LeftCosets - Maple Help

GroupTheory

 LeftCosets
 construct the left cosets of a subgroup of a group
 RightCosets
 construct the right cosets of a subgroup of a group

 Calling Sequence LeftCosets( H, G ) RightCosets( H, G )

Parameters

 G - a permutation group or a Cayley table group H - a subgroup of G

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${12}$ (2)
 > $H≔\mathrm{SylowSubgroup}\left(2,G\right)$
 ${H}{≔}{\mathrm{}}$ (3)
 > $\mathrm{GroupOrder}\left(H\right)$
 ${4}$ (4)
 > $\mathrm{lc}≔\mathrm{LeftCosets}\left(H,G\right)$
 ${\mathrm{lc}}{≔}\left\{\left(\right){·}⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩{,}\left(\left({2}{,}{4}{,}{3}\right)\right){·}⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩{,}\left(\left({2}{,}{3}{,}{4}\right)\right){·}⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩\right\}$ (5)
 > $\mathrm{nops}\left(\mathrm{lc}\right)=\frac{\mathrm{GroupOrder}\left(G\right)}{\mathrm{GroupOrder}\left(H\right)}$
 ${3}{=}{3}$ (6)

Since the subgroup H is normal in G, the left and right cosets coincide.

 > $\mathrm{map}\left(\mathrm{Representative},\mathrm{lc}\right)$
 $\left\{\left(\right){,}\left({2}{,}{4}{,}{3}\right){,}\left({2}{,}{3}{,}{4}\right)\right\}$ (7)
 > $\mathrm{map}\left(\mathrm{Representative},\mathrm{RightCosets}\left(H,G\right)\right)$
 $\left\{\left(\right){,}\left({2}{,}{4}{,}{3}\right){,}\left({2}{,}{3}{,}{4}\right)\right\}$ (8)
 > $\mathrm{IsNormal}\left(H,G\right)$
 ${\mathrm{true}}$ (9)

Compatibility

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