The direct product of two magmas A and B is the set of pairs (a,b), with a in A and b in B, and with binary operation defined componentwise.

The DirectProduct( A, B ) command returns the Cayley table of the direct product of the magmas A and B.

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

$A≔⟨⟨⟨1\|2⟩\,⟨2\|1⟩⟩⟩$

$A≔\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$

$B≔⟨⟨⟨1\|1\|2⟩\,⟨2\|3\|2⟩\,⟨3\|2\|1⟩⟩⟩$

$B≔\left[\begin{array}{ccc}1& 1& 2\\ 2& 3& 2\\ 3& 2& 1\end{array}\right]$

$\mathrm{DirectProduct}\left(A\,B\right)$

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

$\mathrm{AreIsomorphic}\left(\mathrm{DirectProduct}\left(A\,B\right)\,\mathrm{DirectProduct}\left(B\,A\right)\right)$

$\mathrm{true}$

$C≔⟨⟨⟨1\|2\|3⟩\,⟨2\|3\|1⟩\,⟨3\|1\|2⟩⟩⟩$

$C≔\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ 3& 1& 2\end{array}\right]$

$\mathrm{AreIsomorphic}\left(\mathrm{DirectProduct}\left(\mathrm{DirectProduct}\left(A\,B\right)\,C\right)\,\mathrm{DirectProduct}\left(A\,\mathrm{DirectProduct}\left(B\,C\right)\right)\right)$

$\mathrm{true}$

The Magma[DirectProduct] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.