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.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$H≔\mathrm{Subgroup}\left(\left\{\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)\right\}\,G\right)$

$H≔⟨\left(1\,2\right)\left(3\,4\right)⟩$

$C≔\mathrm{RightCoset}\left(H\,\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\right]\right)\right)$

$C≔⟨\left(1\,2\right)\left(3\,4\right)⟩·\left(\left(1\,2\,3\right)\right)$

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

$2$

$\mathrm{numelems}\left(C\right)$

$2$

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

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

$\mathrm{Representative}\left(C\right)$

$\left(1\,2\,3\right)$

$\mathrm{evalb}\left(\mathrm{Perm}\left(\left[\left[1\,3\,2\right]\right]\right)\mathbf{in}C\right)$

$\mathrm{false}$

$M≔⟨⟨1\left|2\right|3\left|4\right|5|6⟩\,⟨2\left|1\right|6\left|5\right|4|3⟩\,⟨3\left|5\right|1\left|6\right|2|4⟩\,⟨4\left|6\right|5\left|1\right|3|2⟩\,⟨5\left|3\right|4\left|2\right|6|1⟩\,⟨6\left|4\right|2\left|3\right|1|5⟩⟩$

$M≔\left[\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 2& 1& 6& 5& 4& 3\\ 3& 5& 1& 6& 2& 4\\ 4& 6& 5& 1& 3& 2\\ 5& 3& 4& 2& 6& 1\\ 6& 4& 2& 3& 1& 5\end{array}\right]$

$G≔\mathrm{CayleyTableGroup}\left(M\right)$

$G≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 6\; elements\; >}$

$H≔\mathrm{Subgroup}\left(\left[2\right]\&comma;G\right)$

$H≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 1\; generator\; >}$

$\mathrm{RC}≔\mathrm{RightCoset}\left(H\&comma;3\right)$

$\mathrm{RC}≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 1\; generator\; >}·3$

$\mathrm{numelems}\left(H\right)=\mathrm{numelems}\left(\mathrm{RC}\right)$

$2=2$

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

$\left\{3\&comma;6\right\}$

$\mathrm{LC}≔\mathrm{LeftCoset}\left(3\&comma;H\right)$

$\mathrm{LC}≔3·\mathrm{<\; a\; Cayley\; table\; group\; with\; 2\; elements\; >}$

$\mathrm{numelems}\left(H\right)=\mathrm{numelems}\left(\mathrm{LC}\right)$

$2=2$

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

$\left\{3\&comma;5\right\}$

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

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