  
  [1X2 Usage[0X
  
  Unless  otherwise  specified,  all  the  functions described below taking an
  argument  [9Xn[0m  do whatever the manual says they do up to homological degree [9Xn[0m.
  These  functions  are idempotent in the sense that called a second time with
  the  same  argument  [9Xn[0m,  they  do  nothing, but called with a bigger [9Xn[0m, they
  continue computing from where the previous calculations finished.
  
  
  [1X2.1 Cohomology Objects[0X
  
  The  computation  of  group  cohomology  involves  several calculations, the
  results of which are reused in later calculations, and are thus collected in
  an object of type [9XCObject[0m, which is created with the following command.
  
  [1X2.1-1 CohomologyObject[0m
  
  [2X> CohomologyObject( [0X[3XG, M[0X[2X ) ________________________________________[0Xoperation
  [2X> CohomologyObject( [0X[3XG[0X[2X ) ___________________________________________[0Xoperation
  [6XReturns:[0X  a [9XCObject[0m.
  
  This  function  creates  a  [9XCObject[0m  having components the p-group [9XG[0m and the
  [5XMeatAxe[0m  module  [9XM[0m,  which should be a kG-module where G the group [9XG[0m and k a
  field of characteristic p. Note that [5XMeatAxe[0m modules know what k is, but not
  what  G  is,  which is why this operation requires the user to specify G but
  not k.
  
  Fortunately,  most  users don't need to know anything about [5XMeatAxe[0m modules,
  being  interested primarily in the case where k=GF(p) and M=k is the trivial
  kG-module.  In  this  situation,  the second invocation creates a cohomology
  object  having  components  the  p-group G and the trivial [5XMeatAxe[0m kG-module
  k=GF(p).
  
  We  emphasize  that  in  the  first  invocation,  k  can  be  any  field  of
  characteristic  p  and  [9XM[0m  can  be  any  [5XMeatAxe[0m  module  over  kG, and that
  [9XProjectiveResolution[0m  works  when [9XM[0m is an arbitrary [5XMeatAxe[0m module, but that
  all  the functions dealing with the ring-structure of H*(G,k) require that [9XM[0m
  be the trivial module.
  
  The  cohomology  object is used to store, in addition to the items mentioned
  above, the boundary maps, the Betti numbers, the multiplication table, etc.
  
  
  [1X2.2 Minimal Projective Resolutions[0X
  
  Given  a  p-group  G,  a field k of characteristic p, and a kG-module M, the
  function  below  computes the beginning of the minimal projective resolution
  of M
  
  
       P_n -> ... -> P_2 -> P_1 -> P_0 -> M -> 0
  
  
  where  P_i  is  the direct sum (kG)^(b_i) for certain numbers b_i, the [13XBetti
  numbers[0m  of  the  resolution.  The  minimal kG-projective resolution of M is
  unique  up  to  chain isomorphism. Because of the minimality of P the groups
  Ext^i(M,N)  are  simply  Hom(P_i,N),  and  if  M  and N are both the trivial
  kG-module k, then H^i(G,k)=Ext^i(k,k)=k^(b_i).
  
  [1X2.2-1 ProjectiveResolution[0m
  
  [2X> ProjectiveResolution( [0X[3XC, n[0X[2X ) ____________________________________[0Xoperation
  [6XReturns:[0X  a list containing the Betti numbers b_0, b_1,..., b_n.
  
  Given  a  cohomology  object  [9XC[0m  having  components  G  and M, this function
  computes  the first [9Xn[0m+1 terms of the minimal projective resolution P of M of
  the  form  P_i=(kG)^(b_i)  for  i=0,1,...,n and returns the numbers b_i as a
  list.
  
  [1X2.2-2 BoundaryMap[0m
  
  [2X> BoundaryMap( [0X[3XC, n[0X[2X ) _____________________________________________[0Xoperation
  [6XReturns:[0X  the [9Xn[0mth boundary map.
  
  Given   the  cohomology  object  [9XC[0m,  this  function  computes  a  projective
  resolution  to  degree [9Xn[0m if it hasn't been computed already, and returns the
  [9Xn[0mth boundary map P_n -> P_n-1.
  
  The map returned is a b_n x b_n-1|G| matrix, having in the ith row the image
  of the element 1_G from the ith direct summand of P_n.
  
