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Quelle schurextensions.tex Sprache: Latech |
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%% %W schurextension.tex GAP documentation Dörte Feichtenschlager %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Schur extensions for p-power-poly-pcp-groups} 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 \cite{EF11}). For a group $G = F/R$ the Schur extension $H$ is defined as $H = F/[F,R]$ (see \cite{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 %display{nontext} $$ \eqalign{ &\, g_i^p = rel[i][i],\cr &\, t_i^{expo} = rel[n+i][n+i],\cr &\, g_i^{g_j} = rel[j][i],\cr &\, t_i^{g_j} = rel[j][n+i],\cr &\, t_i^{t_j} = 1. } $$ %display{text} %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. %enddisplay Then we compute a consistent pp-presentation of the corresponding Schur extensions of with generators $g_1, ..., g_n, t_1, ..., t_d, c_1, ... c_m$ and relations %display{nontext} $$ \eqalign{ &\, g_i^p=rel[i][i],\cr &\, t_i^{expo} = rel[n+i][n+i],\cr &\, c_i^{expo\_vec[i]} = rel[n+d+i,n+d+i],\cr &\, g_i^{g_j} = rel[j][i], \cr &\, t_i^{g_j} = rel[j][n+i],\cr &\, t_i^{t_j} = rel[n+j][n+i],\cr &\, c_i^{g_j} = 1, \cr &\, c_i^{t_j} = 1, \cr &\, c_i^{c_j} = 1. } $$ %display{text} %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. %enddisplay where the $t_i$'s commute modulo $< c_1, ..., c_m>$ and the $c_i$'s are central. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Computing Schur extensions} \>SchurExtParPres( <G> ) computes the Schur extensions corresponding to the <p>-power-poly-pcp-groups <G> and returns them as <p>-power-poly-pcp-groups. \>SchurExtParPres( <ParPres> ) F 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 %display{tex} {\tt PPPPcpGroups}, %enddisplay "PPPPcpGroups". \beginexample 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" ) \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Computing other invariants from Schur extensions} \>AbelianInvariantsMultiplier( <G> ) F 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 elements $d_i$, depending on the underlying parameter, such that $M(G) \cong C_{d_1} \times \ldots \times C_{d_k}$. \beginexample 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 ] \endexample \>SchurMultiplicatorPPPPcps( <G> )!{for p-power-poly-pcp-groups} F computes the Schur multiplicators of the <p>-power-poly-pcp-groups <G> and then returns the corresponding %display{tex} {\tt PPPPcpGroups}, %enddisplay "PPPPcpGroups". \beginexample gap> G := PPPPcpGroups( 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> SchurMultiplicatorPPPPcps( G ); < P-Power-Poly-pcp-groups with 2 generators of relative orders [ 3,9*3^x ] > \endexample \>AbelianInvariants( <G> )!{for p-power-poly-pcp-groups} F computes the abelian invariants of the <p>-power-poly-pcp-groups <G> and returns them as a list of list describing the parametrised elements. \beginexample 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 ] \endexample \>ZeroCohomologyPPPPcps( <G>[, <p>] ) F computes the zero-th-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 $\Z$. \beginexample 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 ] \endexample \>FirstCohomologyPPPPcps( <G>[, <p>] ) F 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 $\Z$. \beginexample 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 ); [ ] \endexample \>SecondCohomologyPPPPcps( <G>[, <p>] ) F 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 $\Z$. \beginexample 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 ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Info classes for the computation of the Schur extension} The following info classes are available \>`InfoConsistencyRelPPowerPoly' V \beginitems `level 1' & shows which consistency relations are computed and gives the result; \enditems the default value is 0. \>`InfoCollectingPPowerPoly' V \beginitems `level 1' & shows what is done during collecting; \enditems the default value is 0. ¤ Dauer der Verarbeitung: 0.24 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28