Elements - Maple Help

GroupTheory

 Elements
 get the elements of an object

 Calling Sequence Elements( S )

Parameters

 S - a group data structure, an orbit, a coset, or conjugacy class

Description

 • The Elements command computes the set of elements of an object.
 • The input object S may be a group object, an orbit, a coset, or a conjugacy class.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $E≔\mathrm{Elements}\left(G\right)$
 ${E}{≔}\left\{\left({2}{,}{4}{,}{3}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right){,}\left({1}{,}{3}{,}{2}\right){,}\left({1}{,}{4}{,}{2}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left(\right){,}\left({1}{,}{4}{,}{3}\right){,}\left({1}{,}{2}{,}{4}\right){,}\left({1}{,}{3}{,}{4}\right){,}\left({1}{,}{2}{,}{3}\right){,}\left({2}{,}{3}{,}{4}\right)\right\}$ (2)
 > $\mathrm{nops}\left(E\right)=\mathrm{GroupOrder}\left(G\right)$
 ${12}{=}{12}$ (3)

Check the parity of the elements of G.

 > $\mathrm{map}\left(\mathrm{PermParity},E\right)$
 $\left\{{1}\right\}$ (4)
 > $\mathrm{orb}≔\mathrm{Orbit}\left(2,G\right)$
 ${\mathrm{orb}}{≔}{{2}}^{{{\mathbf{A}}}_{{4}}}$ (5)
 > $\mathrm{Elements}\left(\mathrm{orb}\right)$
 $\left\{{1}{,}{2}{,}{3}{,}{4}\right\}$ (6)
 > $H≔\mathrm{SylowSubgroup}\left(2,G\right)$
 ${H}{≔}{\mathrm{}}$ (7)
 > $\mathrm{rc}≔\mathrm{RightCosets}\left(H,G\right):$
 > $c≔\mathrm{rc}\left[-1\right]$
 ${c}{≔}⟨\left({1}{,}{4}\right)\left({2}{,}{3}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)⟩{·}\left(\left({2}{,}{3}{,}{4}\right)\right)$ (8)
 > $\mathrm{Elements}\left(c\right)$
 $\left\{\left({1}{,}{3}{,}{2}\right){,}\left({1}{,}{4}{,}{3}\right){,}\left({1}{,}{2}{,}{4}\right){,}\left({2}{,}{3}{,}{4}\right)\right\}$ (9)
 > $\mathrm{cc}≔\mathrm{ConjugacyClasses}\left(G\right)$
 ${\mathrm{cc}}{≔}\left[{\left(\right)}^{{{\mathbf{A}}}_{{4}}}{,}{\left(\left({1}{,}{2}\right)\left({3}{,}{4}\right)\right)}^{{{\mathbf{A}}}_{{4}}}{,}{\left(\left({2}{,}{3}{,}{4}\right)\right)}^{{{\mathbf{A}}}_{{4}}}{,}{\left(\left({2}{,}{4}{,}{3}\right)\right)}^{{{\mathbf{A}}}_{{4}}}\right]$ (10)
 > $\mathrm{GroupOrder}\left(G\right)=\mathrm{add}\left(i,i=\mathrm{map}\left(\mathrm{nops}@\mathrm{Elements},\mathrm{cc}\right)\right)$
 ${12}{=}{12}$ (11)

Compatibility

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