right)\right),'\mathrm{stylesheet}'=\left['\mathrm{shape}'="7gon",'\mathrm{color}'="green"\right]\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}:$

The self-loops indicate those supersingular primes $p$ for which the Monster has an element of order ${p}^{2}$.

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

The Gruenberg-Kegel graph of a Frobenius group is never connected.

 > $G≔\mathrm{FrobeniusGroup}\left(2238,1\right):$
 > $\mathrm{GKG}≔\mathrm{GruenbergKegelGraph}\left(G\right)$
 ${\mathrm{GKG}}{≔}{\mathrm{Graph 2: an undirected unweighted graph with 3 vertices and 1 edge\left(s\right)}}$ (2)
 > $\mathrm{GraphTheory}:-\mathrm{ConnectedComponents}\left(\mathrm{GKG}\right)$
 $\left[\left[{2}{,}{3}\right]{,}\left[{373}\right]\right]$ (3)

Compatibility

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