  See the file [9Xdoc/example.*[0m for an example of the usage and interpretation of
  the result of this function.
  
  
  [1X2.3 Cohomology Generators and Relators[0X
  
  See [CTVZ03] for the details of the calculation of cohomology products using
  composition  of  chain  maps.  See  also  the  file [9Xdoc/explanation.*[0m for an
  explanation of the implementation.
  
  [1X2.3-1 CohomologyGenerators[0m
  
  [2X> CohomologyGenerators( [0X[3XC, n[0X[2X ) ____________________________________[0Xoperation
  [6XReturns:[0X  a  list  containing  the  degrees  of  the  elements  of  a set of
            generators of the cohomology ring.
  
  Given  a  cohomology  object [9XC[0m having group component G and module component
  the trivial kG-module, this function computes a set of generators of H*(G,k)
  having  degree  [9Xn[0m or less, and stores them in [9XC[0m. The function returns a list
  of the degrees of these generators.
  
  The  actual  cohomology  generators are represented by maps P_i -> k for 0le
  ile n and are stored in [9XC[0m as matrices. Only their degrees are returned.
  
  [1X2.3-2 CohomologyRelators[0m
  
  [2X> CohomologyRelators( [0X[3XC, n[0X[2X ) ______________________________________[0Xoperation
  [6XReturns:[0X  a list of generators and a list of relators.
  
  Given a cohomology object [9XC[0m having group component G and module component k,
  this  function  computes  a  set  of  generators of the ideal of relators in
  H*(G,k), all having degree n or less.
  
  More specifically, the function returns two lists, the first list containing
  the  variables  [9Xz[0m,  [9Xy[0m,  [9Xx[0m, ... corresponding to the generators of H*(G,k) if
  there  are  fewer  than 12 generators and containing the variables [9Xx_1[0m, [9Xx_2[0m,
  [9Xx_3[0m,  ...  otherwise.  The  second  list  is  a  list  of polynomials in the
  variables from the first list.
  
  These  two  lists should be interpreted as follows. A degree [9Xn[0m approximation
  of the cohomology ring H*(G,k) is given by the polynomial ring over k in the
  non-commuting  variables  from  the first list, (having degrees given by the
  list  returned by [9XCohomologyGenerators[0m in section [14X2.3-1[0m ) and subject to the
  relators in the second list. See section [14X2.6[0m for more details still.
  
  For example, consider the following commands.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> C:=CohomologyObject(DihedralGroup(8));[0X
    [4X<object>[0X
    [4Xgap> CohomologyGenerators(C,10);[0X
    [4X[ 1, 1, 2 ][0X
    [4Xgap> CohomologyRelators(C,10);[0X
    [4X[ [ z, y, x ], [ z*y+y^2 ] ][0X
  [4X------------------------------------------------------------------[0X
  
  This  tells us that for G=D_8 and k=GF(p) the cohomology ring H*(G,k) is the
  graded-commutative polynomial ring in the variables z, y, x of degrees 1, 1,
  2, subject to the relation zy+y^2. But since H*(G,k) is commutative, k being
  of  characteristic  2, we have H*(G,k)=k[z,y,x]/(zy+y^2). This result can be
  further improved by taking z=z+y, giving H*(G,k)=k[z,y,x]/(zy).
  
  Observe  that  in  this  case,  we  knew  in advance that there was a set of
  generators  for H*(G,k) all having degree less than 10, and that there was a
  set  of  generators of the ideal of relators all having degree less than 10.
  See see section [14X2.6[0m for details.
  
  While  this isn't likely to occur, we point out that if there are 12 or more
  generators  and  some  of the indeterminates [9Xx_1[0m, [9Xx_2[0m, [9Xx_3[0m, ... have already
  been  named,  say  by  a  previous  call  to  [9XCohomologyRelators[0m, then these
  variables  will  retain  their  old  names.  If this is confusing, you could
  restart [5XGAP[0m and do it again.
  
  Finally,  [9XCohomologyRelators[0m  is  [13Xnot[0m  idempotent for efficiency reasons, so
  sadly,  if  you don't uncover all the relators the first time, you will have
  to start all over from the beginning.
  
  
  [1X2.4 Tests for Completion[0X
  
  A  test  or series of tests for completion of the calculation will hopefully
  be implemented soon. See [CTVZ03] for the details.
  
