group(deprecated)/cosets - Help

group

 cosets
 find a complete list of right coset representatives for a subgroup of a permutation group or a group given by generators and relations

 Calling Sequence cosets(sbgrl) cosets(pg, sbpg)

Parameters

 sbgrl - subgroup of a group given by generators and relations pg, sbpg - permutation groups of same degree

Description

 • 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.

Examples

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

 > $\mathrm{with}\left(\mathrm{group}\right):$
 > $g≔\mathrm{grelgroup}\left(\left\{a,b,c\right\},\left\{\left[a,b,c,a,\frac{1}{b}\right],\left[b,c,a,b,\frac{1}{c}\right],\left[c,a,b,c,\frac{1}{a}\right]\right\}\right):$
 > $\mathrm{cosets}\left(\mathrm{subgrel}\left(\left\{y=\left[a,b,c\right]\right\},g\right)\right)$
 $\left\{\left[{}\right]{,}\left[{a}\right]{,}\left[{a}{,}{b}\right]\right\}$ (1)
 > $\mathrm{pg1}≔\mathrm{permgroup}\left(7,\left\{\left[\left[1,2\right]\right],\left[\left[1,2,3,4,5,6,7\right]\right]\right\}\right):$
 > $\mathrm{pg2}≔\mathrm{permgroup}\left(7,\left\{\left[\left[1,2,3\right]\right],\left[\left[3,4,5,6,7\right]\right]\right\}\right):$
 > $\mathrm{cosets}\left(\mathrm{pg1},\mathrm{pg2}\right)$
 $\left\{\left[{}\right]{,}\left[\left[{6}{,}{7}\right]\right]\right\}$ (2)

The cosets function can be used to produce all of the elements of a group by finding the cosets of the identity element of the group:

 > $\mathrm{pg}≔\mathrm{permgroup}\left(4,\left\{\left[\left[1,2\right]\right],\left[\left[1,4\right]\right]\right\}\right):$
 > $\mathrm{ident}≔\mathrm{permgroup}\left(4,\left\{\left[\right]\right\}\right):$
 > $\mathrm{cosets}\left(\mathrm{pg},\mathrm{ident}\right)$
 $\left\{\left[{}\right]{,}\left[\left[{1}{,}{2}\right]\right]{,}\left[\left[{1}{,}{4}\right]\right]{,}\left[\left[{2}{,}{4}\right]\right]{,}\left[\left[{1}{,}{2}{,}{4}\right]\right]{,}\left[\left[{1}{,}{4}{,}{2}\right]\right]\right\}$ (3)