Subsections

• Construction of Subgroups
• Construction of Quotient Groups
• Construction of Extensions
• Construction of Standard Groups

Construct the subgroup H of the pc-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:

• An element liftable to G;
• A sequence of integers representing an element of G;
• A subgroup of G;
• A set or sequence of (a), (b), or (c).

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.

The generators of H consist of the words specified directly by terms 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 (unless H is trivial).

The inclusion map from H to G is returned as well.

Construct the subgroup N of the pc-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 well.

The map from the subgroup H of G to G.

Construct the quotient Q of the pc-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 pc-group G, construct the quotient of G by N.

    Workspace: RngIntElt                Default: 1000000

    Metabelian: BoolElt                 Default: false

    Exponent: RngIntElt                 Default: 0

    Print: RngIntElt                    Default: 0

Given a pc-group G, a prime p, and a positive integer c, this function constructs a consistent power-conjugate presentation for the largest p-quotient H of G having lower exponent-p class at most min(c, 63). If c is given as zero, then the limit 63 is placed on the class. The function returns both the p-quotient H as a pc-group and the homomorphism from G to H.

For further details about the pQuotient function, see the online help nodes on finitely presented groups.

pQuotient( F, p, c : parameters ) : GrpFP, RngIntElt, RngIntElt -> GrpPC, Map

    Workspace: RngIntElt                Default: 1000000

    Metabelian: BoolElt                 Default: false

    Exponent: RngIntElt                 Default: 0

    Print: RngIntElt                    Default: 0

Given a finitely presented group F, a prime p, and a positive integer c, this function constructs a consistent power-conjugate presentation for the largest p-quotient H of F having lower exponent-p class at most min(c, 63). If c is given as zero, then the limit 63 is placed on the class. The function returns both the p-quotient H defined by a pc-presentation and the homomorphism from F to H.

For further details about the pQuotient function, see the online help nodes on finitely presented groups.

The direct product K of the pc-groups G and H. The second argument returned is a sequence containing the inclusion maps I_G: G -> K and I_H: H -> K. The third argument returned is a sequence containing the projection maps P_G: K -> G and P_H: K -> H.

DirectProduct(Q) : [GrpPC] -> GrpPC, [ Map ], [ Map ]

The direct product of pc-groups in the non-empty sequence Q, and the inclusion and projection maps.

The split extension K of the pc-group G by the pc-group H, where the action of H on G is given by the homomorphism phi: H -> Aut(G). The extension K will have a normal subgroup G isomorphic to G, while the quotient group K/G is isomorphic to H.

The homomorphism phi is given by the sequence of maps f. Suppose that the pc-generators for H are h_1, ..., h_s. The i-th entry of f defines the action of h_i on G. That is, f[i](x) = h_i^(-1).x.h_i, for x in G.

Extension(M, H) : ModGrp, GrpPC -> GrpPC

The split extension K of the G-module M by the pc-group H. We use the action of H on M to define the action of H on an elementary abelian p-group of order p^d where M is a d-dimensional module over GF(p), p prime.

Extension(G, H, f, t) : GrpPC, GrpPC, [Map], [GrpPCElt] -> GrpPC

Extension(G, H, f, t) : GrpPC, GrpPC, [Map], Tup -> GrpPC

The non-split extension K of the pc-group G by the pc-group H, where the action of H on G is given by the homomorphism phi: H -> Aut(G). The extension K will have a normal subgroup G isomorphic to G, while the quotient group K/G is isomorphic to H.

The homomorphism phi is given by the sequence of maps f. Suppose that the pc-generators for H are h_1, ..., h_s. The i-th entry of f defines the action of h_i on G. That is, f[i](x) = h_i^(-1).x.h_i, for x in G.

The specification of t involves giving the relations h_j^(-1)h_ih_j = w_(ij), where w_(ij) is a word in K for 1 <= j < i <= s. For i = j, we need the relation (h_i)^(p_i) = w_(ii), where w_(ii) is a word in K for 1 <= i <= s. Each w_(ij) is the RHS of the relation from H with the tail x_(ij). The tails are given by the sequence t in the order t = [x_(11), x_(21), x_(22), x_(31), ... , x_(ss) ]. Alternatively, t can be given as a set of tuples < i, j, x_(ij) > for non-trivial x_(ij).

Note that if x_(ij) = Id(G), for 1 <= i <= s and 1 <= j <= i, then K will just be the split extension of G and H.

Extension(M, H, t) : ModGrp, GrpPC, [ModGrpElt] -> GrpPC

The non-split extension K of the G-module M by the pc-group H. We use the action of H on M to define the action of H on an elementary abelian p-group of order p^d where M is a d-dimensional module over GF(p), p prime.

The specification of t is similar to that for t in the preceding description.

The wreath product of the pc-groups G and H, where the regular permutation representation of H is used to define the action.

WreathProduct(G, H, f) : GrpPC, GrpPC, Map -> GrpPC

WreathProduct(G, H, f) : GrpPC, GrpPC, [GrpPermElt] -> GrpPC

The wreath product of the pc-groups G and H where the action of H is given by f, which may be either a suitable map or a sequence of permutations defining an isomorphism from H into P, a permutation representation of H. The isomorphism phi: H -> P is defined by H.i -> f[i] for i = 1, ..., s.

A number of functions are provided which construct presentations for various standard finite groups.

AbelianGroup(GrpPC, Q) : Cat, [RngIntElt] -> GrpPC

Construct the abelian group defined by the sequence Q = [n_1, ..., n_r] of positive integers as a pc-group. The function returns the abelian group which is the direct product of the cyclic groups C_(n_1) x C_(n_2) x ... x C_(n_r).

The cyclic group of order n as a pc-group.

The dihedral group of order 2 * n as a pc-group.

Construct an extra-special group G of order p^(2n + 1) as a pc-group. If p = 2, then G is the central product of n copies of the quaternion group Q_8. If p > 2, then G is the central product of n copies of the extra-special group of order p^3 and exponent p.

> H := DihedralGroup(GrpPerm, 5);
> G := WreathProduct(DihedralGroup(GrpPC, 3), DihedralGroup(GrpPC, 5),
>       [H.2, H.1]);