5 Basic methods and functions for pcp-groups

Pcp-groups are groups in the **GAP** sense and hence all generic **GAP** methods for groups can be applied for pcp-groups. However, for a number of group theoretic questions **GAP** does not provide generic methods that can be applied to pcp-groups. For some of these questions there are functions provided in **Polycyclic**.

In this chapter we describe some important basic functions which are available for pcp-groups. A number of higher level functions are outlined in later sections and chapters.

Let \(U, V\) and \(N\) be subgroups of a pcp-group.

`5.1-1 \=`

`‣ \=` ( U, V ) | ( method ) |

decides if `U` and `V` are equal as sets.

`‣ Size` ( U ) | ( method ) |

returns the size of `U`.

`‣ Random` ( U ) | ( method ) |

returns a random element of `U`.

`‣ Index` ( U, V ) | ( method ) |

returns the index of `V` in `U` if `V` is a subgroup of `U`. The function does not check if `V` is a subgroup of `U` and if it is not, the result is not meaningful.

`5.1-5 \in`

`‣ \in` ( g, U ) | ( method ) |

checks if `g` is an element of `U`.

`‣ Elements` ( U ) | ( method ) |

returns a list containing all elements of `U` if `U` is finite and it returns the list [fail] otherwise.

`‣ ClosureGroup` ( U, V ) | ( method ) |

returns the group generated by `U` and `V`.

`‣ NormalClosure` ( U, V ) | ( method ) |

returns the normal closure of `V` under action of `U`.

`‣ HirschLength` ( U ) | ( method ) |

returns the Hirsch length of `U`.

`‣ CommutatorSubgroup` ( U, V ) | ( method ) |

returns the group generated by all commutators \([u,v]\) with \(u\) in `U` and \(v\) in `V`.

`‣ PRump` ( U, p ) | ( method ) |

returns the subgroup \(U'U^p\) of `U` where `p` is a prime number.

`‣ SmallGeneratingSet` ( U ) | ( method ) |

returns a small generating set for `U`.

`‣ IsSubgroup` ( U, V ) | ( function ) |

tests if `V` is a subgroup of `U`.

`‣ IsNormal` ( U, V ) | ( function ) |

tests if `V` is normal in `U`.

`‣ IsNilpotentGroup` ( U ) | ( method ) |

checks whether `U` is nilpotent.

`‣ IsAbelian` ( U ) | ( method ) |

checks whether `U` is abelian.

`‣ IsElementaryAbelian` ( U ) | ( method ) |

checks whether `U` is elementary abelian.

`‣ IsFreeAbelian` ( U ) | ( property ) |

checks whether `U` is free abelian.

A subgroup of a pcp-group \(G\) can be defined by a set of generators as described in Section 4.3. However, many computations with a subgroup \(U\) need an *induced generating sequence* or *igs* of \(U\). An igs is a sequence of generators of \(U\) whose list of exponent vectors form a matrix in upper triangular form. Note that there may exist many igs of \(U\). The first one calculated for \(U\) is stored as an attribute.

An induced generating sequence of a subgroup of a pcp-group \(G\) is a list of elements of \(G\). An igs is called *normed*, if each element in the list is normed. Moreover, it is *canonical*, if the exponent vector matrix is in Hermite Normal Form. The following functions can be used to compute induced generating sequence for a given subgroup `U` of `G`.

`‣ Igs` ( U ) | ( attribute ) |

`‣ Igs` ( gens ) | ( function ) |

`‣ IgsParallel` ( gens, gens2 ) | ( function ) |

returns an induced generating sequence of the subgroup `U` of a pcp-group. In the second form the subgroup is given via a generating set `gens`. The third form computes an igs for the subgroup generated by `gens` carrying `gens2` through as shadows. This means that each operation that is applied to the first list is also applied to the second list.

