Cosets

Subsections

• Coset Tables and Transversals
• Action on a Coset Space

Coset Tables and Transversals

Transversal(G, H) : GrpPC, GrpPC -> {@ GrpPCElt @}, Map

RightTransversal(G, H) : GrpPC, GrpPC -> {@ GrpPCElt @}, Map

Given a group G and a subgroup H of G, this function returns

• An indexed set of elements T of G forming a right transversal for G over H; and
• The corresponding transversal mapping phi: G -> T. If T = [t_1, ..., t_r] and g in G, phi is defined by phi(g) = t_i, where g in H * t_i.

CosetTable(G, H) : GrpPC, GrpPC -> Map

Given a group G and a subgroup H of G of index r, return a mapping M:< {1..r}, G > -> {1..r} describing the action of G on the (right) cosets of H.

Transversal(G, H, K) : GrpPC, GrpPC, GrpPC -> {@ GrpPCElt @}, Map

An indexed set of representatives for the double cosets HuK in G, and the corresponding transversal mapping. The algorithm used is described in [Sla01].

Action on a Coset Space

CosetAction(G, H) : Grp, Grp -> Hom(Grp), GrpPerm, GrpPC

Given a subgroup H of the group G, construct the permutation representation of G given by the action of G on the (right) coset space cos(G, H). The function returns:

• The natural homomorphism f: G -> L;
• The induced group L;
• The kernel K of the action (a subgroup of G).

CosetImage(G, H) : Grp, Grp -> GrpPerm

Given a subgroup H of the group G, construct the image L of G given by the action of G on the (right) coset space cos(G, H). L is returned as a permutation group.

CosetKernel(G, H) : Grp, Grp -> Grp

Given a subgroup H of the group G, construct the kernel of the action of G on the (right) coset space cos(G, H).