The DirectProduct command takes a sequence of zero or more groups as input, and returns a group data structure representing the direct product of these groups.

An element of the direct product is a list [s1,s2,...] where s1 is an element from G1, s2 is from G2, and so on.

Therefore, the generators defined by DirectProduct are of the form [s1,e2,e3,..], where s1 is a generator from G1, e2 is the identity from G2, e3 is the identity from G3, and so on. Similarly, we have the generators [e1,s2,e3,..],[e1,e2,s3,...] and so forth.

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

$G≔\mathrm{DirectProduct}\left(\mathrm{Alt}\left(4\,'\mathrm{form}'=''fpgroup''\right)\,\mathrm{DihedralGroup}\left(5\right)\,\mathrm{PSL}\left(2\,3\right)\right)$

$G≔⟨s\,t\mid s^2\,t^3\,ststst\,t^{-1}{s}^{-1}ts{t}^{-1}{s}^{-1}ts⟩\times {\mathbf{D}}_5\times \mathbf{PSL}\left(2\,3\right)$

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

$1440$

$\mathrm{DirectFactor}\left(G\,2\right)$

${\mathbf{D}}_5$

$e≔\mathrm{RandomElement}\left(G\right)$

$e≔\left[\left[\frac{1}{s}\,t\,s\,t\right]\,\left(1\,3\,5\,2\,4\right)\,\left(1\,3\right)\left(2\,4\right)\right]$

$\mathrm{\phi }≔\mathrm{CanonicalProjection}\left(G\,2\right)$

$\mathrm{\phi }≔\mathrm{<a\; group\; morphism>}$

$f≔\mathrm{\phi }\left(e\right)$

$f≔\left(1\&comma;3\&comma;5\&comma;2\&comma;4\right)$

$f\mathbf{in}\mathrm{DihedralGroup}\left(5\right)$

$\mathrm{true}$

$G≔\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(\mathrm{\infty }\right)\&comma;\mathrm{Symm}\left(3\right)\right)$

$G≔⟨g\mid ⟩\times {\mathbf{S}}_3$

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

$\mathrm{\infty }$

$G≔\mathrm{CustomGroup}\left(\left\{1\right\}\&comma;\mathrm{`=`}\left(\mathrm{`.`}\&comma;\left(a\&comma;b\right)\mapsto a+b\mathbf{mod}2\right)\&comma;\mathrm{`=`}\left(\mathrm{`/`}\&comma;a\mapsto a\right)\right)$

$G≔\mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}$

$H≔\mathrm{DirectProduct}\left(G\&comma;G\right)$

$H≔\mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}\times \mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}$

$\mathrm{Generators}\left(H\right)$

$\left[\left[1\&comma;0\right]\&comma;\left[0\&comma;1\right]\right]$

$\mathrm{AreIsomorphic}\left(H\&comma;\mathrm{Group}\left(\left\{\left[\left[1\&comma;2\right]\&comma;\left[3\&comma;4\right]\right]\&comma;\left[\left[1\&comma;3\right]\&comma;\left[2\&comma;4\right]\right]\right\}\right)\right)$

$\mathrm{true}$

$\mathrm{Elements}\left(\mathrm{DirectProduct}\left(G\&comma;G\&comma;G\right)\right)$

$\left\{\left[0\&comma;0\&comma;0\right]\&comma;\left[0\&comma;0\&comma;1\right]\&comma;\left[0\&comma;1\&comma;0\right]\&comma;\left[0\&comma;1\&comma;1\right]\&comma;\left[1\&comma;0\&comma;0\right]\&comma;\left[1\&comma;0\&comma;1\right]\&comma;\left[1\&comma;1\&comma;0\right]\&comma;\left[1\&comma;1\&comma;1\right]\right\}$

$\mathrm{K1}≔\mathrm{DirectProduct}\left(G\&comma;\mathrm{DirectProduct}\left(\mathrm{SymmetricGroup}\left(3\right)\&comma;\mathrm{CyclicGroup}\left(4\&comma;'\mathrm{form}'=''fpgroup''\right)\right)\right)$

$\mathrm{K1}≔\mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}\times {\mathbf{S}}_3\times ⟨\mathrm{g0}\mid {\mathrm{g0}}^4⟩$

$\mathrm{K2}≔\mathrm{DirectProduct}\left(\mathrm{DirectProduct}\left(G\&comma;\mathrm{SymmetricGroup}\left(3\right)\right)\&comma;\mathrm{CyclicGroup}\left(4\right)\&comma;'\mathrm{form}'=''fpgroup''\right)$

$\mathrm{K2}≔\mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}\times {\mathbf{S}}_3\times C_4$

$\mathrm{AreIsomorphic}\left(\mathrm{K1}\&comma;\mathrm{K2}\right)$

$\mathrm{true}$

$\mathrm{K1}≔\mathrm{DirectProduct}\left(H\&comma;\mathrm{SymmetricGroup}\left(3\right)\right)$

$\mathrm{K1}≔\mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}\times \mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}\times {\mathbf{S}}_3$

$\mathrm{K2}≔\mathrm{DirectProduct}\left(\mathrm{SymmetricGroup}\left(3\right)\&comma;H\right)$

$\mathrm{K2}≔{\mathbf{S}}_3\times \mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}\times \mathrm{<\; a\; custom\; group\; with\; 1\; generator\; >}$

$\mathrm{AreIsomorphic}\left(\mathrm{K1}\&comma;\mathrm{K2}\right)$

$\mathrm{true}$

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

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