In this chapter we describe how the consistent pp-presentations of infinite coclass sequences can be used to compute a pp-presentation for the corresponding Schur extensions (see [EF11]).
For a group G = F/R the Schur extension H is defined as H = F/[F,R] (see [EN08]).
So for a parameter x that can take values in the positive integers, let (G_x = F/R_x | x ∈ ℕ), for ℕ the positive integers, describe an infinite coclass sequence of finite p-groups G_x of coclass r. Then for each value for the parameter x, the group G_x has a consistent polycyclic presentation with generators g_1, ..., g_n, t_1, ..., t_d and relations
\begin{array}{rl}& g_i^p = rel[i][i],\\ & t_i^{expo} = rel[n+i][n+i],\\ & g_i^{g_j} = rel[j][i],\\ & t_i^{g_j} = rel[j][n+i],\\ & t_i^{t_j} = 1.\end{array}
Then we compute a consistent pp-presentation of the corresponding Schur extensions with generators g_1, ..., g_n, t_1, ..., t_d, c_1, ..., c_m and relations
\begin{array}{rl}& g_i^p=rel[i][i],\\ & t_i^{expo} = rel[n+i][n+i],\\ & c_i^{expo\_vec[i]} = rel[n+d+i][n+d+i],\\ & g_i^{g_j} = rel[j][i], \\ & t_i^{g_j} = rel[j][n+i],\\ & t_i^{t_j} = rel[n+j][n+i],\\ & c_i^{g_j} = 1, \\ & c_i^{t_j} = 1, \\ & c_i^{c_j} = 1,\end{array}
where the t_i's commute modulo ⟨ c_1, ..., c_m⟩ and the c_i's are central.
‣ SchurExtParPres( G ) | ( operation ) |
computes the Schur extensions corresponding to the p-power-poly-pcp-groups G and returns them as p-power-poly-pcp-groups.
‣ SchurExtParPres( ParPres ) | ( operation ) |
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 such that it can be used as input for PPPPcpGroups (3.2-1).
gap> SchurExtParPres( ParPresGlobalVar_2_1[1] ); rec( cc := fail, d := 1, expo := [ 2, [ 0, 2 ], true, [ 1, 1 ] ], expo_vec := [ [ 2, [ 2 ] ], [ 2, [ ], true, [ infinity, infinity ] ], [ 2, [ ], true, [ infinity, infinity ] ], [ 2, [ ], true, [ infinity, infinity ] ] ], m := 4, n := 2, name := "SchurExt_D", prime := 2, rel := [ [ [ [ 7, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ] ], [ [ [ 2, 1 ], [ 3, [ 2, [ -1, 2 ], true ] ], [ 6, [ 2, [ 1, -2 ], true, [ 0, 0 ] ] ] ], [ [ 3, [ 2, [ 1 ], true, [ 0, 0 ] ] ], [ 5, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ] ], [ [ [ 3, [ 2, [ -1, 2 ], true ] ], [ 4, [ 2, [ 1 ], true, [ 0, 0 ] ] ], [ 6, [ 2, [ 2, -2 ], true, [ 0, 1 ] ] ] ], [ [ 3, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 4, [ 2, [ 1 ], true, [ 0, 0 ] ] ], [ 6, [ 2, [ 0, 2 ] ] ] ] ], [ [ [ 4, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 4, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 4, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 4, [ 2, [ ], true, [ infinity, infinity ] ] ] ] ], [ [ [ 5, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 5, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 5, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 5, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 5, [ 2, [ ], true, [ infinity, infinity ] ] ] ] ], [ [ [ 6, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 6, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 6, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 6, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 6, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 6, [ 2, [ ], true, [ infinity, infinity ] ] ] ] ], [ [ [ 7, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 7, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 7, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 7, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 7, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 7, [ 2, [ 1 ], true, [ 0, 0 ] ] ] ], [ [ 7, [ 2, [ ], true, [ infinity, infinity ] ] ] ] ] ] )
‣ AbelianInvariantsMultiplier( G ) | ( function ) |
computes the abelian invariants of the Schur multiplicators M(G) of the p-power-poly-pcp-groups G. The output is a list [d_1, ..., d_k] consisting of elements d_i, depending on the underlying parameter, such that M(G) ≅ C_d_1 × ... × C_d_k.
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] ); < P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] > gap> AbelianInvariantsMultiplier( G ); [ [ 2, [ 2 ] ] ]
‣ SchurMultiplicatorPPPPcps( G ) | ( operation ) |
computes the Schur multiplicators of the p-power-poly-pcp-groups G and then returns them as p-power-poly-pcp-groups (see PPPPcpGroups (3.2-1)).
gap> G := PPPPcpGroups( ParPresGlobalVar_3_1[1] ); < P-Power-Poly-pcp-groups with 5 generators of relative orders [ 3,3,3,3*3^x, 3*3^x ] > gap> SchurMultiplicatorPPPPcps( G ); < P-Power-Poly-pcp-groups with 2 generators of relative orders [ 3,9*3^x ] >
‣ AbelianInvariants( G ) | ( function ) |
computes the abelian invariants of the p-power-poly-pcp-groups G and returns them as a list of lists 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, [ 1 ], true, [ 0, 0 ] ], [ 2, [ 1 ], true, [ 0, 0 ] ] ]
‣ ZeroCohomologyPPPPcps( G[, p] ) | ( operation ) |
computes the zeroth cohomology groups H^0(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 [a_1,..., a_k] where the cohomology group is isomorphic to C_a_1 × ... × C_a_k with C_i a cyclic group of order i (for i > 0) and C_0 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> ZeroCohomologyPPPPcps( G, 2 ); [ 2 ]
‣ FirstCohomologyPPPPcps( G[, p] ) | ( operation ) |
computes the first cohomology groups H^1(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 [a_1,..., a_k] where the cohomology group is isomorphic to C_a_1 × ... × C_a_k with C_i a cyclic group of order i (for i > 0) and C_0 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 ); [ ]
‣ SecondCohomologyPPPPcps( G[, p] ) | ( operation ) |
computes the second cohomology groups H^2(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 [a_1,..., a_k] where the cohomology group is isomorphic to C_a_1 × ... × C_a_k with C_i a cyclic group of order i (for i > 0) and C_0 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 ]
The following info classes are available:
‣ InfoConsistencyRelPPowerPoly | ( info class ) |
level 1shows which consistency relations are computed and gives the result;
The default value is 0.
‣ InfoCollectingPPowerPoly | ( info class ) |
level 1shows what is done during collecting;
The default value is 0.
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