GAP (polycyclic) - Chapter 8: Cohomology for pcp-groups

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8 Cohomology for pcp-groups
8.1 Cohomology records
8.1-1 CRRecordByMats
8.1-2 CRRecordBySubgroup
8.2 Cohomology groups
8.2-1 OneCoboundariesCR
8.2-2 TwoCohomologyModCR
8.3 Extended 1-cohomology
8.3-1 OneCoboundariesEX
8.3-2 OneCocyclesEX
8.3-3 OneCohomologyEX
8.4 Extensions and Complements
8.4-1 ComplementCR
8.4-2 ComplementsCR
8.4-3 ComplementClassesCR
8.4-4 ComplementClassesEfaPcps
8.4-5 ComplementClasses
8.4-6 ExtensionCR
8.4-7 ExtensionsCR
8.4-8 ExtensionClassesCR
8.4-9 SplitExtensionPcpGroup
8.5 Constructing pcp groups as extensions

### 8 Cohomology for pcp-groups

The GAP 4 package Polycyclic provides methods to compute the first and second cohomology group for a pcp-group \(U\) and a finite dimensional \(ℤ U\) or \(FU\) module \(A\) where \(F\) is a finite field. The algorithm for determining the first cohomology group is outlined in [Eic00].

As a preparation for the cohomology computation, we introduce the cohomology records. These records provide the technical setup for our cohomology computations.

#### 8.1 Cohomology records

Cohomology records provide the necessary technical setup for the cohomology computations for polycyclic groups.

##### 8.1-1 CRRecordByMats

`‣ CRRecordByMats`( U, mats ) | ( function )

creates an external module. Let U be a pcp group which acts via the list of matrices mats on a vector space of the form \(ℤ^n\) or \(\mathbb{F}_p^n\). Then this function creates a record which can be used as input for the cohomology computations.

##### 8.1-2 CRRecordBySubgroup

`‣ CRRecordBySubgroup`( U, A ) | ( function )

`‣ CRRecordByPcp`( U, pcp ) | ( function )

creates an internal module. Let U be a pcp group and let A be a normal elementary or free abelian normal subgroup of U or let pcp be a pcp of a normal elementary of free abelian subfactor of U. Then this function creates a record which can be used as input for the cohomology computations.

The returned cohomology record C contains the following entries:

factor

  a pcp of the acting group. If the module is external, then this is Pcp(U). If the module is internal, then this is Pcp(U, A) or Pcp(U, GroupOfPcp(pcp)).

mats, invs and one

  the matrix action of factor with acting matrices, their inverses and the identity matrix.

dim and char

  the dimension and characteristic of the matrices.

relators and enumrels

  the right hand sides of the polycyclic relators of factor as generator exponents lists and a description for the corresponding left hand sides.

central

  is true, if the matrices mats are all trivial. This is used locally for efficiency reasons.

And additionally, if \(C\) defines an internal module, then it contains:

group

  the original group U.

normal

  this is either Pcp(A) or the input pcp.

extension

  information on the extension of A by U/A.

#### 8.2 Cohomology groups

Let \(U\) be a pcp-group and \(A\) a free or elementary abelian pcp-group and a \(U\)-module. By \(Z^i(U, A)\) be denote the group of \(i\)-th cocycles and by \(B^i(U, A)\) the \(i\)-th coboundaries. The factor \(Z^i(U,A) / B^i(U,A)\) is the \(i\)-th cohomology group. Since \(A\) is elementary or free abelian, the groups \(Z^i(U, A)\) and \(B^i(U, A)\) are elementary or free abelian groups as well.

The Polycyclic package provides methods to compute first and second cohomology group for a polycyclic group U. We write all involved groups additively and we use an explicit description by bases for them. Let \(C\) be the cohomology record corresponding to \(U\) and \(A\).