`‣ Ngs` ( U ) | ( attribute ) |

`‣ Ngs` ( igs ) | ( function ) |

returns a normed induced generating sequence of the subgroup `U` of a pcp-group. The second form takes an igs as input and norms it.

`‣ Cgs` ( U ) | ( attribute ) |

`‣ Cgs` ( igs ) | ( function ) |

`‣ CgsParallel` ( gens, gens2 ) | ( function ) |

returns a canonical generating sequence of the subgroup `U` of a pcp-group. In the second form the function takes an igs as input and returns a canonical generating sequence. The third version takes a generating set and computes a canonical generating sequence carrying `gens2` through as shadows. This means that each operation that is applied to the first list is also applied to the second list.

For a large number of methods for pcp-groups `U` we will first of all determine an `igs` for `U`. Hence it might speed up computations, if a known `igs` for a group `U` is set *a priori*. The following functions can be used for this purpose.

`‣ SubgroupByIgs` ( G, igs ) | ( function ) |

`‣ SubgroupByIgs` ( G, igs, gens ) | ( function ) |

returns the subgroup of the pcp-group `G` generated by the elements of the induced generating sequence `igs`. Note that `igs` must be an induced generating sequence of the subgroup generated by the elements of the `igs`. In the second form `igs` is a igs for a subgroup and `gens` are some generators. The function returns the subgroup generated by `igs` and `gens`.

`‣ AddToIgs` ( igs, gens ) | ( function ) |

`‣ AddToIgsParallel` ( igs, gens, igs2, gens2 ) | ( function ) |

`‣ AddIgsToIgs` ( igs, igs2 ) | ( function ) |

sifts the elements in the list \(gens\) into \(igs\). The second version has the same functionality and carries shadows. This means that each operation that is applied to the first list and the element `gens` is also applied to the second list and the element `gens2`. The third version is available for efficiency reasons and assumes that the second list `igs2` is not only a generating set, but an igs.

A subfactor of a pcp-group \(G\) is again a polycyclic group for which a polycyclic presentation can be computed. However, to compute a polycyclic presentation for a given subfactor can be time-consuming. Hence we introduce *polycyclic presentation sequences* or *Pcp* to compute more efficiently with subfactors. (Note that a subgroup is also a subfactor and thus can be handled by a pcp)

A pcp for a pcp-group \(U\) or a subfactor \(U / N\) can be created with one of the following functions.

`‣ Pcp` ( U[, flag] ) | ( function ) |

`‣ Pcp` ( U, N[, flag] ) | ( function ) |

returns a polycyclic presentation sequence for the subgroup `U` or the quotient group `U` modulo `N`. If the parameter `flag` is present and equals the string "snf", the function can only be applied to an abelian subgroup `U` or abelian subfactor `U`/`N`. The pcp returned will correspond to a decomposition of the abelian group into a direct product of cyclic groups.

A pcp is a component object which behaves similar to a list representing an igs of the subfactor in question. The basic functions to obtain the stored values of this component object are as follows. Let \(pcp\) be a pcp for a subfactor \(U/N\) of the defining pcp-group \(G\).

`‣ GeneratorsOfPcp` ( pcp ) | ( function ) |

this returns a list of elements of \(U\) corresponding to an igs of \(U/N\).

`5.4-3 \[\]`

`‣ \[\]` ( pcp, i ) | ( method ) |

returns the `i`-th element of `pcp`.

`‣ Length` ( pcp ) | ( method ) |

returns the number of generators in `pcp`.

`‣ RelativeOrdersOfPcp` ( pcp ) | ( function ) |

the relative orders of the igs in `U/N`.

`‣ DenominatorOfPcp` ( pcp ) | ( function ) |

returns an igs of `N`.

`‣ NumeratorOfPcp` ( pcp ) | ( function ) |

returns an igs of `U`.

`‣ GroupOfPcp` ( pcp ) | ( function ) |

returns `U`.

`‣ OneOfPcp` ( pcp ) | ( function ) |

returns the identity element of `G`.