  
  [1X2.5 Cohomology Rings[0X
  
  Whereas  the operations in sections [14X2.3-1[0m and [14X2.3-2[0m calculate a presentation
  for  the  cohomology  ring, the operation below creates the ring in [5XGAP[0m as a
  structure constant algebra.
  
  See [CTVZ03] for the details of the calculation of cohomology products using
  composition  of  chain  maps.  See  also  the  file [9Xdoc/explanation.*[0m for an
  explanation of the implementation.
  
  [1X2.5-1 CohomologyRing[0m
  
  [2X> CohomologyRing( [0X[3XC, n[0X[2X ) __________________________________________[0Xoperation
  [2X> CohomologyRing( [0X[3XG, n[0X[2X ) __________________________________________[0Xoperation
  [6XReturns:[0X  the cohomology ring of G.
  
  Given  a cohomology object [9XC[0m with group component G and module component the
  trivial  kG-module,  this  function  returns  the degree [9Xn[0m truncation of the
  cohomology  ring  H*(G,k).  See  [14X2.6[0m for what this means exactly. The object
  returned is a structure constant algebra.
  
  Users  interested  only  in working with the cohomology ring of a group as a
  [5XGAP[0m  object, and not in calculating generators, relators, induced maps, etc,
  can use the second invocation of this function, which returns the cohomology
  ring   of   the   group   [9XG[0m  immediately,  throwing  away  all  intermediate
  calculations.
  
  Observe   that  the  object  returned  is  a  degree  [9Xn[0m  truncation  of  the
  infinite-dimensional   cohomology  ring.  A  consequence  of  this  is  that
  multiplying  two elements whose product has degree greater than [9Xn[0m results in
  zero, whether or not the product is really zero.
  
  Observe  also that calling [9XCohomologyRing[0m a second time with a bigger [9Xn[0m does
  [13Xnot[0m  extend the previous ring, but rather, recalculates the entire ring from
  the  beginning.  Extending  the  previous  ring  appears not to be worth the
  effort  for  technical  reasons,  since  almost  everything would need to be
  recalculated again anyway.
  
  Recall that H*(G,k) is a graded algebra, the components being the cohomology
  groups  H^i(G,k).  The  following  functions  were  intended  to be used for
  cohomology  rings,  but  in  principle,  they  work for any graded structure
  constant algebra.
  
  [1X2.5-2 IsHomogeneous[0m
  
  [2X> IsHomogeneous( [0X[3Xe[0X[2X ) ______________________________________________[0Xoperation
  [6XReturns:[0X  [9Xtrue[0m or [9Xfalse[0m.
  
  Given  an  element [9Xe[0m of a cohomology ring H*(G,k), this operation determines
  whether or not [9Xe[0m is homogeneous, that is, whether [9Xe[0m is contained in H^i(G,k)
  for some [9Xi[0m.
  
  [1X2.5-3 Degree[0m
  
  [2X> Degree( [0X[3Xe[0X[2X ) ________________________________________________________[0Xmethod
  [6XReturns:[0X  the degree of [9Xe[0m.
  
  This  function  returns the degree of the possibly non-homogeneous element [9Xe[0m
  of  a  cohomology ring H*(G,k). Specifically, if H*(G,k) = A_0 + A_1 + A_2 +
  ...  where A_i = H^i(G,k) then this function returns the minimum n such that
  [9Xe[0m is in A_0 + A_1 + ... + A_n.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> A:=CohomologyRing(DihedralGroup(8),10);[0X
    [4X<algebra of dimension 66 over GF(2)>[0X
    [4Xgap> b:=Basis(A);[0X
    [4XCanonicalBasis( <algebra of dimension 66 over GF(2)> )[0X
    [4Xgap> x:=b[2]+b[4];[0X
    [4Xv.2+v.4[0X
    [4Xgap> IsHomogeneous(x);[0X
    [4Xfalse[0X
    [4Xgap> Degree(x);[0X
    [4X2 [0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.5-4 LocateGeneratorsInCohomologyRing[0m
  
  [2X> LocateGeneratorsInCohomologyRing( [0X[3XC[0X[2X ) ____________________________[0Xfunction
  [6XReturns:[0X  a list containing the cohomology generators.
  
  Having  already  called  [9XCohomologyRing[0m (see [14X2.5-1[0m), this function returns a
  list  of  elements  of  the cohomology ring which together with the identity
  element generate the cohomology ring.
  
