  
  [1X8 [33X[0;0YCohomology for pcp-groups[133X[101X
  
  [33X[0;0YThe  [5XGAP[105X  4  package  [5XPolycyclic[105X  provides  methods to compute the first and
  second cohomology group for a pcp-group [22XU[122X and a finite dimensional [22Xℤ U[122X or [22XFU[122X
  module  [22XA[122X where [22XF[122X is a finite field. The algorithm for determining the first
  cohomology group is outlined in [Eic00].[133X
  
  [33X[0;0YAs a preparation for the cohomology computation, we introduce the cohomology
  records.  These  records  provide  the  technical  setup  for our cohomology
  computations.[133X
  
  
  [1X8.1 [33X[0;0YCohomology records[133X[101X
  
  [33X[0;0YCohomology  records provide the necessary technical setup for the cohomology
  computations for polycyclic groups.[133X
  
  [1X8.1-1 CRRecordByMats[101X
  
  [33X[1;0Y[29X[2XCRRecordByMats[102X( [3XU[103X, [3Xmats[103X ) [32X function[133X
  
  [33X[0;0Ycreates  an external module. Let [3XU[103X be a pcp group which acts via the list of
  matrices [3Xmats[103X on a vector space of the form [22Xℤ^n[122X or [22XF_p^n[122X. Then this function
  creates a record which can be used as input for the cohomology computations.[133X
  
  [1X8.1-2 CRRecordBySubgroup[101X
  
  [33X[1;0Y[29X[2XCRRecordBySubgroup[102X( [3XU[103X, [3XA[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCRRecordByPcp[102X( [3XU[103X, [3Xpcp[103X ) [32X function[133X
  
  [33X[0;0Ycreates  an  internal  module.  Let  [3XU[103X  be a pcp group and let [3XA[103X be a normal
  elementary  or  free  abelian  normal subgroup of [3XU[103X or let [3Xpcp[103X be a pcp of a
  normal elementary of free abelian subfactor of [3XU[103X. Then this function creates
  a record which can be used as input for the cohomology computations.[133X
  
  [33X[0;0YThe returned cohomology record [3XC[103X contains the following entries:[133X
  
  [8X[3Xfactor[103X[8X[108X
        [33X[0;6Ya  pcp  of  the  acting group. If the module is external, then this is
        [3XPcp(U)[103X.  If  the  module is internal, then this is [3XPcp(U, A)[103X or [3XPcp(U,
        GroupOfPcp(pcp))[103X.[133X
  
  [8X[3Xmats[103X[8X, [3Xinvs[103X[8X and [3Xone[103X[8X[108X
        [33X[0;6Ythe  matrix  action of [3Xfactor[103X with acting matrices, their inverses and
        the identity matrix.[133X
  
  [8X[3Xdim[103X[8X and [3Xchar[103X[8X[108X
        [33X[0;6Ythe dimension and characteristic of the matrices.[133X
  
  [8X[3Xrelators[103X[8X and [3Xenumrels[103X[8X[108X
        [33X[0;6Ythe right hand sides of the polycyclic relators of [3Xfactor[103X as generator
        exponents  lists  and  a  description  for the corresponding left hand
        sides.[133X
  
  [8X[3Xcentral[103X[8X[108X
        [33X[0;6Yis  true,  if  the matrices [3Xmats[103X are all trivial. This is used locally
        for efficiency reasons.[133X
  
  [33X[0;0YAnd additionally, if [22XC[122X defines an internal module, then it contains:[133X
  
  [8X[3Xgroup[103X[8X[108X
        [33X[0;6Ythe original group [3XU[103X.[133X
  
  [8X[3Xnormal[103X[8X[108X
        [33X[0;6Ythis is either [3XPcp(A)[103X or the input [3Xpcp[103X.[133X
  
  [8X[3Xextension[103X[8X[108X
        [33X[0;6Yinformation on the extension of [3XA[103X by [3XU/A[103X.[133X
  
  
  [1X8.2 [33X[0;0YCohomology groups[133X[101X
  
  [33X[0;0YLet  [22XU[122X  be  a  pcp-group  and [22XA[122X a free or elementary abelian pcp-group and a
  [22XU[122X-module. By [22XZ^i(U, A)[122X be denote the group of [22Xi[122X-th cocycles and by [22XB^i(U, A)[122X
  the [22Xi[122X-th coboundaries. The factor [22XZ^i(U,A) / B^i(U,A)[122X is the [22Xi[122X-th cohomology
  group.  Since  [22XA[122X  is  elementary  or  free abelian, the groups [22XZ^i(U, A)[122X and
  [22XB^i(U, A)[122X are elementary or free abelian groups as well.[133X
  
