The LeftCosets( H, G ) command returns the set of left cosets of the subgroup H of the permutation group G.

The RightCosets( H, G ) command returns the set of right cosets of the subgroup H of the permutation group G.

In each case, the collection of cosets (left or right) is returned as a set.

The group G must be an instance of either a permutation group or a Cayley table group, and H must be a subgroup of G.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

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

$12$

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

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

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

$4$

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

$\mathrm{lc}≔\left\{\left(\right)·⟨\left(1\&comma;3\right)\left(2\&comma;4\right)\&comma;\left(1\&comma;4\right)\left(2\&comma;3\right)⟩\&comma;\left(\left(2\&comma;4\&comma;3\right)\right)·⟨\left(1\&comma;3\right)\left(2\&comma;4\right)\&comma;\left(1\&comma;4\right)\left(2\&comma;3\right)⟩\&comma;\left(\left(2\&comma;3\&comma;4\right)\right)·⟨\left(1\&comma;3\right)\left(2\&comma;4\right)\&comma;\left(1\&comma;4\right)\left(2\&comma;3\right)⟩\right\}$

$\mathrm{nops}\left(\mathrm{lc}\right)=\frac{\mathrm{GroupOrder}\left(G\right)}{\mathrm{GroupOrder}\left(H\right)}$

$3=3$

$\mathrm{map}\left(\mathrm{Representative}\&comma;\mathrm{lc}\right)$

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

$\mathrm{map}\left(\mathrm{Representative}\&comma;\mathrm{RightCosets}\left(H\&comma;G\right)\right)$

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

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

$\mathrm{true}$

The GroupTheory[LeftCosets] and GroupTheory[RightCosets] commands were introduced in Maple 17.

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