<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <!-- %% --> <!-- %A grppc.xml GAP documentation Alexander Hulpke --> <!-- %A Bettina Eick --> <!-- %% --> <!-- %% --> <!-- %Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland --> <!-- %Y Copyright (C) 2002 The GAP Group --> <!-- %% -->
<Chapter Label="Pc Groups">
<Heading>Pc Groups</Heading>
<!-- % --> <!-- % Wunsch von Herrn Neubueser: PrintPcPresentation --> <!-- % -->
Pc groups are polycyclic groups that use the polycyclic presentation for element arithmetic. This presentation gives them a <Q>natural</Q> pcgs,
the <Ref Attr="FamilyPcgs"/> with respect to which pcgs operations as
described in chapter <Ref Chap="Polycyclic Groups"/> are
particularly efficient.
<P/>
Let <M>G</M> be a polycyclic group with pcgs <M>P = (g_1, \ldots, g_n)</M>
and corresponding relative orders <M>(r_1, \ldots, r_n)</M>. Recall that the
<M>r_i</M> are positive integers or infinity and let <M>I</M> be the set of
indices <M>i</M> with <M>r_i</M> a positive integer.
Then <M>G</M> has a finite presentation on the generators
<M>g_1, \ldots, g_n</M> with relations of the following form.
<Table Align="lcl">
<Row>
<Item><M>g_i^{{r_i}}</M></Item>
<Item>=</Item>
<Item><M>g_{{i+1}}^{a(i,i,i+1)} \cdots g_n^{a(i,i,n)}</M></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item>for <M>1 \leq i \leq n</M> and <M>i \in I</M></Item>
</Row>
<Row>
<Item><M>g_i^{{-1}} g_j g_i</M></Item>
<Item>=</Item>
<Item><M>g_{{i+1}}^{a(i,j,i+1)} \cdots g_n^{a(i,j,n)}</M></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item>for <M>1 \leq i < j \leq n</M></Item>
</Row>
</Table>
For infinite groups we need additionally
<Table Align="lcl">
<Row>
<Item><M>g_i^{{-1}} g_j^{{-1}} g_i</M></Item>
<Item>=</Item>
<Item><M>g_{{i+1}}^{b(i,j,i+1)} \cdots g_n^{b(i,j,n)}</M></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item>for <M>1 \leq i < j \leq n</M> and <M>j \not \in I</M></Item>
</Row>
<Row>
<Item><M>g_i g_j g_i^{{-1}}</M></Item>
<Item>=</Item>
<Item><M>g_{{i+1}}^{c(i,j,i+1)} \cdots g_n^{c(i,j,n)}</M></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item>for <M>1 \leq i < j \leq n</M> and <M>i \not \in I</M></Item>
</Row>
<Row>
<Item><M>g_i g_j^{{-1}} g_i^{{-1}}</M></Item>
<Item>=</Item>
<Item><M>g_{{i+1}}^{d(i,j,i+1)} \cdots g_n^{d(i,j,n)}</M></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item>for <M>1 \leq i < j \leq n</M> and <M>i, j, \not \in I</M></Item>
</Row>
</Table>
Here the right hand sides are assumed to be words in normal
form; that is, for <M>k \in I</M> we have for all exponents
<M>0 \leq a(i,j,k), b(i,j,k), c(i,j,k), d(i,j,k) < r_k</M>.
<P/>
A finite presentation of this type is called a <E>power-conjugate
presentation</E> and a <E>pc group</E> 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 <E>power-commutator</E> presentation. Both types of presentation
are abbreviated as <E>pc presentation</E>.
Note that a pc presentation is a rewriting system.
<P/>
Clearly, whenever a group <M>G</M> with pcgs <M>P</M> is given, then we can
write down the corresponding pc presentation. On the other hand,
one may just write down a presentation on <M>n</M> abstract generators
<M>g_1, \ldots, g_n</M> with relations of the above form and define a
group <M>H</M> by this.
Then the subgroups <M>C_i = \langle g_i, \ldots, g_n \rangle</M> of <M>H</M>
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 <M>H</M> is <E>confluent</E>.
Note that &GAP; 4 can only work correctly with pc groups defined by a
confluent pc presentation.
<P/>
At the current state of implementations the &GAP; library contains
methods to compute with finite polycyclic groups, while the &GAP;
package <Package>Polycyclic</Package> by Bettina Eick and Werner Nickel
allows also computations with infinite polycyclic groups which are given
by a pc-presentation.