  [33X[0;0YThe  [5XPolycyclic[105X  package  provides  methods  to  compute  first  and  second
  cohomology  group  for  a  polycyclic  group [3XU[103X. We write all involved groups
  additively  and  we  use an explicit description by bases for them. Let [22XC[122X be
  the cohomology record corresponding to [22XU[122X and [22XA[122X.[133X
  
  [33X[0;0YLet  [22Xf_1,  ...,  f_n[122X  be  the elements in the entry [22Xfactor[122X of the cohomology
  record  [22XC[122X. Then we use the following embedding of the first cocycle group to
  describe 1-cocycles and 1-coboundaries: [22XZ^1(U, A) -> A^n : δ ↦ (δ(f_1), ...,
  δ(f_n))[122X[133X
  
  [33X[0;0YFor  the  second  cohomology  group we recall that each element of [22XZ^2(U, A)[122X
  defines  an  extension  [22XH[122X  of  [22XA[122X  by [22XU[122X. Thus there is a pc-presentation of [22XH[122X
  extending  the  pc-presentation  of  [22XU[122X  given  by the record [22XC[122X. The extended
  presentation  is  defined by tails in [22XA[122X; that is, each relator in the record
  entry  [22Xrelators[122X  is  extended by an element of [22XA[122X. The concatenation of these
  tails  yields  a  vector  in  [22XA^l[122X  where [22Xl[122X is the length of the record entry
  [22Xrelators[122X  of  [22XC[122X.  We use these tail vectors to describe [22XZ^2(U, A)[122X and [22XB^2(U,
  A)[122X. Note that this description is dependent on the chosen presentation in [22XC[122X.
  However,  the  factor  [22XZ^2(U,  A)/  B^2(U,  A)[122X  is independent of the chosen
  presentation.[133X
  
  [33X[0;0YThe  following  functions  are available to compute explicitly the first and
  second cohomology group as described above.[133X
  
  [1X8.2-1 OneCoboundariesCR[101X
  
  [33X[1;0Y[29X[2XOneCoboundariesCR[102X( [3XC[103X ) [32X function[133X
  [33X[1;0Y[29X[2XOneCocyclesCR[102X( [3XC[103X ) [32X function[133X
  [33X[1;0Y[29X[2XTwoCoboundariesCR[102X( [3XC[103X ) [32X function[133X
  [33X[1;0Y[29X[2XTwoCocyclesCR[102X( [3XC[103X ) [32X function[133X
  [33X[1;0Y[29X[2XOneCohomologyCR[102X( [3XC[103X ) [32X function[133X
  [33X[1;0Y[29X[2XTwoCohomologyCR[102X( [3XC[103X ) [32X function[133X
  
  [33X[0;0YThe  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.[133X
  
  [8X[3Xgcc[103X[8X[108X
        [33X[0;6Ythe basis of the cocycles of [3XC[103X.[133X
  
  [8X[3Xgcb[103X[8X[108X
        [33X[0;6Ythe basis of the coboundaries of [3XC[103X.[133X
  
  [8X[3Xfactor[103X[8X[108X
        [33X[0;6Ya  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 [3XA[103X is free abelian and thus we use a
        description of this additive map as record. This record contains[133X
  
        [8X[3Xgens[103X[8X[108X
              [33X[0;12Ya base for the image.[133X
  
        [8X[3Xrels[103X[8X[108X
              [33X[0;12Yrelative orders for the image.[133X
  
        [8X[3Ximgs[103X[8X[108X
              [33X[0;12Ythe images for the elements in [3Xgcc[103X.[133X
  
        [8X[3Xprei[103X[8X[108X
              [33X[0;12Ypreimages for the elements in [3Xgens[103X.[133X
  
        [8X[3Xdenom[103X[8X[108X
              [33X[0;12Ythe kernel of the map; that is, another basis for [3Xgcb[103X.[133X
  
  [33X[0;0YThere  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[133X
  
  [1X8.2-2 TwoCohomologyModCR[101X
  
  [33X[1;0Y[29X[2XTwoCohomologyModCR[102X( [3XC[103X, [3Xlat[103X ) [32X function[133X
  
