5 Schur extensions for p-power-poly-pcp-groups

computes the Schur extensions corresponding to the p-power-poly-pcp-groups G and returns them as p-power-poly-pcp-groups.

computes a consistent pp-presentation of Schur extensions of the groups defined by the record ParPres which describes p-power-poly-pcp-groups. The output is a record rec(rel, expo, n, d, m, prime, cc, expo_vec, name), which describes the Schur extensions as p-power-poly-pcp-groups; it is encoded in a form that it can be used as input for PPPPcpGroups.

gap> SchurExtParPres( ParPresGlobalVar_2_1[1] );
rec( prime := 2, 
  rel := [ [ [ [ 7, 1 ] ] ], [ [ [ 2, 1 ], [ 3, -1+2*2^x ], [ 6, 1-2*2^x ] ], 
          [ [ 3, 1 ], [ 5, 1 ] ] ], 
      [ [ [ 3, -1+2*2^x ], [ 4, 1 ], [ 6, 2-2*2^x ] ], [ [ 3, 1 ] ], 
          [ [ 4, 1 ], [ 6, 2*2^x ] ] ], 
      [ [ [ 4, 1 ] ], [ [ 4, 1 ] ], [ [ 4, 1 ] ], [ [ 4, 0 ] ] ], 
      [ [ [ 5, 1 ] ], [ [ 5, 1 ] ], [ [ 5, 1 ] ], [ [ 5, 1 ] ], [ [ 5, 0 ] ] ]
        , 
      [ [ [ 6, 1 ] ], [ [ 6, 1 ] ], [ [ 6, 1 ] ], [ [ 6, 1 ] ], [ [ 6, 1 ] ], 
          [ [ 6, 0 ] ] ], 
      [ [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 1 ] ], 
          [ [ 7, 1 ] ], [ [ 7, 0 ] ] ] ], n := 2, d := 1, m := 4, 
  expo := 2*2^x, expo_vec := [ 2, 0, 0, 0 ], cc := fail, name := "SchurExt_D" 
 )

5.2 Computing other invariants from Schur extensions

computes the abalian invariants of the Schur multiplicators M(G) of the p-power-poly-pcp-groups G. The output is a list [d₁, ·.·, d_k] consisting elements d_i, depending on the underlying parameter, such that M(G) ≅ C_d1 ×…×C_dk.

gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
SchurMultiplicatorsStructurePPPPcps( G );
[ 2 ]

computes the Schur multiplicators of the p-power-poly-pcp-groups G and then returns the corresponding PPPPcpGroups.

gap> G := PPPPcpGroup( ParPresGlobalVar_3_1[1] );
< P-Power-Poly pcp-group with 5 generators of relative orders [ 3,3,3,3*3^x,3*3^x ] >
gap> SchurMultiplicator( G );
< P-Power-Poly pcp-groups with 2 generators of relative orders [ 3,9*3^x ] >

computes the abelian invariants of the p-power-poly-pcp-groups G and returns them as a list of list describing the parametrised elements.

gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> AbelianInvariants( G );
[ 2, 2 ]

computes the zero-th-cohomology groups H⁰(G,R) of the p-power-poly-pcp-groups G with coefficients in R, where R ≅ GF(p) if the prime p is given or R ≅ Z otherwise. The action of G on R is taken to be trivial. The function returns a list of integers [a₁,…, a_k] where the cohomology group is isomorphic to C_a1 ×…×C_ak with C_i a cyclic group of order i (for i > 0) and C₀ is interpreted as Z.

gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> ZeroCohomologyPPPPcp( G, 2 );
[ 2 ]

computes the first-cohomology groups H¹(G,R) of the p-power-poly-pcp-groups G with coefficients in R, where R ≅ GF(p) if the prime p is given or R ≅ Z otherwise. The action of G on R is taken to be trivial. The function returns a list of integers [a₁,…, a_k] where the cohomology group is isomorphic to C_a1 ×…×C_ak with C_i a cyclic group of order i (for i > 0) and C₀ is interpreted as Z.

gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> FirstCohomologyPPPPcps( G );
[  ]

computes the second-cohomology groups H²(G,R) of the p-power-poly-pcp-groups G with coefficients in R, where R ≅ GF(p) if the prime p is given or R ≅ Z otherwise. The action of G on R is taken to be trivial. The function returns a list of integers [a₁,…, a_k] where the cohomology group is isomorphic to C_a1 ×…×C_ak with C_i a cyclic group of order i (for i > 0) and C₀ is interpreted as Z.

gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> SecondCohomologyPPPPcps( G, 2 );
[ 2, 2, 2 ]

5.3 Info classes for the computation of the Schur extension

The following info classes are available

level 1

shows which consistency relations are computed and gives the result;

the default value is 0.

level 1

shows what is done during collecting;

the default value is 0.

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