  This  function  is a wrapper for [9XCohomologyGenerators[0m (see [14X2.3-1[0m). It points
  out  which  elements  of  the cohomology ring correspond with the generators
  found by [9XCohomologyGenerators[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> C:=CohomologyObject(SmallGroup(8,4));[0X
    [4X<object>[0X
    [4Xgap> A:=CohomologyRing(C,10);[0X
    [4X<algebra of dimension 17 over GF(2)>[0X
    [4Xgap> L:=LocateGeneratorsInCohomologyRing(C);[0X
    [4X[ v.2, v.3, v.7 ][0X
    [4Xgap> A=Subalgebra(A,Concatenation(L,[One(A)]));[0X
    [4Xtrue[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X2.6 What Happens if [9Xn[1X Isn't Big Enough?[0X
  
  Since   P   is   a   [13Xminimal[0m  projective  resolution,  we  have  H^i(G,k)  =
  Hom_{kG}(P_i,k)  where  P_i  = (kG)^b_i so that H^i(G,k) has a natural basis
  consisting  of the maps sending the element 1_G of the jth direct summand of
  P_i  to 1_k and all other direct summands to 0, for j=1,2,...,b_i, where b_i
  is the kG-rank of P_i.
  
  The  command  [9XCohomologyRing(C,n)[0m  forms the vector space whose basis is the
  concatenation  of the natural bases of H^i(G,k) for i=1,2,...,n and computes
  all  products of basis elements x and y for which deg x+deg y <= n. Thinking
  of  H*(G,k)  in  terms  of  it's  multiplication  table, this means that the
  function computes the upper left-hand corner of the multiplication table. If
  deg  x  + deg y > n, the product xy is taken to be zero. Therefore, the ring
  returned by [9XCohomologyRing[0m is H*(G,k)/J where J is the ideal of all elements
  of degree >n.
  
  The  ring  determined  by  [9XCohomologyGenerators[0m  and  [9XCohomologyRelators[0m  is
  somewhat  different.  [9XCohomologyGenerators[0m  proceeds inductively, taking all
  natural  basis  elements of H^1(G,k) as generators, and for i=2... n, taking
  all   natural   basis  elements  of  H^i(G,k)  which  are  [13Xnot[0m  products  of
  lower-degree  elements  as generators. Therefore, unless you know that there
  is  an  n  for  which there exists a generating set of H*(G,k) consisting of
  elements  of degree n or less, then you are [13Xnot[0m guaranteed that the elements
  returned  by  the  [9XCohomologyGenerators[0m  generate  H*(G,k)  as  a  ring. The
  knowledge of such an n is the subject of section [14X2.4[0m.
  
  Similarly, [9XCohomologyRelators[0m proceeds inductively until degree n, returning
  a  list  of  polynomials which generate the ideal of relators of degree n or
  less. Again, you have to already know how big n should be.
  
  The  result  of the preceding information is that there is a homomorphism k<
  x_1,x_2,...,  x_m  >/ I -> H*(G,k), where k< x_1,x_2,...,x_m > is the graded
  polynomial  ring  over  k  in  the  non-commuting variables x_1,x_2,...,x_m,
  having degrees the numbers in the list returned by [9XCohomologyGenerators[0m, and
  I  is  the ideal in k< x_1,x_2,..., x_m > generated by the elements returned
  by [9XCohomologyRelators(C,n)[0m.
  
  Therefore, if there is a generator of degree greater than n, then f won't be
  surjective.  Similarly, if there is a relator of degree greater than n which
  is  not  a  consequence of lower degree relators, then f won't be injective.
  See  section  [14X2.4[0m for a discussion on how big n needs to be to ensure that f
  will be an isomorphism.
  
  
  [1X2.7 Induced Maps[0X
  
  Let  f:  H  ->  G  be a group homomorphism. Then f induces a homomorphism on
  cohomology H*(G,k) -> H*(H,k) which is returned by the following function.
  
  [1X2.7-1 InducedHomomorphismOnCohomology[0m
  
  [2X> InducedHomomorphismOnCohomology( [0X[3XC, D, f, n[0X[2X ) ____________________[0Xfunction
  [6XReturns:[0X  the induced homomorphism on cohomology.
  
