[Polycyclic] 7 Higher level methods for pcp-groups

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# 7 Higher level methods for pcp-groups

### Sections

1. Subgroup series in pcp-groups
2. Orbit stabilizer methods for pcp-groups
3. Centralizers, Normalizers and Intersections
4. Finite subgroups
5. Subgroups of finite index and maximal subgroups
6. Further attributes for pcp-groups based on the Fitting subgroup
7. Functions for nilpotent groups
8. Random methods for pcp-groups

This is a description of some higher level functions of the polycyclic package of GAP 4. Throughout this chapter we let G be a pc-presented group and we consider algorithms for subgroups U and V of G. For background and a description of the underlying algorithms we refer to Eic01b.

## 7.1 Subgroup series in pcp-groups

Many algorithm for pcp-groups work by induction using some series through the group. In this section we prove a number of useful series for pcp-groups. An efa series is a normal series with elementary or free abelian factors. See Eic00 for outlines on the algorithms of a number of the available series.

- `PcpSeries(` U`)`

  returns the polycyclic series of U defined by an igs of U.

- `EfaSeries(` U`)`

  returns a normal series of U with elementary or free abelian factors.

- `SemiSimpleEfaSeries(` U`)`

  returns an efa series of U such that every factor in the series is semisimple as a module for U over a finite field or over the rationals.

- `DerivedSeries(` U`)`

  the derived series of U.

- `RefinedDerivedSeries(` U`)`

  the derived series of U refined to an efa series such that in each abelian factor of the derived series the free abelian factor is at the top.

- `RefinedDerivedSeriesDown(` U`)`

  the derived series of U refined to an efa series such that in each abelian factor of the derived series the free abelian factor is at the bottom.

- `LowerCentralSeries(` U`)`

  the lower central series of U. If U does not have a largest nilpotent quotient group, then this function may not terminate.

- `UpperCentralSeries(` U`)`

  the upper central series of U. This function always terminates, but it may terminate at a proper subgroup of U.

- `TorsionByPolyEFSeries(` U`)`

  returns an efa series of U such that all torsion-free factors are at the top and all finite factors are at the bottom. Such a series might not exist for U and in this case the function returns fail.

      gap> G := ExamplesOfSomePcpGroups(5);
      Pcp-group with orders [ 2, 0, 0, 0 ]
      gap> Igs(G);
      [ g1, g2, g3, g4 ]

      gap> PcpSeries(G);
      [ Pcp-group with orders [ 2, 0, 0, 0 ],
        Pcp-group with orders [ 0, 0, 0 ],
        Pcp-group with orders [ 0, 0 ],
        Pcp-group with orders [ 0 ],
        Pcp-group with orders [  ] ]

      gap> List( PcpSeries(G), Igs );
      [ [ g1, g2, g3, g4 ], [ g2, g3, g4 ], [ g3, g4 ], [ g4 ], [  ] ]

  Algorithms for pcp-groups often use an efa series of G and work down over the factors of this series. Usually, pcp's of the factors are more useful than the actual factors. Hence we provide the following.

- `PcpsBySeries(` ser`)`
- `PcpsBySeries(` ser`,` "snf"`)`

  returns a list of pcp's corresponding to the factors of the series. If the second argument is present, then each pcp corresponds to a decomposition of the abelian groups into direct factors.

- `PcpsOfEfaSeries(` U`)`

  return a list of pcp's corresponding to an efa series of U.

      gap> G := ExamplesOfSomePcpGroups(5);
      Pcp-group with orders [ 2, 0, 0, 0 ]

      gap> PcpsBySeries( DerivedSeries(G));
      [ Pcp [ g1, g2, g3, g4 ] with orders [ 2, 2, 2, 2 ],
        Pcp [ g2^-2, g3^-2, g4^2 ] with orders [ 0, 0, 4 ],
        Pcp [ g4^8 ] with orders [ 0 ] ]
      gap> PcpsBySeries( RefinedDerivedSeries(G));
      [ Pcp [ g1, g2, g3 ] with orders [ 2, 2, 2 ],
        Pcp [ g4 ] with orders [ 2 ],
        Pcp [ g2^2, g3^2 ] with orders [ 0, 0 ],
        Pcp [ g4^2 ] with orders [ 2 ],
        Pcp [ g4^4 ] with orders [ 2 ],
        Pcp [ g4^8 ] with orders [ 0 ] ]

