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.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$E≔\mathrm{Elements}\left(G\right)$

$E≔\left\{\left(2\,4\,3\right)\,\left(1\,4\right)\left(2\,3\right)\,\left(1\,3\,2\right)\,\left(1\,4\,2\right)\,\left(1\,3\right)\left(2\,4\right)\,\left(1\,2\right)\left(3\,4\right)\,\left(\right)\,\left(1\,4\,3\right)\,\left(1\,2\,4\right)\,\left(1\,3\,4\right)\,\left(1\,2\,3\right)\,\left(2\,3\,4\right)\right\}$

$\mathrm{nops}\left(E\right)=\mathrm{GroupOrder}\left(G\right)$

$12=12$

$\mathrm{map}\left(\mathrm{PermParity}\,E\right)$

$\left\{1\right\}$

$\mathrm{orb}≔\mathrm{Orbit}\left(2\,G\right)$

$\mathrm{orb}≔2^{{\mathbf{A}}_4}$

$\mathrm{Elements}\left(\mathrm{orb}\right)$

$\left\{1\,2\,3\,4\right\}$

$H≔\mathrm{SylowSubgroup}\left(2\,G\right)$

$H≔\mathrm{<a\; permutation\; group\; on\; 4\; letters>}$

$\mathrm{rc}≔\mathrm{RightCosets}\left(H\&comma;G\right)\&colon;$

$c≔\mathrm{rc}\left[-1\right]$

$c≔⟨\left(1\&comma;4\right)\left(2\&comma;3\right)\&comma;\left(1\&comma;3\right)\left(2\&comma;4\right)⟩·\left(\left(2\&comma;3\&comma;4\right)\right)$

$\mathrm{Elements}\left(c\right)$

$\left\{\left(1\&comma;3\&comma;2\right)\&comma;\left(1\&comma;4\&comma;3\right)\&comma;\left(1\&comma;2\&comma;4\right)\&comma;\left(2\&comma;3\&comma;4\right)\right\}$

$\mathrm{cc}≔\mathrm{ConjugacyClasses}\left(G\right)$

$\mathrm{cc}≔\left[{\left(\right)}^{{\mathbf{A}}_4}\&comma;{\left(\left(1\&comma;2\right)\left(3\&comma;4\right)\right)}^{{\mathbf{A}}_4}\&comma;{\left(\left(2\&comma;3\&comma;4\right)\right)}^{{\mathbf{A}}_4}\&comma;{\left(\left(2\&comma;4\&comma;3\right)\right)}^{{\mathbf{A}}_4}\right]$

$\mathrm{GroupOrder}\left(G\right)=\mathrm{add}\left(i\&comma;i=\mathrm{map}\left(\mathrm{nops}@\mathrm{Elements}\&comma;\mathrm{cc}\right)\right)$

$12=12$

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

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