  [33X[0;0Ywhere [3XC[103X is a cohomology record and [3Xlat[103X is a basis for a sublattice of a free
  abelian module. The output format is the same as for [10XTwoCohomologyCR[110X.[133X
  
  
  [1X8.3 [33X[0;0YExtended 1-cohomology[133X[101X
  
  [33X[0;0YIn 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.[133X
  
  [1X8.3-1 OneCoboundariesEX[101X
  
  [33X[1;0Y[29X[2XOneCoboundariesEX[102X( [3XC[103X ) [32X function[133X
  
  [33X[0;0Yreturns a record consisting of the entries[133X
  
  [8X[3Xbasis[103X[8X[108X
        [33X[0;6Ya basis for [22XB^1(U, A) ≤ A^n[122X.[133X
  
  [8X[3Xtransf[103X[8X[108X
        [33X[0;6YThere  is  a  derivation  mapping  from [22XA[122X to [22XB^1(U,A)[122X. This mapping is
        described here as transformation from [22XA[122X to [3Xbasis[103X.[133X
  
  [8X[3Xfixpts[103X[8X[108X
        [33X[0;6Ythe fixpoints of [22XA[122X. This is also the kernel of the derivation mapping.[133X
  
  [1X8.3-2 OneCocyclesEX[101X
  
  [33X[1;0Y[29X[2XOneCocyclesEX[102X( [3XC[103X ) [32X function[133X
  
  [33X[0;0Yreturns a record consisting of the entries[133X
  
  [8X[3Xbasis[103X[8X[108X
        [33X[0;6Ya basis for [22XZ^1(U, A) ≤ A^n[122X.[133X
  
  [8X[3Xtransl[103X[8X[108X
        [33X[0;6Ya  special  solution.  This  is  only of interest in case that [22XC[122X is an
        internal  module  and  in this case it gives the translation vector in
        [22XA^n[122X used to obtain complements corresponding to the elements in [22Xbasis[122X.
        If  [22XC[122X  is  not an internal module, then this vector is always the zero
        vector.[133X
  
  [1X8.3-3 OneCohomologyEX[101X
  
  [33X[1;0Y[29X[2XOneCohomologyEX[102X( [3XC[103X ) [32X function[133X
  
  [33X[0;0Yreturns the combined information on the first cohomology group.[133X
  
  
  [1X8.4 [33X[0;0YExtensions and Complements[133X[101X
  
  [33X[0;0YThe  natural  applications  of  first  and  second  cohomology  group is the
  determination of extensions and complements. Let [22XC[122X be a cohomology record.[133X
  
  [1X8.4-1  ComplementCR[101X
  
  [33X[1;0Y[29X[2X ComplementCR[102X( [3XC[103X, [3Xc[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the complement corresponding to the 1-cocycle [3Xc[103X. In the case that [3XC[103X
  is  an  external  module, we construct the split extension of [22XU[122X with [22XA[122X first
  and then determine the complement. In the case that [3XC[103X is an internal module,
  the  vector  [3Xc[103X  must  be an element of the affine space corresponding to the
  complements as described by [10XOneCocyclesEX[110X.[133X
  
  [1X8.4-2  ComplementsCR[101X
  
  [33X[1;0Y[29X[2X ComplementsCR[102X( [3XC[103X ) [32X function[133X
  
  [33X[0;0Yreturns  all complements using the correspondence to [22XZ^1(U,A)[122X. Further, this
  function returns fail, if [22XZ^1(U,A)[122X is infinite.[133X
  
  [1X8.4-3  ComplementClassesCR[101X
  
  [33X[1;0Y[29X[2X ComplementClassesCR[102X( [3XC[103X ) [32X function[133X
  
  [33X[0;0Yreturns  complement  classes  using the correspondence to [22XH^1(U,A)[122X. Further,
  this function returns fail, if [22XH^1(U,A)[122X is infinite.[133X
  
  [1X8.4-4  ComplementClassesEfaPcps[101X
  
  [33X[1;0Y[29X[2X ComplementClassesEfaPcps[102X( [3XU[103X, [3XN[103X, [3Xpcps[103X ) [32X function[133X
  