      gap> PcpsBySeries( RefinedDerivedSeries(G), "snf");
      [ Pcp [ g1, g2, g3 ] with orders [ 2, 2, 2 ],
        Pcp [ g4 ] with orders [ 2 ],
        Pcp [ g2^2, g3^2 ] with orders [ 0, 0 ],
        Pcp [ g4^2 ] with orders [ 2 ],
        Pcp [ g4^4 ] with orders [ 2 ],
        Pcp [ g4^8 ] with orders [ 0 ] ]

      gap>  PcpsOfEfaSeries( G );
      [ Pcp [ g1 ] with orders [ 2 ],
        Pcp [ g2 ] with orders [ 0 ],
        Pcp [ g3 ] with orders [ 0 ],
        Pcp [ g4 ] with orders [ 0 ] ]

  ## 7.2 Orbit stabilizer methods for pcp-groups

  Let U be a pcp-group which acts on a set Omega. One of the fundamental problems in algorithmic group theory is the determination of orbits and stabilizers of points in Omega under the action of U. We distinguish two cases: the case that all considered orbits are finite and the case that there are infinite orbits. In the latter case, an orbit cannot be listed and a description of the orbit and its corresponding stabilizer is much harder to obtain.

  If the considered orbits are finite, then the following two functions can be applied to compute the considered orbits and their corresponding stabilizers.

- `PcpOrbitStabilizer(` point`,` gens`,` acts`,` oper`)`
- `PcpOrbitsStabilizers(` points`,` gens`,` acts`,` oper`)`

  The input gens can be an igs or a pcp of a pcp-group U. The elements in the list gens act as the elements in the list acts via the function oper on the given points; that is, oper( point, acts[i] ) applies the ith generator to a given point. Thus the group defined by acts must be a homomorphic image of the group defined by gens. The first function returns a record containing the orbit as component 'orbit' and and igs for the stabilizer as component 'stab'. The second function returns a list of records, each record contains 'repr' and 'stab'. Both of these functions run forever on infinite orbits.

      gap> G := DihedralPcpGroup( 0 );
      Pcp-group with orders [ 2, 0 ]
      gap> mats := [ [[-1,0],[0,1]], [[1,1],[0,1]] ];;
      gap> pcp := Pcp(G);
      Pcp [ g1, g2 ] with orders [ 2, 0 ]
      gap> PcpOrbitStabilizer( [0,1], pcp, mats, OnRight );
      rec( orbit := [ [ 0, 1 ] ],
           stab := [ g1, g2 ],
           word := [ [ [ 1, 1 ] ], [ [ 2, 1 ] ] ] )

  If the considered orbits are infinite, then it may not always be possible to determine a description of the orbits and their stabilizers. However, as shown in EOs01 and Eic02, it is possible to determine stabilizers and check if two elements are contained in the same orbit if the given action of the polycyclic group is a unimodular linear action on a vector space. The following functions are available for this case.

- `StabilizerIntegralAction(` U`,` mats`,` v`)`
- `OrbitIntegralAction(` U`,` mats`,` v`,` w`)`

  The first function computes the stabilizer in U of the vector v where the pcp group U acts via mats on an integral space and v and w are elements in this integral space. The second function checks whether v and w are in the same orbit and the function returns either false or a record containing an element in U mapping v to w and the stabilizer of v.

- `NormalizerIntegralAction(` U`,` mats`,` B`)`
- `ConjugacyIntegralAction(` U`,` mats`,` B`,` C`)`

  The first function computes the normalizer in U of the lattice with the basis B, where the pcp group U acts via mats on an integral space and B is a subspace of this integral space. The second functions checks whether the two lattices with the bases B and C are contained in the same orbit under U. The function returns either false or a record with an element in U mapping B to C and the stabilizer of B.