<P/> <!-- %% The Pc groups were designed and implemented by Frank Celler and --> <!-- %% Werner Nickel, the code for Two-Cohomology and Extensions is due to --> <!-- %% Bettina Eick who also wrote the Random Isomorphism test together with Hans --> <!-- %% Ulrich Besche. -->
Algorithms for pc groups use the methods for polycyclic groups described in
chapter <Ref Chap="Polycyclic Groups"/>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="The Family Pcgs">
<Heading>The Family Pcgs</Heading>
Clearly, the generators of a power-conjugate presentation of
a pc group <M>G</M> form a pcgs of the pc group. This pcgs is called
the <E>family pcgs</E>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Elements of Pc Groups">
<Heading>Elements of Pc Groups</Heading>
<ManSection>
<Heading>Comparison of elements of pc groups</Heading>
<Meth Name="\=" Arg='pcword1, pcword2' Label="for pcwords"/>
<Meth Name="\<" Arg='pcword1, pcword2' Label="for pcwords"/>
<Description>
<Index Subkey="for pcwords">equality</Index>
<Index Subkey="for pcwords">smaller</Index>
The elements of a pc group <M>G</M> are always represented as words
in normal form with respect to the family pcgs of <M>G</M>.
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.
</Description>
</ManSection>
<ManSection>
<Heading>Arithmetic operations for elements of pc groups</Heading>
<Meth Name="\*" Arg='pcword1, pcword2' Label="for pcwords"/>
<Attr Name="Inverse" Arg='pcword' Label="for a pcword"/>
<Description>
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).
<P/>
To multiply two elements in a pc group, we
first concatenate the corresponding words and then use an algorithm
called <E>collection</E> to transform the new word into a word in normal
form.
<P/>
<Example><![CDATA[
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
]]></Example>
</Description>
</ManSection>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Pc Groups versus Fp Groups">
<Heading>Pc Groups versus Fp Groups</Heading>
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.
<Description>
It is possible to convert a pc group to a fp group in &GAP;.
The function <Ref Func="IsomorphismFpGroupByPcgs"/> computes the
power-commutator presentation defined by <A>pcgs</A>.
The string <A>str</A> can be used to give a name to the generators
of the fp group.
<P/>
<Example><![CDATA[
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 ]
]]></Example>
</Description>
</ManSection>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Constructing Pc Groups">
<Heading>Constructing Pc Groups</Heading>
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;.
<P/>
(If you have already a suitable pcgs in another representation, use
<Ref Attr="PcGroupWithPcgs"/>, see below.)
<P/>
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 <Ref Func="PcGroupFpGroup"/>.
Note that this does not work for arbitrary presentations of polycyclic
groups, see Chapter <Ref Sect="Quotient Methods"/> for further information.
<P/>
Another way is to create and manipulate a collector of a pc group by hand
and to use it to define a pc group.
This is the most technical way and has little error checking and thus is
intended mostly for experts who want to create a pc presentation in a
particular way.
&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 <Ref Oper="SingleCollector"/>)
and the combinatorial collector for finite <M>p</M>-groups.
See <Cite Key="Sims94"/> for further information on collecting strategies.
<P/>
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.
<P/>
For performance reasons it is beneficial to enforce a <Q>syllable</Q>
representation in the free group
(see <Ref Sect="Representations for Associative Words"/>).
<P/>
Note that in the end, the collector has to be converted to a group,
see <Ref Oper="GroupByRws"/>.
<P/>
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 <Ref Attr="IsomorphismRefinedPcGroup"/> and <Ref Attr="RefinedPcGroup"/>.
<Description>
initializes a single collector or a combinatorial collector,
where <A>fgrp</A> must be a free group and <A>relorders</A> must be a list
of the relative orders of the pc series.
<P/>
A combinatorial collector can only be set up for a finite <M>p</M>-group.
Here, the relative orders <A>relorders</A> must all be equal and a prime.
</Description>
</ManSection>
<ManSection>
<Oper Name="SetConjugate" Arg='coll, j, i, w'/>
<Description>
Let <M>f_1, \ldots, f_n</M> be the generators of the underlying free group
of the collector <A>coll</A>.
<P/>
For <A>i</A> <M><</M> <A>j</A>,
<Ref Oper="SetConjugate"/> sets the conjugate <M>f_j^{{f_i}}</M> to equal
<A>w</A>, which is assumed to be a canonical word in <M>f_{{i+1}}, \ldots, f_n</M>.
No check of the arguments is performed.
</Description>
</ManSection>
<ManSection>
<Oper Name="SetCommutator" Arg='coll, j, i, w'/>
<Description>
Let <M>f_1, \ldots, f_n</M> be the generators of the underlying free group
of the collector <A>coll</A>.
<P/>
For <A>i</A> <M><</M> <A>j</A>,
<Ref Oper="SetCommutator"/> sets the commutator of <M>f_j</M> and <M>f_i</M>
to equal <A>w</A>, which is assumed to be a canonical word in
<M>f_{{i+1}}, \ldots, f_n</M>.
No check of the arguments is performed.
