The operations of this section all assume that the complete coset tables for the subgroups H and K of the fp-group G can be constructed.
Given a subgroup H of finite index in the fp-group G, and a word u of G, construct the subgroup obtained by conjugating H by u. Note that this operation only requires that generating words for H in G be known: the complete coset table is not required.
Given subgroups H and K, both of finite index in the fp-group G, return the subgroup which is the intersection of H and K.
Given a subgroup H of finite index in the fp-group G, construct the core of H in G.
If H has been defined as a subgroup of the fp-group G, this function returns generators of H as words in the generators of G (assuming they are known).
Given a subgroup H of finite index in the fp-group G, construct a maximal overgroup of H in G. A maximal overgroup of H is a maximal subgroup of G that contains H. If H is already maximal, the group G is returned.
Given a subgroup H of finite index in the fp-group G, construct a minimal overgroup of H in G. A minimal overgroup of a subgroup H is a subgroup K of G such that K contains H as a maximal subgroup. If H is already maximal in G, the group G is returned.
Given a subgroup H of finite index in the fp-group G, construct the normal closure of H in G.
Given a subgroup H of finite index in the fp-group G, construct the normalizer of H in G.
Given a subgroup H of finite index in the fp-group G, return the Schreier generators for H as a set of words in G.
Given a subgroup H of finite index in the fp-group G, construct a (right) Schreier system of coset representatives for H in G. The function returns(a) the Schreier system as a set of words in G;
(b) the corresponding Schreier coset function.
Given subgroups H and K, both of finite index in the fp-group G, return an indexed set of words which comprise a set of representatives for the double cosets HuK of H and K in G, as well as a map from G to the representatives. It should be noted that this function is evaluated by first constructing the right cosets of H in G and then computing the orbits of the cosets under the action of the generators of the subgroup K. Thus, the applicability of this function is limited to subgroups H whose cosets may be enumerated by the Todd-Coxeter procedure.
> p4g<r, s> := Group< r, s | r^2 = s^4 = (r*s^-1*r*s)^2 = 1 >;
> p4g;
GrpFP: p4g on 2 generators
Relations
r^2 = Id(p4g)
s^4 = Id(p4g)
(r * s^-1 * r * s)^2 = Id(p4g)
> h := sub< p4g | (s^-1 * r)^4, s * r >;
> k := sub< p4g | (s^-1 * r)^2, (s * r)^2 >;
> Index(p4g, h);
8
> Index(p4g, k);
8
> n := NormalClosure(p4g, h);
> n;
Finitely presented group n on 5 generators
Index in group p4g is 2
Generators as words in group p4g
n.1 = s * r
n.2 = (s^-1 * r)^4
n.3 = r^-1 * s * r^2
n.4 = r * s
n.5 = s^2 * r * s^-1
> m := MinimalOvergroup(p4g, h);
> m;
Finitely presented group m on 3 generators
Index in group p4g is 4 = 2^2
Generators as words in group p4g
m.1 = s * r
m.2 = (s^-1 * r)^4
m.3 = (r * s)^2
> n := MaximalOvergroup(p4g, k);
> n;
Finitely presented group n on 4 generators
Index in group p4g is 2
Generators as words in group p4g
n.1 = r
n.2 = s^-2
n.3 = s * r * s^2 * r * s^-2 * r * s^-2 * r * s^-1
n.4 = s * r * s
> l := h meet k;
> l;
Finitely presented group l
Subgroup of group p4g defined by coset table
> j := h^s;
> j;
Finitely presented group j on 2 generators
Generators as words in group p4g
j.1 = r * s
j.2 = (s^-2 * r * s)^4
> j eq h;
false
> IsConjugate(p4g, h, j);
true r^-1
> c := Core(p4g, h);
> c;
Finitely presented group C on 17 generators
Index in group p4g is 32 = 2^5
Generators as words in group p4g
C.1 = s^4
C.2 = r * s^4 * r
C.3 = (s * r)^4
C.4 = (s * r * s^-1 * r)^2
C.5 = (s^-1 * r * s * r)^2
C.6 = (s^-1 * r)^4
C.7 = s * r * s^2 * r * s^-1 * r * s^-2 * r
C.8 = s * r * s^4 * r * s^-1
C.9 = s^2 * r * s * r * s^-2 * r * s^-1 * r
C.10 = s^2 * r * s^-1 * r * s^-2 * r * s * r
C.11 = s^-1 * r * s^2 * r * s * r * s^-2 * r
C.12 = s^-1 * r * s^4 * r * s
C.13 = r * s * r * s^4 * r * s^-1 * r
C.14 = r * s^-1 * r * s^4 * r * s * r
C.15 = s^2 * r * s^2 * r * s^-2 * r * s^-2 * r
C.16 = s^2 * r * s^4 * r * s^-2
C.17 = r * s^2 * r * s^4 * r * s^-2 * r
The operations of this subsection all assume that the complete coset tables for
the subgroups H and K of the fp-group G can be constructed.
