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

This function finds all the right cosets of the given subgroup in the given group, assigns integers consecutively to these cosets, constructs a permutation on these coset numbers for each group generator, and returns the permutation group generated by these permutations. Thus the permutation group will be a homomorphic image of (but not necessarily isomorphic to) the original group. A permgroup is returned whose generators are named the same as the original group generators.

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

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

$g≔\mathrm{grelgroup}\left(\left\{x\,y\right\}\,\left\{\left[x\,x\,y\,x\,y\,y\,y\right]\,\left[y\,y\,x\,y\,x\,x\,x\right]\right\}\right)\:$

$\mathrm{sg}≔\mathrm{subgrel}\left(\left\{y=\left[y\right]\right\}\,g\right)\:$

$\mathrm{permrep}\left(\mathrm{sg}\right)$

$\mathrm{permgroup}\left(8\,\left\{x=\left[\left[1\,2\,3\,7\,5\,6\,4\right]\right]\,y=\left[\left[2\,5\,8\,6\,7\,3\,4\right]\right]\right\}\right)$