</Description>
</ManSection>
<ManSection>
<Oper Name="SetPower" Arg='coll, i, w'/>
<Description>
Let <M>f_1, \ldots, f_n</M> be the generators of the underlying free group
of the collector <A>coll</A>,
and let <M>r_i</M> be the corresponding relative orders.
<P/>
<Ref Oper="SetPower"/> sets the power <M>f_i^{{r_i}}</M> to equal <A>w</A>,
which is assumed to be a canonical word in <M>f_{{i+1}}, \ldots, f_n</M>.
No check of the arguments is performed.
</Description>
</ManSection>
<Description>
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.
</Description>
</ManSection>
<ManSection>
<Prop Name="IsConfluent" Arg='G' Label="for pc groups"/>
<Description>
checks whether the pc group <A>G</A> has been built from a collector with
a confluent power-commutator presentation.
<P/>
<Example><![CDATA[
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
]]></Example>
</Description>
</ManSection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Computing Pc Groups">
<Heading>Computing Pc Groups</Heading>
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.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Saving a Pc Group">
<Heading>Saving a Pc Group</Heading>
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">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Operations for Pc Groups">
<Heading>Operations for Pc Groups</Heading>
All the operations described in Chapters <Ref Chap="Groups"/>
and <Ref Chap="Polycyclic Groups"/> apply to a pc group.
Nearly all methods for pc groups are
methods for groups with pcgs as described
in Chapter <Ref Chap="Polycyclic Groups"/>.
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.
Section <Ref Sect="2-Cohomology"/> describes operations and methods for
arbitrary finite groups.
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="2-Cohomology and Extensions">
<Heading><M>2</M>-Cohomology and Extensions</Heading>
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, <M>H</M> is an extension of <M>G</M> by <M>M</M>,
if there exists a normal subgroup <M>N</M> in <M>H</M> which is isomorphic
to <M>M</M> such that <M>H/N</M> is isomorphic to <M>G</M>.
<P/>
Pc groups are particularly suited for such applications, since the
<M>2</M>-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.
<P/>
To define the elementary abelian group <M>M</M> together with an action of
<M>G</M> on <M>M</M> we consider <M>M</M> as a MeatAxe module for <M>G</M>
over a finite field (section <Ref Oper="IrreducibleModules"/>
describes functions that can be used to obtain certain modules).
For further information on meataxe modules see Chapter
<Ref Chap="The MeatAxe"/>.
Note that the matrices defining the module must correspond to the pcgs
of the group <A>G</A>.
<P/>
There exists an action of the subgroup of <E>compatible pairs</E> in
<M>Aut(G) \times Aut(M)</M> which acts on the second cohomology group,
see <Ref Func="CompatiblePairs"/>.
<M>2</M>-cocycles which lie in the same orbit under this action define
isomorphic extensions of <M>G</M>. However, there may be isomorphic
extensions of <M>G</M> corresponding to cocycles in different orbits.
<P/>
See also the &GAP; package <Package>GrpConst</Package> by Hans Ulrich Besche
and Bettina Eick that contains methods to construct up to isomorphism
the groups of a given order.
<P/>
Finally we note that for the computation of split extensions it is not
necessary that <A>M</A> must correspond to an elementary abelian group. Here
it is possible to construct split extensions of arbitrary pc groups,
see <Ref Oper="SplitExtension"/>.
<Description>
returns the split extension of <A>G</A> by the <A>G</A>-module <A>M</A>.
See also <Ref Oper="SplitExtension" Label="with specified homomorphism"/>
for its 3-argument version.
</Description>
</ManSection>
<Description>
returns the module of an extension <A>E</A> of <A>G</A> by <A>M</A>.
This is the normal subgroup of <A>E</A> which corresponds to <A>M</A>.
<P/>
<Example><![CDATA[
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 ] ]
]]></Example>
<P/>
Note that the extensions returned by <Ref Oper="Extensions"/> are computed
up to equivalence, but not up to isomorphism.
</Description>
</ManSection>
<ManSection>
<Oper Name="SplitExtension" Arg='G, aut, N' Label="with specified homomorphism"/>
<Description>
returns the split extensions of the pc group <A>G</A> by the pc group
<A>N</A>.
<A>aut</A> should be a homomorphism from <A>G</A> into Aut(<A>N</A>).
<P/>
In the following example we construct the holomorph of <M>Q_8</M> as split
extension of <M>Q_8</M> by <M>S_4</M>.
<P/>
<Example><![CDATA[
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>
]]></Example>
</Description>
</ManSection>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Coding a Pc Presentation">
<Heading>Coding a Pc Presentation</Heading>
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 <Cite Key="BescheEick98"/>. This method is used with the small
groups library (see <Ref BookName="smallgrp" Chap="The Small Groups Library"/>.
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">
</Section>
</Chapter>
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