u in H : GrpFPElt, GrpFP -> BoolElt
Given a subgroup H of finite index in the fp-group G, and a word u of G, return true if u is an element of H.
Given a subgroup H of finite index in the fp-group G, and a word u of G, return true if u is not an element of H.
Given subgroups H and K, both of finite index in the fp-group G, return true if H and K are equal.
Given subgroups H and K, both of finite index in the fp-group G, return true if H and K are not equal.
Given subgroups H and K, both of finite index in the fp-group G, return true if H is contained in K.
Given subgroups H and K, both of finite index in the fp-group G, return true if H is not contained in K.
Given subgroups H and K, both of finite index in the fp-group G, return true and a conjugating element if H and K are conjugate subgroups of G and false otherwise.
Given a subgroup H of finite index in the fp-group G, return true if H is a normal subgroup of G.
Given a subgroup H of finite index in the fp-group G, return true if H is a maximal subgroup of G.
Given a subgroup H of finite index in the fp-group G, return true if H is a self-normalizing subgroup of G.
> J2<a, b, c> := Group<a, b, c | a^3, b^3, c^3, a*b*a*b^-1*a^-1*b^-1, (c*a)^5,
> (c*b)^5, (c*b^-1*c*b)^2,
> a^-1*b*a*c*a^-1*b*a*c^-1*a^-1*b^-1*a*c^-1,
> a*b*a^-1*c*a*b*a^-1*c^-1*a*b^-1*a^-1*c^-1>;
>
> Seen := { 0, 1 };
> Found := { };
> Sgs := [ ];
> for i := 1 to 30 do
> u := Random(J2, 1, 1);
> v := Random(J2, 3, 5);
> H := sub< J2 | u, v >;
> Indx := Index(J2, H);
> if Indx notin Seen then
> Include(~Seen, Indx);
> if not IsMaximal(J2, H) then
> H := MaximalOvergroup(J2, H);
> end if;
> if Indx notin Found then
> Include(~Sgs, H);
> Include(~Seen, Indx);
> Include(~Found, Indx);
> end if;
> end if;
> end for;
> Sgs;
[
Finitely presented group on 3 generators
Index in group J2 is 315 = 3^2 * 5 * 7
Generators as words in group J2
$.1 = b^-1
$.2 = a^-2 * c * a^-1
$.3 = c,
Finitely presented group on 3 generators
Index in group J2 is 1008 = 2^4 * 3^2 * 7
Generators as words in group J2
$.1 = b^-1
$.2 = c^-1 * a^-1 * c^-1 * b
$.3 = a * c * b * c^-1 * a^-1 * c * b * c^-1,
Finitely presented group on 3 generators
Index in group J2 is 100 = 2^2 * 5^2
Generators as words in group J2
$.1 = c
$.2 = (b * a^-1)^2
$.3 = a * b^-1
]
Thus after taking 30 2-generator random subgroups, we have obtained
three maximal subgroups, including the two largest maximal subgroups.