CayleyTable - Maple Help

GroupTheory

 CayleyTable
 construct the Cayley table of a group

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

Parameters

 G - a small group E - (optional) list ; an ordering of the elements of G

Description

 • The Cayley table of a (small) group $G$ specifies the binary operation that defines $G$.
 • The CayleyTable( G ) command returns the Cayley table of the group G, in which the elements of G have been labeled by the integers 1..n, where n is the order of G.
 • If G is a group originally specified by its Cayley table, then the CayleyTable command simply returns it.  For other groups, the Cayley table is computed, if possible.
 • 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.
 • The DrawCayleyTable command allows you to visualize the Cayley table of a small group using colors and can be formatted by using a wide variety of options.
 • Note that computing the Cayley table 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

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SymmetricGroup}\left(3\right)$
 ${G}{≔}{{\mathbf{S}}}_{{3}}$ (1)
 > $\mathrm{CayleyTable}\left(G\right)$
 $\left[\begin{array}{cccccc}{1}& {2}& {3}& {4}& {5}& {6}\\ {2}& {1}& {4}& {3}& {6}& {5}\\ {3}& {5}& {1}& {6}& {2}& {4}\\ {4}& {6}& {2}& {5}& {1}& {3}\\ {5}& {3}& {6}& {1}& {4}& {2}\\ {6}& {4}& {5}& {2}& {3}& {1}\end{array}\right]$ (2)

Generate all elements of G using the combinat package.

 > $\mathrm{permutation_lists}≔\mathrm{combinat}:-\mathrm{permute}\left(3\right)$
 ${\mathrm{permutation_lists}}{≔}\left[\left[{1}{,}{2}{,}{3}\right]{,}\left[{1}{,}{3}{,}{2}\right]{,}\left[{2}{,}{1}{,}{3}\right]{,}\left[{2}{,}{3}{,}{1}\right]{,}\left[{3}{,}{1}{,}{2}\right]{,}\left[{3}{,}{2}{,}{1}\right]\right]$ (3)

Translate these elements into their Group Theory representation.

 > $\mathrm{permutations}≔\mathrm{map}\left(\mathrm{Perm},\mathrm{permutation_lists}\right)$
 ${\mathrm{permutations}}{≔}\left[\left(\right){,}\left({2}{,}{3}\right){,}\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right){,}\left({1}{,}{3}{,}{2}\right){,}\left({1}{,}{3}\right)\right]$ (4)

Get the CayleyTable of G where the elements are ordered as in the list permutations.

 > $\mathrm{CayleyTable}\left(G,\mathrm{elements}=\mathrm{permutations}\right)$
 $\left[\begin{array}{cccccc}{1}& {2}& {3}& {4}& {5}& {6}\\ {2}& {1}& {4}& {3}& {6}& {5}\\ {3}& {5}& {1}& {6}& {2}& {4}\\ {4}& {6}& {2}& {5}& {1}& {3}\\ {5}& {3}& {6}& {1}& {4}& {2}\\ {6}& {4}& {5}& {2}& {3}& {1}\end{array}\right]$ (5)

Compatibility

 • The GroupTheory[CayleyTable] command was introduced in Maple 17.