2. Usage
  
  Unless  otherwise  specified,  all  the  functions described below taking an
  argument  n  do whatever the manual says they do up to homological degree n.
  These  functions  are idempotent in the sense that called a second time with
  the  same  argument  n,  they  do  nothing, but called with a bigger n, they
  continue computing from where the previous calculations left off.
  
  
  2.1 Cohomology Objects
  
  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 CObject, which is created with the following command.
  
  2.1-1 CohomologyObject
  
  > CohomologyObject( G, M ) ________________________________________operation
  > CohomologyObject( G ) ___________________________________________operation
  Returns:  a CObject.
  
  This  function  creates  a  CObject  having components the p-group G and the
  MeatAxe  module M, which should be a kG-module with G the same group and k a
  field  of  characteristic  p.  Note that MeatAxe modules record k but not G,
  which is why this operation requires the user to specify G but not k.
  
  Fortunately,  most  users don't need to know anything about MeatAxe modules,
  being  interested  primarily in the case where k=GF(p), and M=k, the trivial
  kG-module.  In  this  situation,  the second invocation creates a cohomology
  object  having  components  the  p-group G and the trivial MeatAxe kG-module
  k=GF(p).
  
  We  emphasize  that  in  the  first  invocation,  k  can  be  any  field  of
  characteristic  p  and  M  can  be  any  MeatAxe  module  over  kG, and that
  ProjectiveResolution  works  when M is an arbitrary MeatAxe module, but that
  all  the functions dealing with the ring-structure of H*(G,k) require that M
  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.
  
  
  2.2 Minimal Projective Resolutions
  
  Given  a  p-group  G,  a field k of characteristic p, and a kG-module M, the
  function  below  computes  the  first  n  terms  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 Betti
  numbers  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).
  
  2.2-1 ProjectiveResolution
  
  > ProjectiveResolution( C, n ) ____________________________________operation
  Returns:  a list containing the Betti numbers b_0, b_1,..., b_n.
  
  Given  a  cohomology  object  C  having  components  G  and M, this function
  computes  the first n+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.
  
  2.2-2 BoundaryMap
  
  > BoundaryMap( C, n ) _____________________________________________operation
  Returns:  the nth boundary map.
  
  Given   the  cohomology  object  C,  this  function  computes  a  projective
  resolution  to  degree n if it hasn't been computed already, and returns the
  nth boundary map.
  
  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 doc/example.* for an example of the usage and interpretation of
  the result of this function.
  
  
  2.3 Cohomology Generators and Relators
  
  See [CTVZ03] for the details of the calculation of cohomology products using
  composition  of  chain  maps.  See  also  the  file doc/explanation.* for an
  explanation of the implementation.
  
  2.3-1 CohomologyGenerators
  
  > CohomologyGenerators( C, n ) ____________________________________operation
  Returns:  a  list  containing  the  degrees  of  the  elements  of  a set of
            generators of the cohomology ring.
  
  Given  a  cohomology  object C having group component G and module component
  the trivial kG-module, this function computes a set of generators of H*(G,k)
  having  degree  n or less, and stores them in C. The function returns a list
  of the degrees of these generators.
  
  The  actual  cohomology  generators are represented by maps P_n -> k and are
  stored in C as matrices. Only their degrees are returned.
  
  2.3-2 CohomologyRelators
  
  > CohomologyRelators( C, n ) ______________________________________operation
  Returns:  a list of generators and a list of relators.
  
  Given a cohomology object C 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 multidegree n or less.
  
  More specifically, the function returns two lists, the first list containing
  the  variables  z,  y,  x, ... corresponding to the generators of H*(G,k) if
  there  are  fewer  than 12 generators and containing the variables x_1, x_2,
  x_3,  ...  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 n 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 CohomologyGenerators in section 2.3-1 ) and subject to the
  relators in the second list. See section 2.6 for more details still.
  
  For example, consider the following commands.
  