  This  function returns the induced homomorphism H*(G,k) -> H*(H,k) where the
  groups H and G are the components of the cohomology objects [9XC[0m and [9XD[0m and f: H
  ->  G  is  a  group  homomorphism. If the cohomology rings have not yet been
  calculated,  they  will  be computed to degree n, and in this case, they can
  then be accessed by calling [9XCohomologyRing[0m (see [14X2.5-1[0m).
  
  [1X2.7-2 SubgroupInclusion[0m
  
  [2X> SubgroupInclusion( [0X[3XH, G[0X[2X ) ________________________________________[0Xfunction
  [6XReturns:[0X  the inclusion H-> G
  
  This  function  returns the group homomorphism H-> G when H is a subgroup of
  G.    The    returned   map   can   be   used   as   the   [9Xf[0m   argument   of
  [9XInducedHomomorphismOnCohomology[0m,  in  which case the induced homomorphism is
  the restriction map Res: H*(G,k) -> H*(H,k).
  
  The  following  example calculates the homomorphism on cohomology induced by
  the  inclusion of the cyclic group of size 4 into the dihedral group of size
  8.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> G:=DihedralGroup(8);H:=Subgroup(G,[G.2]);[0X
    [4X<pc group of size 8 with 3 generators>[0X
    [4XGroup([ f2 ])[0X
    [4Xgap> C:=CohomologyObject(H);D:=CohomologyObject(G);[0X
    [4X<object>[0X
    [4X<object>[0X
    [4Xgap> i:=SubgroupInclusion(H,G);[0X
    [4X[ f2 ] -> [ f2 ][0X
    [4Xgap> Res:=InducedHomomorphismOnCohomology(C,D,i,10);;[0X
    [4Xgap> A:=CohomologyRing(D,10);[0X
    [4X<algebra of dimension 66 over GF(2)>[0X
    [4Xgap> LocateGeneratorsInCohomologyRing(D);[0X
    [4X[ v.2, v.3, v.6 ][0X
    [4Xgap> A.1^Res; A.2^Res; A.3^Res; A.6^Res;[0X
    [4Xv.1[0X
    [4X0*v.1[0X
    [4Xv.2[0X
    [4Xv.3[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X2.8 Massey Products[0X
  
  See  [Kra66]  for  the  definitions  and  [Bor01]  for  the  details  of the
  calculation  using the Yoneda cocomplex. See also the file [9Xdoc/explanation.*[0m
  for an explanation of the implementation.
  
  [1X2.8-1 MasseyProduct[0m
  
  [2X> MasseyProduct( [0X[3Xx1, x2, ..., xn[0X[2X ) _________________________________[0Xfunction
  [6XReturns:[0X  the Massey product < x1, x2, ... , xn>.
  
  Given  elements x1, x2, ... , xn of the ring returned by [9XCohomologyRing[0m (see
  [14X2.5[0m)  this  function computes the n-fold Massey product < x1, x2, ... , xn >
  provided  that  the  lower-degree  Massey  products < xi, x{i+1}, ... , xj >
  vanish for all 1 <= i < j <= n and returns [9Xfail[0m otherwise.
  
  As an example, recall that the cohomology rings of the cyclic groups C_3 and
  C_9  of sizes 3 and 9 over k=GF(3) are both given by k< z,y >/(z^2), so they
  are  isomorphic  as rings. However, the following example shows that < z, z,
  z > is non-zero in H*(C_3,k) but is zero in H*(C_9,k).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> A:=CohomologyRing(CyclicGroup(3),10);[0X
    [4X<algebra of dimension 11 over GF(3)>[0X
    [4Xgap> z:=Basis(A)[2];[0X
    [4Xv.2[0X
    [4Xgap> MasseyProduct(z,z);[0X
    [4X0*v.1[0X
    [4Xgap> MasseyProduct(z,z,z);[0X
    [4Xv.3[0X
    [4Xgap> A:=CohomologyRing(CyclicGroup(9),10);[0X
    [4X<algebra of dimension 11 over GF(3)>[0X
    [4Xgap> z:=Basis(A)[2];[0X
    [4Xv.2[0X
    [4Xgap> MasseyProduct(z,z);[0X
    [4X0*v.1[0X
    [4Xgap> MasseyProduct(z,z,z);[0X
    [4X0*v.1[0X
    [4Xgap> MasseyProduct(z,z,z,z,z,z,z,z,z);[0X
    [4Xv.3[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
