LeftCoset - Maple Help

GroupTheory

 LeftCoset
 construct the left coset of a subgroup
 RightCoset
 construct the right coset of a subgroup

 Calling Sequence LeftCoset( g, H ) RightCoset( H, g )

Parameters

 g - an element of some group containing H H - a subgroup of a permutation group

Description

 • Let $H$ be a subgroup of a group $G$, and let $g$ be a member of $G$. The left coset $\mathrm{gH}$ is defined to be the subset $\left\{\mathrm{gh}:h\in H\right\}$ of $G$. Similarly, the right coset $\mathrm{Hg}$ is the subset $\left\{\mathrm{hg}:h\in H\right\}$ of $G$.
 • The LeftCoset( g, H ) command returns a data structure representing the left coset $\mathrm{gH}$ of a subgroup H of a permutation group G. The RightCoset( H, g ) command returns a data structure representing the right coset $\mathrm{Hg}$ of a subgroup H of a permutation group G.
 • The data structures representing (left or right) cosets respond to the following methods.

 Representative( c ) returns the representative of the coset c numelems( c ) returns the number of members of the coset c member( x, c ) or x in c returns true if x belongs to the coset c Elements( c ) returns the elements of the coset c, as a set Subgroup( c ) returns the subgroup of the coset c

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $H≔\mathrm{Subgroup}\left(\left\{\left[\left[1,2\right],\left[3,4\right]\right]\right\},G\right)$
 ${H}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right)⟩$ (2)
 > $C≔\mathrm{RightCoset}\left(H,\left[\left[1,2,3\right]\right]\right)$
 ${C}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right)⟩{·}\left(\left({1}{,}{2}{,}{3}\right)\right)$ (3)
 > $\mathrm{GroupOrder}\left(H\right)$
 ${2}$ (4)
 > $\mathrm{numelems}\left(C\right)$
 ${2}$ (5)
 > $\mathrm{Elements}\left(C\right)$
 $\left\{\left({1}{,}{3}{,}{4}\right){,}\left({1}{,}{2}{,}{3}\right)\right\}$ (6)

The symmetric group on 3 letters.

 > $M≔⟨⟨1|2|3|4|5|6⟩,⟨2|1|6|5|4|3⟩,⟨3|5|1|6|2|4⟩,⟨4|6|5|1|3|2⟩,⟨5|3|4|2|6|1⟩,⟨6|4|2|3|1|5⟩⟩$
 ${M}{≔}\left[\begin{array}{cccccc}{1}& {2}& {3}& {4}& {5}& {6}\\ {2}& {1}& {6}& {5}& {4}& {3}\\ {3}& {5}& {1}& {6}& {2}& {4}\\ {4}& {6}& {5}& {1}& {3}& {2}\\ {5}& {3}& {4}& {2}& {6}& {1}\\ {6}& {4}& {2}& {3}& {1}& {5}\end{array}\right]$ (7)
 > $G≔\mathrm{CayleyTableGroup}\left(M\right)$
 ${G}{≔}{\mathrm{< a Cayley table group with 6 elements >}}$ (8)

2 is an involution.

 > $H≔\mathrm{Subgroup}\left(\left[2\right],G\right)$
 ${H}{≔}{\mathrm{< a Cayley table group with 1 generator >}}$ (9)
 > $\mathrm{RC}≔\mathrm{RightCoset}\left(H,3\right)$
 ${\mathrm{RC}}{≔}{\mathrm{< a Cayley table group with 1 generator >}}{·}{3}$ (10)
 > $\mathrm{numelems}\left(H\right)=\mathrm{numelems}\left(\mathrm{RC}\right)$
 ${2}{=}{2}$ (11)
 > $\mathrm{Elements}\left(\mathrm{RC}\right)$
 $\left\{{3}{,}{6}\right\}$ (12)
 > $\mathrm{LC}≔\mathrm{LeftCoset}\left(3,H\right)$
 ${\mathrm{LC}}{≔}{3}{·}{\mathrm{< a Cayley table group with 2 elements >}}$ (13)
 > $\mathrm{numelems}\left(H\right)=\mathrm{numelems}\left(\mathrm{LC}\right)$
 ${2}{=}{2}$ (14)
 > $\mathrm{Elements}\left(\mathrm{LC}\right)$
 $\left\{{3}{,}{5}\right\}$ (15)

Compatibility

 • The GroupTheory[LeftCoset] and GroupTheory[RightCoset] commands were introduced in Maple 17.