Pc Groups
Pc groups are polycyclic groups that use the polycyclic presentation for element arithmetic. This presentation gives them a natural pcgs, the with respect to which pcgs operations as described in chapter are particularly efficient.

Let G be a polycyclic group with pcgs P = (g_1, \ldots, g_n) and corresponding relative orders (r_1, \ldots, r_n). Recall that the r_i are positive integers or infinity and let I be the set of indices i with r_i a positive integer. Then G has a finite presentation on the generators g_1, \ldots, g_n with relations of the following form.

g_i^{{r_i}}
=
g_{{i+1}}^{a(i,i,i+1)} \cdots g_n^{a(i,i,n)}
for 1 \leq i \leq n and i \in I
g_i^{{-1}} g_j g_i
=
g_{{i+1}}^{a(i,j,i+1)} \cdots g_n^{a(i,j,n)}
for 1 \leq i < j \leq n

For infinite groups we need additionally

g_i^{{-1}} g_j^{{-1}} g_i
=
g_{{i+1}}^{b(i,j,i+1)} \cdots g_n^{b(i,j,n)}
for 1 \leq i < j \leq n and j \not \in I
g_i g_j g_i^{{-1}}
=
g_{{i+1}}^{c(i,j,i+1)} \cdots g_n^{c(i,j,n)}
for 1 \leq i < j \leq n and i \not \in I
g_i g_j^{{-1}} g_i^{{-1}}
=
g_{{i+1}}^{d(i,j,i+1)} \cdots g_n^{d(i,j,n)}
for 1 \leq i < j \leq n and i, j, \not \in I

Here the right hand sides are assumed to be words in normal form; that is, for k \in I we have for all exponents 0 \leq a(i,j,k), b(i,j,k), c(i,j,k), d(i,j,k) < r_k.

A finite presentation of this type is called a power-conjugate presentation and a pc group is a polycyclic group defined by a power-conjugate presentation. Instead of conjugates we could just as well work with commutators and then the presentation would be called a power-commutator presentation. Both types of presentation are abbreviated as pc presentation. Note that a pc presentation is a rewriting system.

Clearly, whenever a group G with pcgs P is given, then we can write down the corresponding pc presentation. On the other hand, one may just write down a presentation on n abstract generators g_1, \ldots, g_n with relations of the above form and define a group H by this. Then the subgroups C_i = \langle g_i, \ldots, g_n \rangle of H form a subnormal series whose factors are cyclic or trivial. In the case that all factors are non-trivial, we say that the pc presentation of H is confluent. Note that &GAP; 4 can only work correctly with pc groups defined by a confluent pc presentation.

At the current state of implementations the &GAP; library contains methods to compute with finite polycyclic groups, while the &GAP; package `Polycyclic` by Bettina Eick and Werner Nickel allows also computations with infinite polycyclic groups which are given by a pc-presentation.

Algorithms for pc groups use the methods for polycyclic groups described in chapter .

The family pcgs

Clearly, the generators of a power-conjugate presentation of a pc group G form a pcgs of the pc group. This pcgs is called the family pcgs.

<#Include Label="FamilyPcgs"> <#Include Label="IsFamilyPcgs"> <#Include Label="InducedPcgsWrtFamilyPcgs"> <#Include Label="IsParentPcgsFamilyPcgs">

Elements of pc groups
Comparison of elements of pc groups
equality
smaller
The elements of a pc group G are always represented as words in normal form with respect to the family pcgs of G. Thus it is straightforward to compare elements of a pc group, since this boils down to a mere comparison of exponent vectors with respect to the family pcgs. In particular, the word problem is efficiently solvable in pc groups.
Arithmetic operations for elements of pc groups
However, multiplication and inversion of elements in pc groups is not as straightforward as in arbitrary finitely presented groups where a simple concatenation or reversion of the corresponding words is sufficient (but one cannot solve the word problem).

To multiply two elements in a pc group, we first concatenate the corresponding words and then use an algorithm called collection to transform the new word into a word in normal form.

gap> g := FamilyPcgs( SmallGroup( 24, 12 ) ); Pcgs([ f1, f2, f3, f4 ]) gap> g[4] * g[1]; f1*f3 gap> (g[2] * g[3])^-1; f2^2*f3*f4
Pc groups versus fp groups

In theory pc groups are finitely presented groups. In practice the arithmetic in pc groups is different from the arithmetic in fp groups. Thus for technical reasons the pc groups in &GAP; do not form a subcategory of the fp groups and hence the methods for fp groups cannot be applied to pc groups in general.