Let \(f_1, \ldots, f_n\) be the elements in the entry \(factor\) of the cohomology record \(C\). Then we use the following embedding of the first cocycle group to describe 1-cocycles and 1-coboundaries: \(Z^1(U, A) \to A^n : \delta \mapsto (\delta(f_1), \ldots, \delta(f_n))\)

For the second cohomology group we recall that each element of \(Z^2(U, A)\) defines an extension \(H\) of \(A\) by \(U\). Thus there is a pc-presentation of \(H\) extending the pc-presentation of \(U\) given by the record \(C\). The extended presentation is defined by tails in \(A\); that is, each relator in the record entry \(relators\) is extended by an element of \(A\). The concatenation of these tails yields a vector in \(A^l\) where \(l\) is the length of the record entry \(relators\) of \(C\). We use these tail vectors to describe \(Z^2(U, A)\) and \(B^2(U, A)\). Note that this description is dependent on the chosen presentation in \(C\). However, the factor \(Z^2(U, A)/ B^2(U, A)\) is independent of the chosen presentation.

The following functions are available to compute explicitly the first and second cohomology group as described above.

##### 8.2-1 OneCoboundariesCR

`‣ OneCoboundariesCR`( C ) | ( function )

`‣ OneCocyclesCR`( C ) | ( function )

`‣ TwoCoboundariesCR`( C ) | ( function )

`‣ TwoCocyclesCR`( C ) | ( function )

`‣ OneCohomologyCR`( C ) | ( function )

`‣ TwoCohomologyCR`( C ) | ( function )

The first four functions return bases of the corresponding group. The last two functions need to describe a factor of additive abelian groups. They return the following descriptions for these factors.

gcc

  the basis of the cocycles of C.

gcb

  the basis of the coboundaries of C.

factor

  a description of the factor of cocycles by coboundaries. Usually, it would be most convenient to use additive mappings here. However, these are not available in case that A is free abelian and thus we use a description of this additive map as record. This record contains

  gens

    a base for the image.

  rels

    relative orders for the image.

  imgs

    the images for the elements in gcc.

  prei

    preimages for the elements in gens.

  denom

    the kernel of the map; that is, another basis for gcb.

There is an additional function which can be used to compute the second cohomology group over an arbitrary finitely generated abelian group. The finitely generated abelian group should be realized as a factor of a free abelian group modulo a lattice. The function is called as

##### 8.2-2 TwoCohomologyModCR

`‣ TwoCohomologyModCR`( C, lat ) | ( function )

where C is a cohomology record and lat is a basis for a sublattice of a free abelian module. The output format is the same as for `TwoCohomologyCR`.

#### 8.3 Extended 1-cohomology

In some cases more information on the first cohomology group is of interest. In particular, if we have an internal module given and we want to compute the complements using the first cohomology group, then we need additional information. This extended version of first cohomology is obtained by the following functions.

##### 8.3-1 OneCoboundariesEX

`‣ OneCoboundariesEX`( C ) | ( function )

returns a record consisting of the entries

basis

  a basis for \(B^1(U, A) \leq A^n\).

transf

  There is a derivation mapping from \(A\) to \(B^1(U,A)\). This mapping is described here as transformation from \(A\) to basis.

fixpts

  the fixpoints of \(A\). This is also the kernel of the derivation mapping.

##### 8.3-2 OneCocyclesEX

`‣ OneCocyclesEX`( C ) | ( function )

returns a record consisting of the entries

basis

  a basis for \(Z^1(U, A) \leq A^n\).

transl

  a special solution. This is only of interest in case that \(C\) is an internal module and in this case it gives the translation vector in \(A^n\) used to obtain complements corresponding to the elements in \(basis\). If \(C\) is not an internal module, then this vector is always the zero vector.

##### 8.3-3 OneCohomologyEX

`‣ OneCohomologyEX`( C ) | ( function )

returns the combined information on the first cohomology group.

#### 8.4 Extensions and Complements

The natural applications of first and second cohomology group is the determination of extensions and complements. Let \(C\) be a cohomology record.