  [33X[0;0YLet  [22XN[122X  be  a  normal  subgroup  of  [22XU[122X. This function returns the complement
  classes  to  [22XN[122X  in  [22XU[122X.  The  classes  are  computed  by  iteration  over the
  [22XU[122X-invariant  efa  series  of  [22XN[122X  described by [3Xpcps[103X. 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 [22XN[122X in [22XU[122X.)[133X
  
  [1X8.4-5  ComplementClasses[101X
  
  [33X[1;0Y[29X[2X ComplementClasses[102X( [[3XV[103X, ][3XU[103X, [3XN[103X ) [32X function[133X
  
  [33X[0;0YLet  [22XN[122X and [22XU[122X be normal subgroups of [22XV[122X with [22XN ≤ U ≤ V[122X. This function attempts
  to  compute  the [22XV[122X-conjugacy classes of complements to [22XN[122X in [22XU[122X. The algorithm
  proceeds  by  iteration over a [22XV[122X-invariant efa series of [22XN[122X. If at some stage
  in  this  iteration  infinitely  many  complements  are discovered, then the
  algorithm returns fail.[133X
  
  [1X8.4-6 ExtensionCR[101X
  
  [33X[1;0Y[29X[2XExtensionCR[102X( [3XC[103X, [3Xc[103X ) [32X function[133X
  
  [33X[0;0Yreturns the extension corresponding to the 2-cocycle [22Xc[122X.[133X
  
  [1X8.4-7 ExtensionsCR[101X
  
  [33X[1;0Y[29X[2XExtensionsCR[102X( [3XC[103X ) [32X function[133X
  
  [33X[0;0Yreturns  all  extensions using the correspondence to [22XZ^2(U,A)[122X. Further, this
  function returns fail, if [22XZ^2(U,A)[122X is infinite.[133X
  
  [1X8.4-8 ExtensionClassesCR[101X
  
  [33X[1;0Y[29X[2XExtensionClassesCR[102X( [3XC[103X ) [32X function[133X
  
  [33X[0;0Yreturns  extension  classes  using  the correspondence to [22XH^2(U,A)[122X. Further,
  this function returns fail, if [22XH^2(U,A)[122X is infinite.[133X
  
  [1X8.4-9 SplitExtensionPcpGroup[101X
  
  [33X[1;0Y[29X[2XSplitExtensionPcpGroup[102X( [3XU[103X, [3Xmats[103X ) [32X function[133X
  
  [33X[0;0Yreturns the split extension of [3XU[103X by the [22XU[122X-module described by [3Xmats[103X.[133X
  
  
  [1X8.5 [33X[0;0YConstructing pcp groups as extensions[133X[101X
  
  [33X[0;0YThis  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.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X# get the group and its matrix action[128X[104X
    [4X[25Xgap>[125X [27XG := UnitriangularPcpGroup(3,0);[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xmats := G!.mats;[127X[104X
    [4X[28X[ [ [ 1, 1, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],[128X[104X
    [4X[28X  [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 1 ] ],[128X[104X
    [4X[28X  [ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ][128X[104X
    [4X[28X[128X[104X
    [4X[28X# set up the cohomology record[128X[104X
    [4X[25Xgap>[125X [27XC := CRRecordByMats(G,mats);;[127X[104X
    [4X[28X[128X[104X
    [4X[28X# compute the second cohomology group[128X[104X
    [4X[25Xgap>[125X [27Xcc := TwoCohomologyCR(C);;[127X[104X
    [4X[28X[128X[104X
    [4X[28X# the abelian invariants of H^2(G,M)[128X[104X
    [4X[25Xgap>[125X [27Xcc.factor.rels;[127X[104X
    [4X[28X[ 2, 0, 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[28X# construct an extension which corresponds to a cocycle that has[128X[104X
    [4X[28X# infinite image in H^2(G,M)[128X[104X
    [4X[25Xgap>[125X [27Xc := cc.factor.prei[2];[127X[104X
    [4X[28X[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XH := ExtensionCR( CR, c);[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[28X# check that the extension does not split - get the normal subgroup[128X[104X
    [4X[25Xgap>[125X [27XN := H!.module;[127X[104X
    [4X[28XPcp-group with orders [ 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[28X# create the interal module[128X[104X
    [4X[25Xgap>[125X [27XC := CRRecordBySubgroup(H,N);;[127X[104X
    [4X[28X[128X[104X
    [4X[28X# use the complements routine[128X[104X
    [4X[25Xgap>[125X [27XComplementClassesCR(C);[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
