DirectProduct - Maple Help

GroupTheory

 DirectProduct
 form the direct product of groups

 Calling Sequence DirectProduct(G1,G2,...)

Parameters

 G1,G2, ... - group data structures

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{DirectProduct}\left(\mathrm{Alt}\left(4,'\mathrm{form}'="fpgroup"\right),\mathrm{DihedralGroup}\left(5\right),\mathrm{PSL}\left(2,3\right)\right)$
 ${G}{≔}⟨{}{s}{,}{t}{}{\mid }{}{{s}}^{{2}}{,}{{t}}^{{3}}{,}{s}{}{t}{}{s}{}{t}{}{s}{}{t}{,}{{t}}^{{-1}}{}{{s}}^{{-1}}{}{t}{}{s}{}{{t}}^{{-1}}{}{{s}}^{{-1}}{}{t}{}{s}{}⟩{×}{{\mathbf{D}}}_{{5}}{×}{\mathbf{PSL}}\left({2}{,}{3}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${1440}$ (2)

Use DirectFactor to access the $k$-th direct factor of a direct product.

 > $\mathrm{DirectFactor}\left(G,2\right)$
 ${{\mathbf{D}}}_{{5}}$ (3)

Access the coordinate projections, as follows.

 > $e≔\mathrm{RandomElement}\left(G\right)$
 ${e}{≔}\left[\left[\frac{{1}}{{s}}{,}{t}{,}{s}{,}{t}\right]{,}\left({1}{,}{3}{,}{5}{,}{2}{,}{4}\right){,}\left({1}{,}{2}{,}{4}\right)\right]$ (4)
 > $\mathrm{φ}≔\mathrm{CanonicalProjection}\left(G,2\right)$
 ${\mathrm{\phi }}{≔}{\mathrm{}}$ (5)
 > $f≔\mathrm{φ}\left(e\right)$
 ${f}{≔}\left({1}{,}{3}{,}{5}{,}{2}{,}{4}\right)$ (6)
 > $f\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∈\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{DihedralGroup}\left(5\right)$
 ${\mathrm{true}}$ (7)
 > $G≔\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(\mathrm{∞}\right),\mathrm{Symm}\left(3\right)\right)$
 ${G}{≔}⟨{}{g}{}{\mid }{}{}⟩{×}{{\mathbf{S}}}_{{3}}$ (8)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${\mathrm{\infty }}$ (9)

Construct the Cyclic Group of order 2.

 > $G≔\mathrm{CustomGroup}\left(\left\{1\right\},\mathrm{.}=\left(\left(a,b\right)→a+b\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}2\right),\mathrm{/}=\left(a→a\right)\right)$
 ${G}{≔}{\mathrm{< a custom group with 1 generator >}}$ (10)

Now form the Klein 4 group.

 > $H≔\mathrm{DirectProduct}\left(G,G\right)$
 ${H}{≔}{\mathrm{< a custom group with 1 generator >}}{×}{\mathrm{< a custom group with 1 generator >}}$ (11)
 > $\mathrm{Generators}\left(H\right)$
 $\left[\left[{1}{,}{0}\right]{,}\left[{0}{,}{1}\right]\right]$ (12)

We verify the isomorphic permutation form of the Klein 4 group.

 > $\mathrm{AreIsomorphic}\left(H,\mathrm{Group}\left(\left\{\left[\left[1,2\right],\left[3,4\right]\right],\left[\left[1,3\right],\left[2,4\right]\right]\right\}\right)\right)$
 ${\mathrm{true}}$ (13)

Consider elements of the DirectProduct.

 > $\mathrm{Elements}\left(\mathrm{DirectProduct}\left(G,G,G\right)\right)$
 $\left\{\left[{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{1}\right]{,}\left[{0}{,}{1}{,}{0}\right]{,}\left[{0}{,}{1}{,}{1}\right]{,}\left[{1}{,}{0}{,}{0}\right]{,}\left[{1}{,}{0}{,}{1}\right]{,}\left[{1}{,}{1}{,}{0}\right]{,}\left[{1}{,}{1}{,}{1}\right]\right\}$ (14)

We verify that DirectProduct is associative and commutative.

 > $\mathrm{K1}≔\mathrm{DirectProduct}\left(G,\mathrm{DirectProduct}\left(\mathrm{SymmetricGroup}\left(3\right),\mathrm{CyclicGroup}\left(4,'\mathrm{form}'="fpgroup"\right)\right)\right)$
 ${\mathrm{K1}}{≔}{\mathrm{< a custom group with 1 generator >}}{×}{{\mathbf{S}}}_{{3}}{×}⟨{}{\mathrm{g0}}{}{\mid }{}{{\mathrm{g0}}}^{{4}}{}⟩$ (15)
 > $\mathrm{K2}≔\mathrm{DirectProduct}\left(\mathrm{DirectProduct}\left(G,\mathrm{SymmetricGroup}\left(3\right)\right),\mathrm{CyclicGroup}\left(4\right),'\mathrm{form}'="fpgroup"\right)$
 ${\mathrm{K2}}{≔}{\mathrm{< a custom group with 1 generator >}}{×}{{\mathbf{S}}}_{{3}}{×}{{C}}_{{4}}$ (16)
 > $\mathrm{AreIsomorphic}\left(\mathrm{K1},\mathrm{K2}\right)$
 ${\mathrm{true}}$ (17)
 > $\mathrm{K1}≔\mathrm{DirectProduct}\left(H,\mathrm{SymmetricGroup}\left(3\right)\right)$
 ${\mathrm{K1}}{≔}{\mathrm{< a custom group with 1 generator >}}{×}{\mathrm{< a custom group with 1 generator >}}{×}{{\mathbf{S}}}_{{3}}$ (18)
 > $\mathrm{K2}≔\mathrm{DirectProduct}\left(\mathrm{SymmetricGroup}\left(3\right),H\right)$
 ${\mathrm{K2}}{≔}{{\mathbf{S}}}_{{3}}{×}{\mathrm{< a custom group with 1 generator >}}{×}{\mathrm{< a custom group with 1 generator >}}$ (19)
 > $\mathrm{AreIsomorphic}\left(\mathrm{K1},\mathrm{K2}\right)$
 ${\mathrm{true}}$ (20)

Compatibility

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