The main feature of a pcp are the possibility to compute exponent vectors without having to determine an explicit pcp-group corresponding to the subfactor that is represented by the pcp. Nonetheless, it is possible to determine this subfactor.

`‣ ExponentsByPcp` ( pcp, g ) | ( function ) |

returns the exponent vector of `g` with respect to the generators of `pcp`. This is the exponent vector of `g`\(N\) with respect to the igs of `U/N`.

`‣ PcpGroupByPcp` ( pcp ) | ( function ) |

let `pcp` be a Pcp of a subgroup or a factor group of a pcp-group. This function computes a new pcp-group whose defining generators correspond to the generators in `pcp`.

gap> G := DihedralPcpGroup(0); Pcp-group with orders [ 2, 0 ] gap> pcp := Pcp(G); Pcp [ g1, g2 ] with orders [ 2, 0 ] gap> pcp[1]; g1 gap> Length(pcp); 2 gap> RelativeOrdersOfPcp(pcp); [ 2, 0 ] gap> DenominatorOfPcp(pcp); [ ] gap> NumeratorOfPcp(pcp); [ g1, g2 ] gap> GroupOfPcp(pcp); Pcp-group with orders [ 2, 0 ] gap> OneOfPcp(pcp); identity

gap> G := ExamplesOfSomePcpGroups(5); Pcp-group with orders [ 2, 0, 0, 0 ] gap> D := DerivedSubgroup( G ); Pcp-group with orders [ 0, 0, 0 ] gap> GeneratorsOfGroup( G ); [ g1, g2, g3, g4 ] gap> GeneratorsOfGroup( D ); [ g2^-2, g3^-2, g4^2 ] # an ordinary pcp for G / D gap> pcp1 := Pcp( G, D ); Pcp [ g1, g2, g3, g4 ] with orders [ 2, 2, 2, 2 ] # a pcp for G/D in independent generators gap> pcp2 := Pcp( G, D, "snf" ); Pcp [ g2, g3, g1 ] with orders [ 2, 2, 4 ] gap> g := Random( G ); g1*g2^-4*g3*g4^2 # compute the exponent vector of g in G/D with respect to pcp1 gap> ExponentsByPcp( pcp1, g ); [ 1, 0, 1, 0 ] # compute the exponent vector of g in G/D with respect to pcp2 gap> ExponentsByPcp( pcp2, g ); [ 0, 1, 1 ]

Pcp's for subfactors of pcp-groups have already been described above. These are usually used within algorithms to compute with pcp-groups. However, it is also possible to explicitly construct factor groups and their corresponding natural homomorphisms.

`‣ NaturalHomomorphismByNormalSubgroup` ( G, N ) | ( method ) |

returns the natural homomorphism \(G \to G/N\). Its image is the factor group \(G/N\).

`5.5-2 \/`

`‣ \/` ( G, N ) | ( method ) |

`‣ FactorGroup` ( G, N ) | ( method ) |

returns the desired factor as pcp-group without giving the explicit homomorphism. This function is just a wrapper for `PcpGroupByPcp( Pcp( G, N ) )`

.

**Polycyclic** provides code for defining group homomorphisms by generators and images where either the source or the range or both are pcp groups. All methods provided by GAP for such group homomorphisms are supported, in particular the following:

`‣ GroupHomomorphismByImages` ( G, H, gens, imgs ) | ( function ) |

returns the homomorphism from the (pcp-) group `G` to the pcp-group `H` mapping the generators of `G` in the list `gens` to the corresponding images in the list `imgs` of elements of `H`.

`‣ Kernel` ( hom ) | ( function ) |

returns the kernel of the homomorphism `hom` from a pcp-group to a pcp-group.

`‣ Image` ( hom ) | ( operation ) |

`‣ Image` ( hom, U ) | ( function ) |

`‣ Image` ( hom, g ) | ( function ) |

returns the image of the whole group, of `U` and of `g`, respectively, under the homomorphism `hom`.

