Construct the subgroup H of the polycyclic group G generated by the elements specified by the terms of the generator list L.A term L[i] of the generator list may consist of any of the following objects:
The collection of words and groups specified by the list must all belong to the group G and H will be constructed as a subgroup of G.
- (a)
- An element liftable to G;
- (b)
- A sequence of integers representing an element of G;
- (c)
- A subgroup of G;
- (d)
- A set or sequence of (a), (b), or (c).
The generators of H consist of the words specified directly by terms of L[i] together with the stored generating words for any groups specified by terms of L[i]. Repetitions of an element and occurrences of the identity element are removed.
The inclusion map from H to G is returned as a second value.
Construct the subgroup N of the polycyclic group G as the normal closure of the subgroup generated by the elements specified by the terms of the generator list L.The possible forms of a term L[i] of the generator list are the same as for the sub-constructor.
The inclusion map from N to G is returned as a second value.
Given an element g belonging to the subgroup H of the group G, rewrite g as an element of G.
Given an element g belonging to the group G, and given a subgroup H of G containing g, rewrite g as an element of H.
Given an element g belonging to the group H, and a group K, such that H and K are subgroups of G, and both H and K contain g, rewrite g as an element of K.
The map from the subgroup H of G to 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 >;
> G;
GrpGPC : G of infinite order on 3 PC-generators
PC-Relations:
b^a = b * c,
b^(a^-1) = b * c^-1
Using the function PCGenerators which is described later, the groups G1, ..., G4 and the corresponding inclusion maps can be defined as follows:
> G_ := []; incl_ := [ PowerStructure(Map) | ];
> for i := 1 to #PCGenerators(G)+1 do
> G_[i], incl_[i] := sub< G | [ g : g in PCGenerators(G) |
> Depth(g) ge i ] >;
> end for;
> for i := 1 to #G_ do
> printf "G_%o = <%o>", i, {@ G!x : x in
> PCGenerators(G_[i]) @}; print "";
> end for;
G_1 = <{@ a, b, c @}>
G_2 = <{@ b, c @}>
G_3 = <{@ c @}>
G_4 = <{@ @}>
Note that we must set the universe of the sequence incl_ to
PowerStructure(Map) manually, since we want to store maps which do not
have a common domain. If we failed to do this, the universe would be chosen
automatically to be the set of all maps from G1 to G when the first map
is inserted into the sequence. Inserting the second map, which does not have
the domain G1, would then cause a runtime error.
Construct the quotient Q of the polycyclic group G by the normal subgroup N, where N is the smallest normal subgroup of G containing the elements specified by the terms of the generator list L.The possible forms of a term L[i] of the generator list are the same as for the sub-constructor.
The quotient group Q and the corresponding natural homomorphism f : G -> Q are returned.
Given a normal subgroup N of the polycyclic group G, construct the quotient of G by N.
For a general description of homomorphisms, we refer to Chapter MAPPINGS. This section describes some special aspects of homomorphisms the domain of which is a polycyclic group.
The kernel of a homomorphism with a domain of type GrpGPC can be computed using the function Kernel, if the codomain is of one of the types GrpGPC, GrpPC (cf. Chapter FINITE SOLUBLE GROUPS), GrpAb (cf. Chapter ABELIAN GROUPS), GrpPerm (cf. Chapter PERMUTATION GROUPS), ModAlg or ModGrp (cf. Chapter MODULES OVER AN ALGEBRA AND GROUP REPRESENTATIONS) or if the codomain is of the type GrpMat (cf. Chapter MATRIX GROUPS OVER GENERAL RINGS) and the image is finite.
In particular, preimages of substructures can be computed in these situations. The kernel of a map will be computed automatically, if the preimage of a substructure of the codomain is to be computed.
The kernel (and hence preimages of substructures) may also be computable, if the codomain is of the type GrpFP (cf. Chapter FINITELY PRESENTED GROUPS) and the domain is nilpotent.
Returns the homomorphism from the polycyclic group P to the group G defined by the assignment S. S can be the one of the following:If the data type of the codomain supports element arithmetic and element comparison, by default the constructed homomorphism is checked by verifying that the would-be images of the polycyclic generators satisfy the defining relations of P and that this assignment is consistent with the assignments made by the user. In this case, it is assured that the returned map is a well-defined homomorphism with the desired images. The most important situation in which it is not possible to perform checking is the case in which the domain is a finitely presented group (FPGroup; cf. Chapter FINITELY PRESENTED GROUPS) which is not free. Checking may be disabled using the parameter Check.