<#Include Label="IsPcGroup">

It is possible to convert a pc group to a fp group in &GAP;. The function computes the power-commutator presentation defined by pcgs. The string str can be used to give a name to the generators of the fp group.

gap> p := FamilyPcgs( SmallGroup( 24, 12 ) ); Pcgs([ f1, f2, f3, f4 ]) gap> iso := IsomorphismFpGroupByPcgs( p, "g" ); [ f1, f2, f3, f4 ] -> [ g1, g2, g3, g4 ] gap> F := Image( iso ); <fp group of size 24 on the generators [ g1, g2, g3, g4 ]> gap> RelatorsOfFpGroup( F ); [ g1^2, g2^-1*g1^-1*g2*g1*g2^-1, g3^-1*g1^-1*g3*g1*g4^-1*g3^-1, g4^-1*g1^-1*g4*g1*g4^-1*g3^-1, g2^3, g3^-1*g2^-1*g3*g2*g4^-1*g3^-1, g4^-1*g2^-1*g4*g2*g3^-1, g3^2, g4^-1*g3^-1*g4*g3, g4^2 ]
Constructing Pc Groups

If necessary, you can supply &GAP; with a pc presentation by hand. (Although this is the most tedious way to input a pc group.) Note that the pc presentation has to be confluent in order to work with the pc group in &GAP;.

(If you have already a suitable pcgs in another representation, use , see below.)

One way is to define a finitely presented group with a pc presentation in &GAP; and then convert this presentation into a pc group, see . Note that this does not work for arbitrary presentations of polycyclic groups, see Chapter for further information.

Another way is to create and manipulate a collector of a pc group by hand and to use it to define a pc group. &GAP; provides different collectors for different collecting strategies; at the moment, there are two collectors to choose from: the single collector for finite pc groups (see ) and the combinatorial collector for finite p-groups. See for further information on collecting strategies.

A collector is initialized with an underlying free group and the relative orders of the pc series. Then one adds the right hand sides of the power and the commutator or conjugate relations one by one. Note that omitted relators are assumed to be trivial.

For performance reasons it is beneficial to enforce a syllable representation in the free group (see ).

Note that in the end, the collector has to be converted to a group, see .

With these methods a pc group with arbitrary defining pcgs can be constructed. However, for almost all applications within &GAP; we need to have a pc group whose defining pcgs is a prime order pcgs, see and .

<#Include Label="PcGroupFpGroup">

initializes a single collector or a combinatorial collector, where fgrp must be a free group and relorders must be a list of the relative orders of the pc series.

A combinatorial collector can only be set up for a finite p-group. Here, the relative orders relorders must all be equal and a prime.
Let f_1, \ldots, f_n be the generators of the underlying free group of the collector coll.

For i < j, sets the conjugate f_j^{{f_i}} to equal w, which is assumed to be a word in f_{{i+1}}, \ldots, f_n.
Let f_1, \ldots, f_n be the generators of the underlying free group of the collector coll.

For i < j, sets the commutator of f_j and f_i to equal w, which is assumed to be a word in f_{{i+1}}, \ldots, f_n.
Let f_1, \ldots, f_n be the generators of the underlying free group of the collector coll, and let r_i be the corresponding relative orders.

sets the power f_i^{{r_i}} to equal w, which is assumed to be a word in f_{{i+1}}, \ldots, f_n.
creates a group from a rewriting system. In the first version it is checked whether the rewriting system is confluent, in the second version this is assumed to be true.
checks whether the pc group G has been built from a collector with a confluent power-commutator presentation.

gap> F := FreeGroup(IsSyllableWordsFamily, 2 );; gap> coll1 := SingleCollector( F, [2,3] ); <<single collector, 8 Bits>> gap> SetConjugate( coll1, 2, 1, F.2 ); gap> SetPower( coll1, 1, F.2 ); gap> G1 := GroupByRws( coll1 ); <pc group of size 6 with 2 generators> gap> IsConfluent(G1); true gap> IsAbelian(G1); true gap> coll2 := SingleCollector( F, [2,3] ); <<single collector, 8 Bits>> gap> SetConjugate( coll2, 2, 1, F.2^2 ); gap> G2 := GroupByRws( coll2 ); <pc group of size 6 with 2 generators> gap> IsAbelian(G2); false
<#Include Label="IsomorphismRefinedPcGroup"> <#Include Label="RefinedPcGroup">

Computing Pc Groups

Another possibility to get a pc group in &GAP; is to convert a polycyclic group given by some other representation to a pc group. For finitely presented groups there are various quotient methods available. For all other types of groups one can use the following functions.

<#Include Label="PcGroupWithPcgs"> <#Include Label="IsomorphismPcGroup"> <#Include Label="IsomorphismSpecialPcGroup">

Saving a Pc Group

As printing a polycyclic group does not display the presentation, one cannot simply print a pc group to a file to save it. For this purpose we need the following function.

<#Include Label="GapInputPcGroup">

Operations for Pc Groups

All the operations described in Chapters and apply to a pc group. Nearly all methods for pc groups are methods for groups with pcgs as described in Chapter . The only method with is special for pc groups is a method to compute intersections of subgroups, since here a pcgs of a parent group is needed and this can only by guaranteed within pc groups.