  ---------------------------  Example  ----------------------------
    
    gap> C:=CohomologyObject(DihedralGroup(8));
    <object>
    gap> CohomologyGenerators(C,10);
    [ 1, 1, 2 ]
    gap> CohomologyRelators(C,10);
    [ [ z, y, x ], [ z*y+y^2 ] ]
    
  ------------------------------------------------------------------
  
  This   tells  us  that  for  G=D_8,  the  cohomology  ring  H*(G,k)  is  the
  graded-commutative  polynomial  ring in the variables z, y, and x of degrees
  1,  1,  and  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 multidegree less than
  10. See see section 2.6 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 x_1, x_2, x_3, ... have already
  been  named,  say  by  a  previous  call  to  CohomologyRelators, then these
  variables  will  retain  their  old  names.  If this is confusing, you could
  restart GAP and do it again.
  
  Finally,  CohomologyRelators  is  not  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.
  
  
  2.4 Tests for Completion
  
  A  test  or series of tests for completion of the calculation will hopefully
  be implemented soon. See [CTVZ03] for the details.
  
  
  2.5 Cohomology Rings
  
  Whereas  the operations in sections 2.3-1 and 2.3-2 calculate a presentation
  for  the  cohomology  ring, the operation below creates the ring in GAP 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 doc/explanation.* for an
  explanation of the implementation.
  
  2.5-1 CohomologyRing
  
  > CohomologyRing( C, n ) __________________________________________operation
  > CohomologyRing( G, n ) __________________________________________operation
  Returns:  the cohomology ring of G.
  
  Given  a cohomology object C with group component G and module component the
  trivial  kG-module,  this  function  returns  the degree n truncation of the
  cohomology  ring  H*(G,k).  See  2.6 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
  GAP  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   G  immediately,  throwing  away  all  intermediate
  calculations.
  
  Observe   that  the  object  returned  is  a  degree  n  truncation  of  the
  infinite-dimensional   cohomology  ring.  A  consequence  of  this  is  that
  multiplying  two elements whose product has degree greater than n results in
  zero, whether or not the product is really zero.
  
  Observe  also that calling CohomologyRing a second time with a bigger n does
  not  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 structure constant 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.
  
  2.5-2 IsHomogeneous
  
  > IsHomogeneous( e ) ______________________________________________operation
  Returns:  true or false.
  
  Given  an  element e of a cohomology ring H*(G,k), this operation determines
  whether or not e is homogeneous, that is, whether e is contained in H^i(G,k)
  for some i.
  
  2.5-3 Degree
  
  > Degree( e ) ________________________________________________________method
  Returns:  the degree of e.
  
  This  function  returns the degree of the possibly non-homogeneous element e
  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
  e is in A_0 + A_1 + ... + A_n.
  
  ---------------------------  Example  ----------------------------
    gap> A:=CohomologyRing(DihedralGroup(8),10);
    <algebra of dimension 66 over GF(2)>
    gap> b:=Basis(A);
    CanonicalBasis( <algebra of dimension 66 over GF(2)> )
    gap> x:=b[2]+b[4];
    v.2+v.4
    gap> IsHomogeneous(x);
    false
    gap> Degree(x);
    2 
  ------------------------------------------------------------------
  
  2.5-4 LocateGeneratorsInCohomologyRing
  
  > LocateGeneratorsInCohomologyRing( C ) ____________________________function
  Returns:  a list containing the cohomology generators.
  
  Having  already  called  CohomologyRing (see 2.5-1), 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 CohomologyGenerators (see 2.3-1). It points
  out  which  elements  of  the cohomology ring correspond with the generators
  found by CohomologyGenerators.
  
  ---------------------------  Example  ----------------------------
    gap> C:=CohomologyObject(SmallGroup(8,4));
    <object>
    gap> A:=CohomologyRing(C,10);
    <algebra of dimension 17 over GF(2)>
    gap> L:=LocateGeneratorsInCohomologyRing(C);
    [ v.2, v.3, v.7 ]
    gap> A=Subalgebra(A,Concatenation(L,[One(A)]));
    true
  ------------------------------------------------------------------
  
  
  2.6 What Happens if n Isn't Big Enough?
  
  Since   P   is   a   minimal  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  CohomologyRing(C,n)  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 CohomologyRing is H*(G,k)/J where J is the ideal of all elements
  of degree >n.
  
