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.

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

$G≔\mathrm{SymmetricGroup}\left(3\right)$

$G≔{\mathbf{S}}_3$

$\mathrm{CayleyTable}\left(G\right)$

$\left[\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 2& 1& 4& 3& 6& 5\\ 3& 5& 1& 6& 2& 4\\ 4& 6& 2& 5& 1& 3\\ 5& 3& 6& 1& 4& 2\\ 6& 4& 5& 2& 3& 1\end{array}\right]$

$\mathrm{permutation\_lists}≔\mathrm{combinat}:-\mathrm{permute}\left(3\right)$

$\mathrm{permutation\_lists}≔\left[\left[1\,2\,3\right]\,\left[1\,3\,2\right]\,\left[2\,1\,3\right]\,\left[2\,3\,1\right]\,\left[3\,1\,2\right]\,\left[3\,2\,1\right]\right]$

$\mathrm{permutations}≔\mathrm{map}\left(\mathrm{Perm}\,\mathrm{permutation\_lists}\right)$

$\mathrm{permutations}≔\left[\left(\right)\,\left(2\,3\right)\,\left(1\,2\right)\,\left(1\,2\,3\right)\,\left(1\,3\,2\right)\,\left(1\,3\right)\right]$

$\mathrm{CayleyTable}\left(G\,\mathrm{elements}=\mathrm{permutations}\right)$

$\left[\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 2& 1& 4& 3& 6& 5\\ 3& 5& 1& 6& 2& 4\\ 4& 6& 2& 5& 1& 3\\ 5& 3& 6& 1& 4& 2\\ 6& 4& 5& 2& 3& 1\end{array}\right]$

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

For more information on Maple 17 changes, see Updates in Maple 17.