GroupTheory/CommutingGraph - Maple Help

GroupTheory

 CommutingGraph
 construct the commuting graph of a group

 Calling Sequence CommutingGraph( G ) CommutingGraph( G, elements = E )

Parameters

 G - a small group E - (optional) list, set, or one of the names all or noncentral

Options

 • elements = list, set, or one of the names all or noncentral
 Specifies a selection of the elements of G to include as vertices of the generated graph.
 If elements is a list or set, these elements are included.
 If elements is noncentral, all elements of G except central elements are included.
 If elements is all (the default), all elements of G are included.

Description

 • For a finite group $G$ and a subset $E$ of its elements, the commuting graph of $G$ and $E$ is the graph whose vertices are elements of $E$ and for which two vertices $p$ and $q$ are adjacent if $\mathrm{pq}=\mathrm{qp}$ in $G$.
 • The CommutingGraph( G ) command returns the commuting graph of G.
 • You can specify a particular ordering for the elements of the group by passing the optional argument elements = E, where E is an explicit list of the members of G.
 • Note that computing the commuting graph of a group requires that all the group elements be computed explicitly, so the command should only be used for groups of modest size.

Examples

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

Draw the commuting graph of the symmetric group of degree 4.

 > $G≔\mathrm{SymmetricGroup}\left(4\right)$
 ${G}{≔}{{\mathbf{S}}}_{{4}}$ (1)
 > $\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CommutingGraph}\left(G\right),\mathrm{style}=\mathrm{spring}\right)$

Draw the commuting graph of the dihedral group of degree 7.

 > $G≔\mathrm{DihedralGroup}\left(7\right)$
 ${G}{≔}{{\mathbf{D}}}_{{7}}$ (2)
 > $\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CommutingGraph}\left(G\right),\mathrm{style}=\mathrm{spring}\right)$

Draw the commuting graph of a Frobenius group of order 72.

 > $G≔\mathrm{FrobeniusGroup}\left(72,1\right)$
 ${G}{≔}{\mathrm{< a permutation group on 9 letters with 5 generators >}}$ (3)
 > $\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CommutingGraph}\left(G\right),\mathrm{style}=\mathrm{spring}\right)$

Compatibility

 • The GroupTheory[CommutingGraph] command was introduced in Maple 2023.