[ref] 44 Pc Groups

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# 44 Pc Groups

### Sections

1. The family pcgs
2. Elements of pc groups
3. Pc groups versus fp groups
4. Constructing Pc Groups
5. Computing Pc Groups
6. Saving a Pc Group
7. Operations for Pc Groups
8. 2-Cohomology and Extensions
9. Coding a Pc Presentation
10. Random Isomorphism Testing

PcGroups are polycyclic groups that use the polycyclic presentation for element arithmetic. This presentation gives them a ``natural'' pcgs, the `FamilyPcgs` (see FamilyPcgs) with respect to which pcgs operations as described in chapter Polycyclic Groups are particularly efficient.

Let G be a polycyclic group with pcgs P = (g1, ¼, gn) and corresponding relative orders (r1, ¼, rn). Recall that the ri are positive integers or infinity and let I be the set of indices i with ri a positive integer. Then G has a finite presentation on the generators g1, ¼, gn with relations of the following form.

giri
=
gi+1a(i,i,i+1) ¼gna(i,i,n)
for 1 £ i £ n and i Î I
gi-1 gj gi
=
gi+1a(i,j,i+1) ¼gna(i,j,n)
for 1 £ i < j £ n

For infinite groups we need additionally

gi-1 gj-1 gi
=
gi+1b(i,j,i+1) ¼gnb(i,j,n)
for 1 £ i < j £ n and j Ï I
gi gj gi-1
=
gi+1c(i,j,i+1) ¼gnc(i,j,n)
for 1 £ i < j £ n and i Ï I
gi gj-1 gi-1
=
gi+1d(i,j,i+1) ¼gnd(i,j,n)
for 1 £ i < j £ n and i, j, Ï I

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

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 g1, ¼, gn with relations of the above form and define a group H by this. Then the subgroups Ci = ági, ¼, gn ñ 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 level of implementation GAP can only deal with finite pc groups. This will be extended in near future.

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

## 44.1 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.

- `FamilyPcgs(` grp`) A`

- `IsFamilyPcgs(` pcgs`) P`

- `InducedPcgsWrtFamilyPcgs(` grp`) A`

- `IsParentPcgsFamilyPcgs(` pcgs`) P`

  This property indicates that the pcgs pcgs is induced with respect to a family pcgs.

  In GAP 3 the family pcgs had been the only pcgs allowed for a pc group. Note that this has changed in GAP 4 where a pc group may have several independent polycyclic generating sequences.

  However, the elementary operations for a non-family pcgs may not be as efficient as the elementary operations for the family pcgs.

  This can have a significant influence on the performance of algorithms for polycyclic groups. Many algorithms require a pcgs that corresponds to an elementary abelian series (see PcgsElementaryAbelianSeries) or even a special pcgs (see Special Pcgs). If the family pcgs has the required properties, it will be used for these purposes, if not GAP has to work with respect to a new pcgs which is not the family pcgs and thus takes longer for elementary calculations like `ExponentsOfPcElement`.

  Therefore, if the family pcgs chosen for arithmetic is not of importance it might be worth to change to another, nicer, pcgs to speed up calculations. This can be achieved, for example, by using the `Range` of the isomorphism obtained by `IsomorphismSpecialPcGroup` (see IsomorphismSpecialPcGroup).

  ## 44.2 Elements of pc groups

  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 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.

- ``pcword`=` pcword``
- ``pcword`<` pcword``

  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 to 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

  ## 44.3 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.

- `IsPcGroup(` G`) C`

  tests whether G is a pc group.

      gap> G := SmallGroup( 24, 12 );
      <pc group of size 24 with 4 generators>
      gap> IsPcGroup( G );
      true
      gap> IsFpGroup( G );
      false

  Note that it is possible to convert a pc group to a fp group in GAP. The following 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.

- `IsomorphismFpGroupByPcgs(` pcgs`,` str`)`

      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 ]

  ## 44.4 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 `PcGroupWithPcgs`, see below PcGroupWithPcgs.)

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

  For performance reasons it is beneficial to enforce a ``syllable'' representation in the free group (see Representations for Associative Words).

- `PcGroupFpGroup(` G`) F`

  creates a PcGroup P from an FpGroup (see Chapter Finitely Presented Groups) G whose presentation is polycyclic. The resulting group P has generators corresponding to the generators of G. They are printed in the same way as generators of G, but they lie in a different family. If the pc presentation of G is not confluent, an error message occurs.

      gap> F := FreeGroup(IsSyllableWordsFamily,"a","b","c","d");;
      gap> a := F.1;; b := F.2;; c := F.3;; d := F.4;;
      gap> rels := [a^2, b^3, c^2, d^2, Comm(b,a)/b, Comm(c,a)/d, Comm(d,a),
      >             Comm(c,b)/(c*d), Comm(d,b)/c, Comm(d,c)];
      [ a^2, b^3, c^2, d^2, b^-1*a^-1*b*a*b^-1, c^-1*a^-1*c*a*d^-1, d^-1*a^-1*d*a,
        c^-1*b^-1*c*b*d^-1*c^-1, d^-1*b^-1*d*b*c^-1, d^-1*c^-1*d*c ]
      gap> G := F / rels;
      <fp group on the generators [ a, b, c, d ]>
      gap> H := PcGroupFpGroup( G );
      <pc group of size 24 with 4 generators>

  Equivalently to the above method one can initiate a collector of a pc group by hand and use it to define a pc group. In GAP there are different collectors for different collecting strategies; at the moment, there are two collectors to choose from: the single collector for finite pc groups and the combinatorial collector for finite p-groups. See Sims94 for further information on collecting strategies.

