The (right) coset table for G over the subgroup H of finite index, relative to the polycyclic generators. This is defined to be a map {1, ..., |G:H|} x G -> {1, ..., |G:H|} describing the action of G on the enumerated set of right cosets of H in G by right multiplication.The underlying set of right coset representatives is identical to the right transversal returned by Transversal and RightTransversal and the same enumeration of the elements is used.
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. The right transversal and its enumeration are identical to those internally used by the function CosetTable.
- (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.
> G<a,b> := DihedralGroup(GrpGPC, 0); > H := sub<G|a*b, b^10>; > Index(G, H); 10 > RT, transmap := Transversal(G, H); > RT; {@ Id(G), b^-1, b^-2, b^-3, b^-4, b^-5, b^-6, b^-7, b^-8, b^-9 @} > transmap; Mapping from: GrpGPC: G to SetIndx: RTFrom this a left transversal is easily obtained:
> LT := {@ x^-1 : x in RT @}; > LT; {@ Id(G), b, b^2, b^3, b^4, b^5, b^6, b^7, b^8, b^9 @}We construct the coset table and define a function RT x G -> RT, describing the action of G on the set of right cosets of H in G.
> ct := CosetTable(G, H); > action := func< r, g | RT[ct(Index(RT, r), g)] >; > action(Id(G), b); b^-9I.e. H * b = Hb - 9.
> action(b^-4, a*b); b^-6I.e. Hb - 4 * (ab) = Hb - 6.
Note that the definition of the function action relies on the fact that the computed right transversal and its enumeration are identical to those internally used by the function CosetTable.
Given a subgroup H of the group G of finite index, construct the permutation representation of G, induced by the action of G on the set of (right) cosets of H in G. The function returns:
- (a)
- The permutation representation f: G -> L ≤(Sym)(|G:H|), induced by the action of G on the set of (right) cosets of H in G;
- (b)
- The epimorphic image L of G under the representation f;
- (c)
- The kernel K of the representation f.
Given a subgroup H of the group G of finite index, construct the permutation group, induced by the action of G on the set of (right) cosets of H in G. The returned group is the epimorphic image L of G under the permutation representation f: G -> L ≤(Sym)(|G:H|), induced by the action of G on the set of (right) cosets of H in G.
Given a subgroup H of the group G of finite index, construct the kernel of the permutation representation f: G -> L ≤(Sym)(|G:H|), induced by the action of G on the set of (right) cosets of H in G.
> F<a,b,c> := FreeGroup(3); > rels := [ b^a=b*c, b^(a^-1)=b*c^-1 ]; > G<a,b,c> := quo<GrpGPC: F | rels>; > > S := sub<G|(a*b)^3, c^7, b^21>; > Index(G, S); 441 > pi, P, K := CosetAction(G, S); > P; Permutation group P acting on a set of cardinality 441 > K; GrpGPC : K of infinite order on 3 PC-generators PC-Relations: K.2^K.1 = K.2 * K.3^63, K.2^(K.-1) = K.2 * K.-3^63 > Index(G, K); 3087We express the generators of the kernel K in terms of the generators of G:
> {@ G!x : x in PCGenerators(K) @}; {@ a^21, b^21, c^7 @}pi(S) is a point stabiliser in the transitive permutation group P of degree 441 and hence should have index 441 in P:
> pi(S); Permutation group acting on a set of cardinality 441 > Index(P, pi(S)); 441