Important: The group package has been deprecated. Use the superseding command GroupTheory[RightCosets] instead.

For groups given by generators and relations, the argument sbgrl should be a subgrel. A set of words in the generators of the group is returned.

For permutation groups, both arguments should be permgroups and sg should be a subgroup of pg. A set of permutations in disjoint cycle notation is returned.

The command with(group,cosets) allows the use of the abbreviated form of this command.

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

$g≔\mathrm{grelgroup}\left(\left\{a\,b\,c\right\}\,\left\{\left[a\,b\,c\,a\,\frac{1}{b}\right]\,\left[b\,c\,a\,b\,\frac{1}{c}\right]\,\left[c\,a\,b\,c\,\frac{1}{a}\right]\right\}\right)\:$

$\mathrm{cosets}\left(\mathrm{subgrel}\left(\left\{y=\left[a\,b\,c\right]\right\}\,g\right)\right)$

$\left\{\left[\right]\,\left[a\right]\,\left[a\,b\right]\right\}$

$\mathrm{pg1}≔\mathrm{permgroup}\left(7\,\left\{\left[\left[1\,2\right]\right]\,\left[\left[1\,2\,3\,4\,5\,6\,7\right]\right]\right\}\right)\:$

$\mathrm{pg2}≔\mathrm{permgroup}\left(7\,\left\{\left[\left[1\,2\,3\right]\right]\,\left[\left[3\,4\,5\,6\,7\right]\right]\right\}\right)\:$

$\mathrm{cosets}\left(\mathrm{pg1}\,\mathrm{pg2}\right)$

$\left\{\left[\right]\,\left[\left[6\,7\right]\right]\right\}$

$\mathrm{pg}≔\mathrm{permgroup}\left(4\,\left\{\left[\left[1\,2\right]\right]\,\left[\left[1\,4\right]\right]\right\}\right)\:$

$\mathrm{ident}≔\mathrm{permgroup}\left(4\,\left\{\left[\right]\right\}\right)\:$

$\mathrm{cosets}\left(\mathrm{pg}\,\mathrm{ident}\right)$

$\left\{\left[\right]\,\left[\left[1\,2\right]\right]\,\left[\left[1\,4\right]\right]\,\left[\left[2\,4\right]\right]\,\left[\left[1\,2\,4\right]\right]\,\left[\left[1\,4\,2\right]\right]\right\}$