      # get a pcp group and a free abelian normal subgroup
      gap> G := ExamplesOfSomePcpGroups(8);
      Pcp-group with orders [ 0, 0, 0, 0, 0 ]
      gap> efa := EfaSeries(G);
      [ Pcp-group with orders [ 0, 0, 0, 0, 0 ],
        Pcp-group with orders [ 0, 0, 0, 0 ],
        Pcp-group with orders [ 0, 0, 0 ],
        Pcp-group with orders [  ] ]
      gap> N := efa[3];
      Pcp-group with orders [ 0, 0, 0 ]
      gap> IsFreeAbelian(N);
      true

      # create conjugation action on N
      gap> mats := LinearActionOnPcp(Igs(G), Pcp(N));
      [ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
        [ [ 0, 0, 1 ], [ 1, -1, 1 ], [ 0, 1, 0 ] ],
        [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
        [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
        [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ]

      # take an arbitrary vector and compute its stabilizer
      gap> StabilizerIntegralAction(G,mats, [2,3,4]);
      Pcp-group with orders [ 0, 0, 0, 0 ]
      gap> Igs(last);
      [ g1, g3, g4, g5 ]

      # check orbits with some other vectors
      gap> OrbitIntegralAction(G,mats, [2,3,4],[3,1,5]);
      rec( stab := Pcp-group with orders [ 0, 0, 0, 0 ], prei := g2 )

      gap> OrbitIntegralAction(G,mats, [2,3,4], [4,6,8]);
      false

      # compute the orbit of a subgroup of Z^3 under the action of G
      gap> NormalizerIntegralAction(G, mats, [[1,0,0],[0,1,0]]);
      Pcp-group with orders [ 0, 0, 0, 0, 0 ]
      gap> Igs(last);
      [ g1, g2^2, g3, g4, g5 ]

  ## 7.3 Centralizers, Normalizers and Intersections

  In this section we list a number of operations for which there are methods installed to compute the corresponding features in polycyclic groups.

- `Centralizer(` U`,` g`)`
- `IsConjugate(` U`,` g`,` h`)`

  These functions solve the conjugacy problem for elements in pcp-groups and they can be used to compute centralizers. The first method returns a subgroup of the given group U, the second method either returns a conjugating element or false if no such element exists.

  The methods are based on the orbit stabilizer algorithms described in EOs01. For nilpotent groups, an algorithm to solve the conjugacy problem for elements is described in Sims94.

- `Centralizer(` U`,` V`)`
- `Normalizer(` U`,` V`)`
- `IsConjugate(` U`,` V`,` W`)`

  These three functions solve the conjugacy problem for subgroups and compute centralizers and normalizers of subgroups. The first two functions return subgroups of the input group U, the third function returns a conjugating element or false if no such element exists.

  The methods are based on the orbit stabilizer algorithms described in Eic02. For nilpotent groups, an algorithm to solve the conjugacy problems for subgroups is described in Lo98.

- `Intersection(` U`,` N`)`

  A general method to compute intersections of subgroups of a pcp-group is described in Eic01b, but it is not yet implemented here. However, intersections of subgroups U, N leqG can be computed if N is normalising U. See Sims94 for an outline of the algorithm.

  ## 7.4 Finite subgroups

  There are various finite subgroups of interest in polycyclic groups. See Eic00 for a description of the algorithms underlying the functions in this section.

- `TorsionSubgroup(` U`)`

  If the set of elements of finite order forms a subgroup, then we call it the torsion subgroup. This function determines the torsion subgroup of U, if it exists, and returns fail otherwise. Note that a torsion subgroup does always exist if U is nilpotent.

- `NormalTorsionSubgroup(` U`)`

  Each polycyclic groups has a unique largest finite normal subgroup. This function computes it for U.

- `IsTorsionFree(` U`)`

  This function checks if U is torsion free. It returns true or false.