##### 8.4-1 ComplementCR

`‣ ComplementCR`( C, c ) | ( function )

returns the complement corresponding to the 1-cocycle c. In the case that C is an external module, we construct the split extension of \(U\) with \(A\) first and then determine the complement. In the case that C is an internal module, the vector c must be an element of the affine space corresponding to the complements as described by `OneCocyclesEX`.

##### 8.4-2 ComplementsCR

`‣ ComplementsCR`( C ) | ( function )

returns all complements using the correspondence to \(Z^1(U,A)\). Further, this function returns fail, if \(Z^1(U,A)\) is infinite.

##### 8.4-3 ComplementClassesCR

`‣ ComplementClassesCR`( C ) | ( function )

returns complement classes using the correspondence to \(H^1(U,A)\). Further, this function returns fail, if \(H^1(U,A)\) is infinite.

##### 8.4-4 ComplementClassesEfaPcps

`‣ ComplementClassesEfaPcps`( U, N, pcps ) | ( function )

Let \(N\) be a normal subgroup of \(U\). This function returns the complement classes to \(N\) in \(U\). The classes are computed by iteration over the \(U\)-invariant efa series of \(N\) described by pcps. If at some stage in this iteration infinitely many complements are discovered, then the function returns fail. (Even though there might be only finitely many conjugacy classes of complements to \(N\) in \(U\).)

##### 8.4-5 ComplementClasses

`‣ ComplementClasses`( [V, ]U, N ) | ( function )

Let \(N\) and \(U\) be normal subgroups of \(V\) with \(N \leq U \leq V\). This function attempts to compute the \(V\)-conjugacy classes of complements to \(N\) in \(U\). The algorithm proceeds by iteration over a \(V\)-invariant efa series of \(N\). If at some stage in this iteration infinitely many complements are discovered, then the algorithm returns fail.

##### 8.4-6 ExtensionCR

`‣ ExtensionCR`( C, c ) | ( function )

returns the extension corresponding to the 2-cocycle \(c\).

##### 8.4-7 ExtensionsCR

`‣ ExtensionsCR`( C ) | ( function )

returns all extensions using the correspondence to \(Z^2(U,A)\). Further, this function returns fail, if \(Z^2(U,A)\) is infinite.

##### 8.4-8 ExtensionClassesCR

`‣ ExtensionClassesCR`( C ) | ( function )

returns extension classes using the correspondence to \(H^2(U,A)\). Further, this function returns fail, if \(H^2(U,A)\) is infinite.

##### 8.4-9 SplitExtensionPcpGroup

`‣ SplitExtensionPcpGroup`( U, mats ) | ( function )

returns the split extension of U by the \(U\)-module described by mats.

#### 8.5 Constructing pcp groups as extensions

This section contains an example application of the second cohomology group to the construction of pcp groups as extensions. The following constructs extensions of the group of upper unitriangular matrices with its natural lattice.

    # get the group and its matrix action
    gap> G := UnitriangularPcpGroup(3,0);
    Pcp-group with orders [ 0, 0, 0 ]
    gap> mats := G!.mats;
    [ [ [ 1, 1, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
      [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 1 ] ],
      [ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ]

    # set up the cohomology record
    gap> C := CRRecordByMats(G,mats);;

    # compute the second cohomology group
    gap> cc := TwoCohomologyCR(C);;

    # the abelian invariants of H^2(G,M)
    gap> cc.factor.rels;
    [ 2, 0, 0 ]

    # construct an extension which corresponds to a cocycle that has
    # infinite image in H^2(G,M)
    gap> c := cc.factor.prei[2];
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1 ]
    gap> H := ExtensionCR( CR, c);
    Pcp-group with orders [ 0, 0, 0, 0, 0, 0 ]

    # check that the extension does not split - get the normal subgroup
    gap> N := H!.module;
    Pcp-group with orders [ 0, 0, 0 ]

    # create the interal module
    gap> C := CRRecordBySubgroup(H,N);;

    # use the complements routine
    gap> ComplementClassesCR(C);
    [  ]

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