  A collector is initiated by underlying free group to the pc presented 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.

- `SingleCollector(` fgrp`,` relorders`)`
- `CombinatorialCollector(` fgrp`,` relorders`)`

  Then the right hand sides of the pc presentation have to be declared. Let f1, ¼, fn be the generators of the underlying free group fgrp.

- `SetConjugate(` coll`,` i`,` j`,` w`)`

  set the conjugate fjfi to equal w where w is a word in fi+1, ¼, fn and i < j.

- `SetCommutator(` coll`,` i`,` j`,` w`)`

  set the commutator of fj and fi to equal w where w is a word in fi+1, ¼, fn and i < j.

- `SetPower(` coll`,` i`,` w`)`

  set the power firi to equal w where w is a word in fi+1, ¼, fn.

  Finally, the collector has to be converted to a group.

- `GroupByRws(` coll`)`
- `GroupByRwsNC(` coll`)`

  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.

- `IsConfluent(` G`)`

  checks whether the pc group G has been build 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

  With the above 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. Hence the following functions are useful.

- `IsomorphismRefinedPcGroup(` G`) A`

  returns an isomorphism from G onto an isomorphic PC group whose family pcgs is a prime order pcgs.

- `RefinedPcGroup(` G`) A`

  returns the range of `IsomorphismRefinedPcGroup(`G`)`.

  ## 44.5 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.

- `PcGroupWithPcgs(` mpcgs`) A`

  creates a new Pc group G whose family pcgs is isomorphic to the (modulo) pcgs mpcgs.

      gap> G := Group( (1,2,3), (3,4,1) );;
      gap> PcGroupWithPcgs( Pcgs(G) );
      <pc group of size 12 with 3 generators>

  If a pcgs is only given by a list of pc elements, `PcgsByPcSequence` (see PcgsByPcSequence) can be used:

      gap> G:=Group((1,2,3,4),(1,2));;
      gap> p:=PcgsByPcSequence(FamilyObj(One(G)),
      > [ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]);
      Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ])
      gap> PcGroupWithPcgs(p);
      <pc group of size 24 with 4 generators>

      gap> G := SymmetricGroup( 5 );
      Sym( [ 1 .. 5 ] )
      gap> H := Subgroup( G, [(1,2,3,4,5), (3,4,5)] );
      Group([ (1,2,3,4,5), (3,4,5) ])
      gap> modu := ModuloPcgs( G, H );
      Pcgs([ (4,5) ])
      gap> PcGroupWithPcgs(modu);
      <pc group of size 2 with 1 generators>

- `IsomorphismPcGroup(` G`) A`

  returns an isomorphism from G onto an isomorphic PC group. The series chosen for this PC representation depends on the method chosen. G must be a polycyclic group of any kind, for example a solvable permutation group.

      gap> G := Group( (1,2,3), (3,4,1) );;
      gap> iso := IsomorphismPcGroup( G );
      Pcgs([ (2,4,3), (1,2)(3,4), (1,3)(2,4) ]) -> [ f1, f2, f3 ]
      gap> H := Image( iso );
      Group([ f1, f2, f3 ])

- `IsomorphismSpecialPcGroup(` G`) A`

  returns an isomorphism from G onto an isomorphic PC group whose family pcgs is a special pcgs. (This can be beneficial to the runtime of calculations.) G may be a polycyclic group of any kind, for example a solvable permutation group.

  ## 44.6 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.

- `GapInputPcGroup(` grp`,` string`) F`

      gap> G := SmallGroup( 24, 12 );
      <pc group of size 24 with 4 generators>
      gap> PrintTo( "save", GapInputPcGroup( G, "H" ) );
      gap> Read( "save" );
      #I A group of order 24 has been defined.
      #I It is called H
      gap> H = G;
      false
      gap> IdSmallGroup( H ) = IdSmallGroup( G );
      true

  ## 44.7 Operations for Pc Groups

  All the operations described in Chapters Groups and Polycyclic Groups apply to a pc group. Nearly all methods for pc groups are methods for groups with pcgs as described in Chapter 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.

  ## 44.8 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. For further information on meataxe modules see Chapter The MeatAxe. Note that the matrices defining the module must correspond to the pcgs of the group G.

- `TwoCoboundaries(` G`,` M`) O`

  returns the group of 2-coboundaries of a pc group G by the G-module M. The generators of M must correspond to Pcgs(G). The group of coboundaries is given as vector space over the field underlying M.