- `FiniteSubgroupClasses(` U`)`

  There exist only finitely many conjugacy classes of finite subgroups in a polycyclic group U and this function can be used to compute them. The algorithm underlying this function proceeds by working down a normal series of U with elementary or free abelian factors. The following function can be used to give the algorithm a specific series.

- `FiniteSubgroupClassesBySeries(` U`,` pcps`)`

      gap> G := ExamplesOfSomePcpGroups(15);
      Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 0 ]
      gap> TorsionSubgroup(G);
      Pcp-group with orders [ 5, 2 ]
      gap> NormalTorsionSubgroup(G);
      Pcp-group with orders [ 5, 2 ]
      gap> IsTorsionFree(G);
      false
      gap> FiniteSubgroupClasses(G);
      [ Pcp-group with orders [ 5, 2 ]^G,
        Pcp-group with orders [ 2 ]^G,
        Pcp-group with orders [ 5 ]^G,
        Pcp-group with orders [  ]^G ]

      gap> G := DihedralPcpGroup( 0 );
      Pcp-group with orders [ 2, 0 ]
      gap> TorsionSubgroup(G);
      fail
      gap> NormalTorsionSubgroup(G);
      Pcp-group with orders [  ]
      gap> IsTorsionFree(G);
      false
      gap> FiniteSubgroupClasses(G);
      [ Pcp-group with orders [ 2 ]^G,
        Pcp-group with orders [ 2 ]^G,
        Pcp-group with orders [  ]^G ]

  ## 7.5 Subgroups of finite index and maximal subgroups

  Here we outline functions to determine various types of subgroups of finite index in polycyclic groups. Again, see Eic00 for a description of the algorithms underlying the functions in this section. Also, we refer to Lo99 for an alternative appraoch.

- `MaximalSubgroupClassesByIndex(` U`,` p`)`

  Each maximal subgroup of a polycyclic group U has p-power index for some prime p. This function can be used to determine the conjugacy classes of all maximal subgroups of p-power index for a given prime p.

- `LowIndexSubgroupClasses(` U`,` n`)`

  There are only finitely many subgroups of a given index in a polycyclic group U. This function computes conjugacy classes of all subgroups of index n in U.

- `LowIndexNormals(` U`,` n`)`

  This function computes the normal subgroups of index n in U.

- `NilpotentByAbelianNormalSubgroup(` U`)`

  This function returns a normal subgroup N of finite index in U such that N is nilpotent-by-abelian. Such a subgroup exists in every polycyclic group and this function computes such a subgroup using LowIndexNormal. However, we note that this function is not very efficient and the function NilpotentByAbelianByFiniteSeries may well be more efficient on this task.

      gap> G := ExamplesOfSomePcpGroups(2);
      Pcp-group with orders [ 0, 0, 0, 0, 0, 0 ]

      gap> MaximalSubgroupClassesPIndex( G, 61 );;
      gap> max := List( last, Representative );;
      gap> List( max, x -> Index( G, x ) );
      [ 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
        61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
        61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
        61, 61, 61, 61, 61, 61, 226981 ]

      gap> LowIndexSubgroupClasses( G, 61 );;
      gap> low := List( last, Representative );;
      gap> List( low, x -> Index( G, x ) );
      [ 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
        61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
        61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
        61, 61, 61, 61, 61, 61 ]

  ## 7.6 Further attributes for pcp-groups based on the Fitting subgroup

  In this section we provide a variety of other attributes for pcp-groups. Most of the methods below are based or related to the Fitting subgroup of the given group. We refer to Eic01 for a description of the underlying methods.

- `FittingSubgroup(` U`)`

  returns the Fitting subgroup of U; that is, the largest nilpotent normal subgroup of U.

- `IsNilpotentByFinite(` U`)`

  checks whether the Fitting subgroup of U has finite index.

- `Centre(` U`)`

  returns the centre of U.

- `FCCentre(` U`)`

  returns the FC-centre of U; that is, the subgroup containing all elements having a finite conjugacy class in U.

- `PolyZNormalSubgroup(` U`)`

  returns a normal subgroup N of U such that N has a polycyclic series with infinite factors only.

