Cosets

Contents

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
(a)
An indexed set of elements T of G forming a right transversal for G over H; and
(b)
The corresponding transversal mapping φ: G -> T. If T = [t1, ..., tr] and g in G, φ is defined by φ(g) = ti, where g∈H * ti.
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].
ShortCosets(p, H, G) : GrpPCElt, GrpPC, GrpPC -> [GrpPCElt]
Computes a set of representatives for the transversal of G modulo H of all cosets that contain p. This computation does not do a full transversal of G modulo H and may therefore be used even if the index of (G:H) is very large.

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 set of (right) cosets of H in G. The function returns:
(a)
The natural homomorphism f: G -> L;
(b)
The induced group L;
(c)
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 set of (right) cosets of H in G. 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 set of (right) cosets of H in G.
V2.28, 13 July 2023