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Since certain group computations are possible or feasible
only for particular group representations,
it is often useful to transfer a group from one category to another.
The functions in this section take a group and return a group isomorphic to it
(or isomorphic to some related group) in another category.
Given a group F in category GrpFP, a prime p and a positive
integer c, construct the largest p-quotient G of F
having lower exponent-p class at most c (or 127, if c is given as 0)
as group in the category GrpPC.
The function also returns the homomorphism from F to G.
The parameters are:
Exponent: RngIntElt Default: 0
If Exponent := m, enforce the exponent law, xm = 1, on the group.
Metabelian: BoolElt Default: false
If Metabelian := true, then a consistent pcp is constructed for the
largest metabelian p-quotient of F having lower exponent-p class
at most c.
Print: RngIntElt Default: 0
This parameter controls the volume of printing. By default its value
is that returned by GetVerbose("pQuotient"), which is
0 unless it has been changed through use of SetVerbose.
The effect is the following:
Print := 0 : No output.
Print := 1 : Report order of p-quotient at each class.
Print := 2 : Report statistics and redundancy information about tails, consistency,
collection of relations and exponent enforcement components of calculation.
Print := 3 : Report in detail on the construction of each class.
Note that the presentation displayed is a power-commutator
presentation (since this is the version stored by the p-quotient).
Workspace: RngIntElt Default: 5000000
The amount of space requested for the p-quotient computation.
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 permutation group L (the image of f);
- (c)
- (if possible) The kernel K of the action (a subgroup of G).
If G is a finitely presented group, then K may be returned undefined.
The permutation representation is obtained by using the Todd-Coxeter
procedure to construct the coset table for H in G. Note that G
may be an infinite group: it is only necessary that the index of H
in G be finite.
Given a subgroup H of the group G, construct
the image of G given by its action on the (right) coset space of H in G,
returning it as a permutation group.
(This is also the second return value of CosetAction(G, H).)
Given a subgroup H of the group G, construct
the kernel of G in its action on the (right) coset space of H in G.
(This is also the third return value of CosetAction(G, H).)
This function may fail if G is a finitely presented group;
it is only available when the index of H in G is very small.
Given a soluble group G, in the category
GrpPerm, GrpMat, GrpAb or GrpPC,
construct a polycyclic group P isomorphic to G. Currently G must be
finite, if it is in the category GrpMat.
In addition to returning P,
the function returns an isomorphism φ: G -> P.
Given a finite soluble group G, in the category
GrpPerm, GrpMat, GrpAb or GrpGPC,
construct a group S given by a power-conjugate presentation, which is
isomorphic to G. In addition to returning S,
the function returns an isomorphism φ: G -> S.
StrongGenerators: BoolElt Default: false
Random: BoolElt Default: true
Max: RngIntElt Default: 100
Run: RngIntElt Default: 20
Given a group G, in the category
GrpPerm, GrpMat, GrpGPC or GrpPC,
construct a finitely presented group F isomorphic to G,
by presenting the group on its given generators.
For groups in the category GrpPerm and GrpMat, the
Todd-Coxeter Schreier algorithm is used to construct the presentation
and a choice of a presentation on the given generators or on the
strong generators is available.
In addition to returning F,
the function returns an isomorphism φ: F -> G,
such that φ(F.i)=G.i for all i.
If the parameter StrongGenerators is set to true ( GrpPerm
and GrpMat only), the presentation will be constructed on the
strong generators of G instead of the given generators.
If strong generators are not already known for G,
they will be constructed;
in this case, the other parameters are also meaningful.
The parameter Random with its associated
parameters Max and Run may be used to apply the Random Schreier
algorithm to construct a probable BSGS before commencing the construction of
the presentation.
We construct a finitely presented group G and a subgroup H,
then find the permutation representation of G
given by its action on the cosets of H.
Since the induced permutation group L has the same order as G,
the representation is faithful,
and the homomorphism f: G -> L is an isomorphism.
> G<a, b> := Group< a, b | a^3, b^3, (b * a)^4,
> ((b^-1)^a * b^-1)^2 * b^a * b >;
> Order(G);
168
> H := sub< G | a^2 * b^2, (a * b)^2 >;
> Index(G, H);
7
> f, L := CosetAction(G, H);
> f;
Mapping from: GrpFP: G to GrpPerm: L
> L;
Permutation group L acting on a set of cardinality 7
(1, 2, 3)(4, 7, 5)
(1, 3, 4)(2, 5, 6)
> Order(L);
168
A permutation representation of Sp(2, 4).
> M := SymplecticGroup(2, 4);
> #M;
60
> Ms := sub< M | M.1 * M.2 >;
> Index(M, Ms);
12
> PG := CosetImage(M, Ms);
> PG;
Permutation group PG acting on a set of cardinality 12
(1, 2, 4)(3, 5, 7)(6, 8, 10)(9, 11, 12)
(1, 3, 2)(4, 6, 8)(5, 7, 9)(10, 12, 11)
> #PG;
60
A finitely presented group isomorphic to PSU(3, 3):
> G := PSU(3, 3);
> F<a, b>, phi := FPGroup(G);
> F;
Finitely presented group F on 2 generators
Relations
a^8 = Id(F)
b^8 = Id(F)
(b * a^-1 * b)^3 = Id(F)
b * a^-1 * b^-1 * a^-1 * b^-1 * a^-1 * b * a^-1 * b * a^-1 = Id(F)
b^-1 * a^-2 * b^-1 * a^-2 * b^-1 * a^-1 * b^-2 * a^-1 = Id(F)
> phi(a) eq G.1 and phi(b) eq G.2;
true
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