- `NilpotentByAbelianByFiniteSeries(` U`)`

  returns a normal series 1 leqF leqA leqU such that F is nilpotent, A/F is abelian and U/A is finite. This series is computed using the Fitting subgroup and the centre of the Fitting factor.

  ## 7.7 Functions for nilpotent groups

  There are (very few) functions which are available for nilpotent groups only. First, there are the different central series. These are available for all groups, but for nilpotent groups they terminate and provide series though the full group. Secondly, the determination of a minimal generating set is available for nilpotent groups only.

- `MinimalGeneratingSet(` U`)`

      gap> G := ExamplesOfSomePcpGroups(14);
      Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 0, 5, 5, 4, 0, 6,
        5, 5, 4, 0, 10, 6 ]
      gap> IsNilpotent(G);
      true

      gap> PcpsBySeries( LowerCentralSeries(G));
      [ Pcp [ g1, g2 ] with orders [ 0, 0 ],
        Pcp [ g3 ] with orders [ 0 ],
        Pcp [ g4 ] with orders [ 0 ],
        Pcp [ g5 ] with orders [ 0 ],
        Pcp [ g6, g7 ] with orders [ 0, 0 ],
        Pcp [ g8 ] with orders [ 0 ],
        Pcp [ g9, g10 ] with orders [ 0, 0 ],
        Pcp [ g11, g12, g13 ] with orders [ 5, 4, 0 ],
        Pcp [ g14, g15, g16, g17, g18 ] with orders [ 5, 5, 4, 0, 6 ],
        Pcp [ g19, g20, g21, g22, g23, g24 ] with orders [ 5, 5, 4, 0, 10, 6 ] ]

      gap> PcpsBySeries( UpperCentralSeries(G));
      [ Pcp [ g1, g2 ] with orders [ 0, 0 ],
        Pcp [ g3 ] with orders [ 0 ],
        Pcp [ g4 ] with orders [ 0 ],
        Pcp [ g5 ] with orders [ 0 ],
        Pcp [ g6, g7 ] with orders [ 0, 0 ],
        Pcp [ g8 ] with orders [ 0 ],
        Pcp [ g9, g10 ] with orders [ 0, 0 ],
        Pcp [ g11, g12, g13 ] with orders [ 5, 4, 0 ],
        Pcp [ g14, g15, g16, g17, g18 ] with orders [ 5, 5, 4, 0, 6 ],
        Pcp [ g19, g20, g21, g22, g23, g24 ] with orders [ 5, 5, 4, 0, 10, 6 ] ]

      gap> MinimalGeneratingSet(G);
      [ g1, g2 ]

  ## 7.8 Random methods for pcp-groups

  Below we introduce a function which computes orbit and stabilizer using a random method. This function tries to approximate the orbit and the stabilizer, but the returned orbit or stabilizer may be incomplete. This function is used in the random methods to compute normalizers and centralizers. Note that determinstic methods for these purposes are also available.

- `RandomOrbitStabilizerPcpGroup(` U`,` point`,` oper`)`

- `RandomCentralizerPcpGroup(` U`,` g`)`
- `RandomCentralizerPcpGroup(` U`,` V`)`

- `RandomNormalizerPcpGroup(` U`,` V`)`

      gap> G := DihedralPcpGroup(0);
      Pcp-group with orders [ 2, 0 ]
      gap> mats := [[[-1, 0],[0,1]], [[1,1],[0,1]]];
      [ [ [ -1, 0 ], [ 0, 1 ] ], [ [ 1, 1 ], [ 0, 1 ] ] ]
      gap> pcp := Pcp(G);
      Pcp [ g1, g2 ] with orders [ 2, 0 ]

      gap> RandomPcpOrbitStabilizer( [1,0], pcp, mats, OnRight ).stab;
      #I  Orbit longer than limit: exiting.
      [  ]

      gap> g := Igs(G)[1];
      g1
      gap> RandomCentralizerPcpGroup( G, g );
      #I  Stabilizer not increasing: exiting.
      Pcp-group with orders [ 2 ]
      gap> Igs(last);
      [ g1 ]

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