`‣ PreImage` ( hom, U ) | ( function ) |

returns the complete preimage of the subgroup `U` under the homomorphism `hom`. If the domain of `hom` is not a pcp-group, then this function only works properly if `hom` is injective.

`‣ PreImagesRepresentative` ( hom, g ) | ( method ) |

returns a preimage of the element `g` under the homomorphism `hom`.

`‣ IsInjective` ( hom ) | ( method ) |

checks if the homomorphism `hom` is injective.

`‣ RefinedPcpGroup` ( G ) | ( function ) |

returns a new pcp-group isomorphic to `G` whose defining polycyclic presentation is refined; that is, the corresponding polycyclic series has prime or infinite factors only. If \(H\) is the new group, then \(H!.bijection\) is the isomorphism \(G \to H\).

`‣ PcpGroupBySeries` ( ser[, flag] ) | ( function ) |

returns a new pcp-group isomorphic to the first subgroup \(G\) of the given series `ser` such that its defining pcp refines the given series. The series must be subnormal and \(H!.bijection\) is the isomorphism \(G \to H\). If the parameter `flag` is present and equals the string "snf", the series must have abelian factors. The pcp of the group returned corresponds to a decomposition of each abelian factor into a direct product of cyclic groups.

gap> G := DihedralPcpGroup(0); Pcp-group with orders [ 2, 0 ] gap> U := Subgroup( G, [Pcp(G)[2]^1440]); Pcp-group with orders [ 0 ] gap> F := G/U; Pcp-group with orders [ 2, 1440 ] gap> RefinedPcpGroup(F); Pcp-group with orders [ 2, 2, 2, 2, 2, 2, 3, 3, 5 ] gap> ser := [G, U, TrivialSubgroup(G)]; [ Pcp-group with orders [ 2, 0 ], Pcp-group with orders [ 0 ], Pcp-group with orders [ ] ] gap> PcpGroupBySeries(ser); Pcp-group with orders [ 2, 1440, 0 ]

By default, a pcp-group is printed using its relative orders only. The following methods can be used to view the pcp presentation of the group.

`‣ PrintPcpPresentation` ( G[, flag] ) | ( function ) |

`‣ PrintPcpPresentation` ( pcp[, flag] ) | ( function ) |

prints the pcp presentation defined by the igs of `G` or the pcp `pcp`. By default, the trivial conjugator relations are omitted from this presentation to shorten notation. Also, the relations obtained from conjugating with inverse generators are included only if the conjugating generator has infinite order. If this generator has finite order, then the conjugation relation is a consequence of the remaining relations. If the parameter `flag` is present and equals the string "all", all conjugate relations are printed, including the trivial conjugate relations as well as those involving conjugation with inverses.

`‣ IsomorphismPcpGroup` ( G ) | ( attribute ) |

returns an isomorphism from `G` onto a pcp-group `H`. There are various methods installed for this operation and some of these methods are part of the **Polycyclic** package, while others may be part of other packages.

For example, **Polycyclic** contains methods for this function in the case that `G` is a finite pc-group or a finite solvable permutation group.

Other examples for methods for IsomorphismPcpGroup are the methods for the case that `G` is a crystallographic group (see **Cryst**) or the case that `G` is an almost crystallographic group (see **AClib**). A method for the case that `G` is a rational polycyclic matrix group is included in the **Polenta** package.

`‣ IsomorphismPcpGroupFromFpGroupWithPcPres` ( G ) | ( function ) |

This function can convert a finitely presented group with a polycyclic presentation into a pcp group.

`‣ IsomorphismPcGroup` ( G ) | ( method ) |

pc-groups are a representation for finite polycyclic groups. This function can convert finite pcp-groups to pc-groups.

`‣ IsomorphismFpGroup` ( G ) | ( method ) |

This function can convert pcp-groups to a finitely presented group.

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