Cosets

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 \(\phi: G \rightarrow T\). If \(T = [t_1, \ldots, t_r]\) 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 \(M:\langle\{1..r\},G\rangle\rightarrow\{1..r\}\) 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 [Slattery, 2001].

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 \(f: G \rightarrow L\);

(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 \(G\).