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GroupTheory

  

GruenbergKegelGraph

  

construct the Gruenberg-Kegel graph of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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 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

withGroupTheory:

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

GKGGruenbergKegelGraphMonster

GKGGraph 1: an undirected unweighted graph with 15 vertices, 23 edge(s), and 3 self-loop(s)

(1)

useGraphTheoryinHighlightVertexGKG,SelfLoopsGKG,'stylesheet'='shape'=pentagon,'color'=redend use:

useGraphTheoryinHighlightVertexGKG,mapop,selectc→nopsc=1,ConnectedComponentsGKG,'stylesheet'='shape'=7gon,'color'=greenend use:

The self-loops indicate those supersingular primes p for which the Monster has an element of order p2.

GraphTheory:-DrawGraphGKG

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

GFrobeniusGroup2238,1:

GKGGruenbergKegelGraphG

GKGGraph 2: an undirected unweighted graph with 3 vertices and 1 edge(s)

(2)

GraphTheory:-ConnectedComponentsGKG

2,3,373

(3)

Compatibility

• 

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

• 

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

See Also

GraphTheory

GroupTheory

use

with