  The  ring  determined  by  CohomologyGenerators  and  CohomologyRelators  is
  somewhat  different.  CohomologyGenerators  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  not  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 not guaranteed that the elements
  returned  by  the  CohomologyGenerators  generate  H*(G,k)  as  a  ring. The
  knowledge of such an n is the subject of section 2.4.
  
  Similarly, CohomologyRelators proceeds inductively until degree n, returning
  a  list of polynomials which generate the ideal of relators of multidegree 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 CohomologyGenerators, and
  I  is  the ideal in k< x_1,x_2,..., x_m > generated by the elements returned
  by CohomologyRelators(C,n).
  
  Therefore, if there is a generator of degree greater than n, then f won't be
  surjective.  Similarly,  if there is a relator of multidegree greater than n
  which  is  not  a  consequence  of  lower  degree  relators, then f won't be
  injective.  See  section  2.4  for  a discussion on how big n needs to be to
  ensure that f be an isomorphism.
  
  
  2.7 Induced Maps
  
  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.
  
  2.7-1 InducedHomomorphismOnCohomology
  
  > InducedHomomorphismOnCohomology( C, D, f, n ) ____________________function
  Returns:  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 C and D 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 CohomologyRing (see 2.5-1).
  
  2.7-2 Inclusion
  
  > Inclusion( H, G ) ________________________________________________function
  Returns:  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   f   argument   of
  InducedHomomorphismOnCohomology,  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.
  
  ---------------------------  Example  ----------------------------
    
    gap> G:=DihedralGroup(8);H:=Subgroup(G,[G.2]);
    <pc group of size 8 with 3 generators>
    Group([ f2 ])
    gap> C:=CohomologyObject(H);D:=CohomologyObject(G);
    <object>
    <object>
    gap> i:=Inclusion(H,G);
    [ f2 ] -> [ f2 ]
    gap> Res:=InducedHomomorphismOnCohomology(C,D,i,10);;
    gap> A:=CohomologyRing(D,10);
    <algebra of dimension 66 over GF(2)>
    gap> LocateGeneratorsInCohomologyRing(D);
    [ v.2, v.3, v.6 ]
    gap> A.1^Res; A.2^Res; A.3^Res; A.6^Res;
    v.1
    0*v.1
    v.2
    v.3
    
  ------------------------------------------------------------------
  
  
  2.8 Massey Products
  
  See  [Kra66]  for  the  definitions  and  [Bor01]  for  the  details  of the
  calculation  using the Yoneda cocomplex. See also the file doc/explanation.*
  for an explanation of the implementation.
  
  2.8-1 MasseyProduct
  
  > MasseyProduct( x1, x2, ..., xn ) _________________________________function
  Returns:  the Massey product < x1, x2, ... , xn>.
  
  Given  elements x1, x2, ... , xn of the ring returned by CohomologyRing (see
  2.5),  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 fail 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).
  
  ---------------------------  Example  ----------------------------
    
    gap> A:=CohomologyRing(CyclicGroup(3),10);
    <algebra of dimension 11 over GF(3)>
    gap> z:=Basis(A)[2];
    v.2
    gap> MasseyProduct(z,z);
    0*v.1
    gap> MasseyProduct(z,z,z);
    v.3
    gap> A:=CohomologyRing(CyclicGroup(9),10);
    <algebra of dimension 11 over GF(3)>
    gap> z:=Basis(A)[2];
    v.2
    gap> MasseyProduct(z,z);
    0*v.1
    gap> MasseyProduct(z,z,z);
    0*v.1
    gap> MasseyProduct(z,z,z,z,z,z,z,z,z);
    v.3
    
  ------------------------------------------------------------------