- (i)
- A list, sequence or indexed set containing the images of the n polycyclic generators P.1, ..., P.n of P. Here, the i-th element of S is interpreted as the image of P.i, i.e. the order of the elements in S is important.
- (ii)
- A list, sequence, enumerated set or indexed set, containing r tuples <xi, yi> or arrow pairs xi - > yi, where xi∈P, yi∈G (i=1, ..., r) and the set {x1, ..., xr} is a generating set for P. In this case, yi is assigned as the image of xi, hence the order of the elements in S is not important. Note that the preimages xi need not be the polycyclic generators of P.
Check: BoolElt Default: trueIf Check is set to false, checking of the homomorphism is disabled.
The direct product K of the polycyclic groups G and H. The second value returned is a sequence containing the inclusion maps IG: G -> K and IH: H -> K. The third value returned is a sequence containing the projection maps PG: K -> G and PH: K -> H.
A number of functions are provided which construct polycyclic presentations for various standard groups.
Construct the abelian group defined by the sequence Q = [n1, ..., nr] as a polycyclic group. The entries ni may be either zero, indicating an infinite cyclic factor, or integers greater than 1. The function returns the polycyclic group which is the direct product of the cyclic groups Z1 x ... x Zr, where Zi is a cyclic group of infinite order if ni = 0, or a cyclic group of order ni if ni > 1.
For n>0, the cyclic group of order n is returned, while for n=0, the infinite cyclic group is returned as a polycyclic group.
For n≥3, the dihedral group of order 2n is returned, while for n=0, the infinite dihedral group is returned as a polycyclic group.
Given a prime p and a positive integer n, construct the elementary abelian group of order pn as a polycyclic group.
Given a prime p and a small positive integer n, construct an extra-special group G of order p2n + 1 as a polycyclic group. The isomorphism type of G can be selected using the parameter Type.Type: MonStgElt Default: "+"Possible values for this parameter are "+" (default) and "-".If Type is set to "+", the function returns for p = 2 the central product of n copies of the dihedral group of order 8, and for p > 2 it returns the unique extra-special group of order p2n + 1 and exponent p.
If Type is set to "-", the function returns for p = 2 the central product of a quaternion group of order 8 and n - 1 copies of the dihedral group of order 8, and for p > 2 it returns the unique extra-special group of order p2n + 1 and exponent p2.
Given a positive integer n, construct the free abelian group of rank n as a polycyclic group.
The free nilpotent group of rank r and class e is returned as a polycyclic group.
> G<a,b> := DihedralGroup(GrpGPC, 4);We obtain a well-defined homomorphism f of G onto a group A = (Z)2 x (Z)2, by assigning two generators of A as the images of a and ab, respectively. Note that in the definition of f we can use the generating set {a, ab} of G, instead of the defining polycyclic generating sequence of G.
> A<u,v> := AbelianGroup([2,2]); > f := hom< G->A | a->u, a*b->v >;We compute the kernel K of f and express the generators of K as elements of G, using the function PCGenerators described later.
> K := Kernel(f);
> PCGenerators(K, G);
{@ b^2 @}
> G1<a,b> := DihedralGroup(GrpGPC, 3);
> G2<u,v> := DihedralGroup(GrpGPC, 0);
> D, incl, proj := DirectProduct(G1, G2);
> D;
GrpGPC : D of infinite order on 4 PC-generators
PC-Relations:
D.1^2 = Id(D),
D.2^3 = Id(D),
D.3^2 = Id(D),
D.2^D.1 = D.2^2,
D.4^D.3 = D.-4
Using the inclusion maps returned by DirectProduct, we
define a subgroup and compute the quotient by its normal closure:
> S := sub<D| incl[1](a)*incl[2](u), incl[1](b)*incl[2](v)>;
> S;
GrpGPC : S of infinite order on 3 PC-generators
PC-Relations:
S.1^2 = Id(S),
S.2^3 = S.3,
S.2^S.1 = S.2^2 * S.-3,
S.3^S.1 = S.-3
> Q, pi := quo<D|S>;
> Q;
GrpGPC : Q of order 2 on 1 PC-generators
PC-Relations:
Q.1^2 = Id(Q)
> Q.1 @@ pi;
D.3