2
-Cohomology and Extensions

One of the most interesting applications of pc groups is the possibility to compute with extensions of these groups by elementary abelian groups; that is, H is an extension of G by M, if there exists a normal subgroup N in H which is isomorphic to M such that H/N is isomorphic to G.

Pc groups are particularly suited for such applications, since the 2-cohomology can be computed efficiently for such groups and, moreover, extensions of pc groups by elementary abelian groups can be represented as pc groups again.

To define the elementary abelian group M together with an action of G on M we consider M as a MeatAxe module for G over a finite field (section describes functions that can be used to obtain certain modules). For further information on meataxe modules see Chapter . Note that the matrices defining the module must correspond to the pcgs of the group G.

There exists an action of the subgroup of compatible pairs in Aut(G) \times Aut(M) which acts on the second cohomology group, see . 2-cocycles which lie in the same orbit under this action define isomorphic extensions of G. However, there may be isomorphic extensions of G corresponding to cocycles in different orbits.

See also the &GAP; package `GrpConst` by Hans Ulrich Besche and Bettina Eick that contains methods to construct up to isomorphism the groups of a given order.

Finally we note that for the computation of split extensions it is not necessary that M must correspond to an elementary abelian group. Here it is possible to construct split extensions of arbitrary pc groups, see .

<#Include Label="TwoCoboundaries"> <#Include Label="TwoCocycles"> <#Include Label="TwoCohomology"> <#Include Label="Extensions"> <#Include Label="Extension">

returns the split extension of G by the G-module M.
returns the module of an extension E of G by M. This is the normal subgroup of E which corresponds to M.

gap> G := SmallGroup( 4, 2 );; gap> mats := List( Pcgs( G ), x -> IdentityMat( 1, GF(2) ) );; gap> M := GModuleByMats( mats, GF(2) );; gap> co := TwoCocycles( G, M );; gap> Extension( G, M, co[2] ); <pc group of size 8 with 3 generators> gap> SplitExtension( G, M ); <pc group of size 8 with 3 generators> gap> Extensions( G, M ); [ <pc group of size 8 with 3 generators>, <pc group of size 8 with 3 generators>, <pc group of size 8 with 3 generators>, <pc group of size 8 with 3 generators>, <pc group of size 8 with 3 generators>, <pc group of size 8 with 3 generators>, <pc group of size 8 with 3 generators>, <pc group of size 8 with 3 generators> ] gap> List(last, IdSmallGroup); [ [ 8, 5 ], [ 8, 2 ], [ 8, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 2 ], [ 8, 3 ], [ 8, 4 ] ]

Note that the extensions returned by are computed up to equivalence, but not up to isomorphism.
<#Include Label="CompatiblePairs"> <#Include Label="ExtensionRepresentatives">

returns the split extensions of the pc group G by the pc group N. aut should be a homomorphism from G into Aut(N).

In the following example we construct the holomorph of Q_8 as split extension of Q_8 by S_4.

gap> N := SmallGroup( 8, 4 ); <pc group of size 8 with 3 generators> gap> IsAbelian( N ); false gap> A := AutomorphismGroup( N ); <group of size 24 with 4 generators> gap> iso := IsomorphismPcGroup( A ); CompositionMapping( Pcgs([ (2,6,5,3), (1,3,5)(2,4,6), (2,5)(3,6), (1,4)(3,6) ]) -> [ f1, f2, f3, f4 ], <action isomorphism> ) gap> H := Image( iso ); Group([ f1, f2, f3, f4 ]) gap> G := Subgroup( H, Pcgs(H){[1,2]} ); Group([ f1, f2 ]) gap> inv := InverseGeneralMapping( iso ); [ f1*f2, f2^2*f3, f4, f3 ] -> [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2, f2, f3 ], Pcgs([ f1, f2, f3 ]) -> [ f2, f1*f2, f3 ], Pcgs([ f1, f2, f3 ]) -> [ f1*f3, f2, f3 ], Pcgs([ f1, f2, f3 ]) -> [ f1, f2*f3, f3 ] ] gap> K := SplitExtension( G, inv, N ); <pc group of size 192 with 7 generators>
Coding a Pc Presentation

If one wants to store a large number of pc groups, then it can be useful to store them in a compressed format, since pc presentations can be space consuming. Here we introduce a method to code and decode pc presentations by integers. To decode a given code the size of the underlying pc group is needed as well. For the full definition and the coding and decoding procedures see . This method is used with the small groups library, see Section .

<#Include Label="CodePcgs"> <#Include Label="CodePcGroup"> <#Include Label="PcGroupCode">

Random Isomorphism Testing

The generic isomorphism test for groups may be applied to pc groups as well. However, this test is often quite time consuming. Here we describe another method to test isomorphism by a probabilistic approach.

<#Include Label="RandomIsomorphismTest">