Cosets and Transversals

Subsections

• Cosets
• Transversals

Cosets

H * g : GrpPerm, GrpPermElt -> Elt

Right coset of the subgroup H of the group G, where g is an element of G.

DoubleCoset(G, H, g, K) : GrpPerm, GrpPerm, GrpPermElt, GrpPerm -> GrpPermDcosElt

The double coset H * g * K of the subgroups H and K of the group G, where g is an element of G.

DoubleCosetRepresentatives(G, H, K) : GrpPerm, GrpPerm, GrpPerm -> SeqEnum

Given a group G and two subgroups H and K of G, return a sequence S containing representatives of the H-K-double cosets in G. The first element of S is guaranteed to be the identity element of G.

x in C : GrpPermElt, Elt -> BoolElt

Returns true if element g of group G lies in the coset C.

x notin C : GrpPermElt, Elt -> BoolElt

Returns true if element g of group G does not lie in the coset C.

C_1 eq C_2 : Elt, Elt -> BoolElt

Returns true if the coset C_1 is equal to the coset C_2.

C_1 ne C_2 : Elt, Elt -> BoolElt

Returns true if the coset C_1 is not equal to the coset C_2.

# C : Elt -> RngIntElt

The cardinality of the coset C.

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

The (right) coset table for G over subgroup H relative to its defining generators.

[Future release] CosetTable(G, f) : Grp, Hom(Grp) -> Hom(Grp)

The coset table for G corresponding to the permutation representation f of G, where f is a homomorphism of G onto a transitive permutation group.

Transversals

Transversal(G, H) : GrpPerm, GrpPerm -> {@ GrpPermElt @}, Map

RightTransversal(G, H) : GrpPerm, GrpPerm -> {@ GrpPermElt @}, Map

Given a permutation 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 -> 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.

TransversalProcess(G, H) : GrpPerm, GrpPerm -> GrpPermTransProc

Given a permutation group G and H, a subgroup of G, create a process to run through a left transversal for H in G. The method used is a backtrack search for a canonical coset representative. TransversalProcess can be used when the index of H in G is too large to allow a full transversal to be created.

TransversalProcessRemaining(P) : GrpPermTransProc -> RngIntElt

The number of coset representatives the process has yet to produce. Initially this will be the index of the subgroup in the group.

TransversalProcessNext(P) : GrpPermTransProc -> GrpPermElt

Advance the process to the next coset representative and return that representative. This may only be used when TransversalProcessRemaining(P) is positive. The first call to TransversalProcessNext will always give the identity element.