- `TwoCocycles(` G`,` M`) O`

  returns the group of 2-cocycles of a pc group G by the G-module M. The generators of M must correspond to Pcgs(G). The group of cocycles is given as vector space over the field underlying M.

- `TwoCohomology(` G`,` M`) O`

  returns a record defining the second cohomology group as factor space of the space of cocycles by the space of coboundaries. G must be a pc group and the generators of M must correspond to the pcgs of G.

      gap> G := SmallGroup( 4, 2 );
      <pc group of size 4 with 2 generators>
      gap> mats := List( Pcgs( G ), x -> IdentityMat( 1, GF(2) ) );
      [ [ <a GF2 vector of length 1> ], [ <a GF2 vector of length 1> ] ]
      gap> M := GModuleByMats( mats, GF(2) );
      rec( field := GF(2), isMTXModule := true, dimension := 1,
        generators := [ <an immutable 1x1 matrix over GF2>,
            <an immutable 1x1 matrix over GF2> ] )
      gap> TwoCoboundaries( G, M );
      [  ]
      gap> TwoCocycles( G, M );
      [ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ],
        [ 0*Z(2), 0*Z(2), Z(2)^0 ] ]
      gap> cc := TwoCohomology( G, M );;
      gap> cc.cohom;
      <linear mapping by matrix, <vector space of dimension 3 over GF(2)> -> ( GF(
      2)^3 )>

- `Extensions(` G`,` M`) O`

  returns all extensions of G by the G-module M up to equivalence as pc groups.

- `Extension(` G`,` M`,` c`) O`
- `ExtensionNC(` G`,` M`,` c`) O`

  returns the extension of G by the G-module M via the cocycle c as pc groups. The `NC` version does not check the resulting group for consistence.

- `SplitExtension(` G`,` M`)`

  returns the split extension of G by the G-module M.

- `ModuleOfExtension(` E`)`

  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 `Extensions` are computed up to equivalence, but not up to isomorphism.

  There exists an action of the subgroup of compatible pairs in Aut(G) ×Aut(M) which acts on the second cohomology group. 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.

- `CompatiblePairs(` G`,` M`[,` D`] ) F`

  returns the group of compatible pairs of the group G with the G-module M as subgroup of the direct product of Aut(G) x Aut(M). Here Aut(M) is considered as subgroup of a general linear group. The optional argument D should be a subgroup of Aut(G) x Aut(M). If it is given, then only the compatible pairs in D are computed.

- `ExtensionRepresentatives(` G`,` M`,` P`) O`

  returns all extensions of G by the G-module M up to equivalence under action of P where P has to be a subgroup of the group of compatible pairs of G with M.

      gap> G := SmallGroup( 4, 2 );;
      gap> mats := List( Pcgs( G ), x -> IdentityMat( 1, GF(2) ) );;
      gap> M := GModuleByMats( mats, GF(2) );;
      gap> A := AutomorphismGroup( G );;
      gap> B := GL( 1, 2 );;
      gap> D := DirectProduct( A, B );
      <group with 6 generators>
      gap> P := CompatiblePairs( G, M, D );
      <group of size 6 with 2 generators>
      gap> ExtensionRepresentatives( G, M, P );
      [ <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> 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> ]

  See also the forthcoming GAP package on `Group Construction Methods`.

  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.

- `SplitExtensions(` G`,` aut`,` N`)`

  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 Q8 as split extension of Q8 by S4.

      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>

  ## 44.9 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 BescheEick98. This method is used with the small groups library, see Section Small Groups.

- `CodePcgs(` pcgs`) F`

  returns the code corresponding to pcgs.

- `CodePcGroup(` G`) F`

  returns the code for a pcgs of G.

- `PcGroupCode(` code`,` size`) F`

  returns a pc group of size size corresponding to code.

- `PcGroupCodeRec(` rec`) F`

  Here rec needs to have entries .code and .order. Then PcGroupCode returns a pc group of size .order corresponding to .code.

      gap> G := SmallGroup( 24, 12 );;
      gap> p := Pcgs( G );;
      gap> code := CodePcgs( p );
      5790338948
      gap> H := PcGroupCode( code, 24 );
      <pc group of size 24 with 4 generators>
      gap> map := GroupHomomorphismByImages( G, H, p, FamilyPcgs(H) );
      Pcgs([ f1, f2, f3, f4 ]) -> Pcgs([ f1, f2, f3, f4 ])
      gap> IsBijective(map);
      true

  ## 44.10 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.

  This method takes a list of groups and a non-negative integer as input. The output is a sublist of the input list where only isomorphic copies have been removed. The integer gives a certain amount of control over the probability to detect all isomorphisms. If it is 0, then nothing will be done at all. The larger the integer is, the larger is the probability of finding all isomorphisms. However, due to the underlying method we can not guarantee that the algorithm finds all isomorphisms, no matter how large n is.

- `RandomIsomorphismTest(` list`,` n`) F`

  Here list must be a list of pc groups and n a non-negative integer. Returns a sublist of list where isomorphic copies detected by the probabilistic test have been removed.

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  GAP 4 manual
  May 2002