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 \((G_x = F/R_x | x \in ℕ)\), 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 \(\langle c_1, \ldots, c_m\rangle\) 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 \(M(G) \cong C_{d_1} \times \ldots \times C_{d_k}\).
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 \(R \cong GF(p)\) if the prime \(p\) is given or \(R \cong ℤ\) otherwise. The action of \(G\) on \(R\) is taken to be trivial. The function returns a list of integers \([a_1,\ldots, a_k]\) where the cohomology group is isomorphic to \(C_{a_1} \times \ldots \times C_{a_k}\) 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 \(R \cong GF(p)\) if the prime \(p\) is given or \(R \cong ℤ\) otherwise. The action of \(G\) on \(R\) is taken to be trivial. The function returns a list of integers \([a_1,\ldots, a_k]\) where the cohomology group is isomorphic to \(C_{a_1} \times \ldots \times C_{a_k}\) 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 \(R \cong GF(p)\) if the prime \(p\) is given or \(R \cong ℤ\) otherwise. The action of \(G\) on \(R\) is taken to be trivial. The function returns a list of integers \([a_1,\ldots, a_k]\) where the cohomology group is isomorphic to \(C_{a_1} \times \ldots \times C_{a_k}\) 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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