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<html><head><title>[SymbCompCC] 5 Schur extensions for p-power-poly-pcp-groups</title></head>
<body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href = "theindex.htm">In <h1>5 Schur extensions for p-power-poly-pcp-groups</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP005.htm#SECT001">Computing Schur extensions</a> <li> <A HREF="CHAP005.htm#SECT002">Computing other invariants from Schur extensions</a> <li> <A HREF="CHAP005.htm#SECT003">Info classes for the computation of the Schur extension</a> </ol><p> <p> 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 <a href="biblio.htm#EF11"><cite>EF11</cite></a>). <p> For a group <i>G</i> = <i>F</i>/<i>R</i> the Schur extension <i>H</i> is defined as <i>H</i> = <i>F</i>/[<i>F</i>,<i>R</i>] (see <a href="biblio.htm#EN08"><cite>EN08</cite></a>). <p> So for a parameter <var>x</var> that can take values in the positive integers, let (<i>G</i><sub><i>x</i></sub> = <i>F</i>/<i>R</i><sub><i>x</i></sub> | <i>x</i> ∈ <b>N</b>), for <b>N</b> the positive integers, descri infinite coclass sequence of finite <i>p</i>-groups <i>G</i><sub><i>X</i></sub> of coclass <i>r</i>. Then for each value for the parameter <var>x</var>, the group <i>G</i><sub><i>x</i></sub> has a consistent polycyclic presentation with generators <i>g</i><sub>1</sub>, ·.·, <i>g</i><sub><i>n</i></sub>, <i>t</i><sub>1</sub>, ·.·, <i>t</i><sub><i> <p> <br clear="all" /><table border="0" width="100%"><tr><td> <table border="0" cellspacing="0" cellp <p> Then we compute a consistent pp-presentation of the corresponding Schur extensions of with generators <i>g</i><sub>1</sub>, ·.·, <i>g</i><sub><i>n</i></sub>, <i>t</i><sub>1</sub>, ·.·, <i>t</i><sub><i> relations <p> <br clear="all" /><table border="0" width="100%"><tr><td> <table border="0" cellspacing="0" cellp <p> where the <i>t</i><sub><i>i</i></sub>'s commute modulo c1, ·.·, cm and the ci's are central. <p> <p> <h2><a name="SECT001">5.1 Computing Schur extensions</a></h2> <p><p> <a name = "SSEC001.2"></a> <li><code>SchurExtParPres( </code><var>G</var><code> )</code> <p> computes the Schur extensions corresponding to the <var>p</var>-power-poly-pcp-groups <var>G</var> and returns them as <var>p</var>-power-poly-pcp-groups. <p> <li><code>SchurExtParPres( </code><var>ParPres</var><code> ) F</code> <p> computes a consistent pp-presentation of Schur extensions of the groups defined by the record <var>ParPres</var> which describes <var>p</var>-power-poly-pcp-groups. The output is a record <var>rec</var>(<var>rel</var>, <var>expo</var>, <var>n</var>, <var>d</var>, <var>m</var>, <var>prime</var>, <var>cc</ which describes the Schur extensions as <var>p</var>-power-poly-pcp-groups; it is encoded in a form that it can be used as input for <a href="CHAP003.htm#SSEC002.1">PPPPcpGroups</a>. <p> <pre> 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" ) </pre> <p> <p> <h2><a name="SECT002">5.2 Computing other invariants from Schur extensions</a></h2> <p><p> <a name = "SSEC002.1"></a> <li><code>AbelianInvariantsMultiplier( </code><var>G</var><code> ) F</code> <p> computes the abelian invariants of the Schur multiplicators <var>M(G)</var> of the <var>p</var>-power-poly-pcp-groups <var>G</var>. The output is a list [<i>d</i><sub>1</sub>, ·.·, <i>d</i><sub><i>k< consisting elements <i>d</i><sub><i>i</i></sub>, depending on the underlying parameter, such that <i>M</i>(<i>G</i>) ≅ <i>C</i><sub><i>d</i><sub>1</sub></sub> ×…×<i>C</i><sub><i>d</i><sub><i>k</i></sub></sub>. <p> <pre> 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 ] </pre> <p> <a name = "SSEC002.2"></a> <li><code>SchurMultiplicatorPPPPcps( </code><var>G</var><code> ) F</code> <p> computes the Schur multiplicators of the <var>p</var>-power-poly-pcp-groups <var>G</var> and then returns the corresponding <a href="CHAP003.htm#SSEC002.1">PPPPcpGroups</a>. <p> <pre> 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 ] > </pre> <p> <a name = "SSEC002.3"></a> <li><code>AbelianInvariants( </code><var>G</var><code> ) F</code> <p> computes the abelian invariants of the <var>p</var>-power-poly-pcp-groups <var>G</var> and return them as a list of list describing the parametrised elements. <p> <pre> 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 ] </pre> <p> <a name = "SSEC002.4"></a> <li><code>ZeroCohomologyPPPPcps( </code><var>G</var><code>[, </code><var>p</var><code>] ) F</code> <p> computes the zero-th-cohomology groups <i>H</i><sup>0</sup>(<i>G</i>,<i>R</i>) of the <var>p</var>-power-poly-pcp-groups <var>G</var> with coefficients in <i>R</i>, where <i>R</i> ≅ <i>GF</i>(<i>p</i> the prime <i>p</i> is given or <i>R</i> ≅ <b>Z</b> otherwise. The action of <i>G</i> on <i>R</i> is taken to be trivial. The function returns a list of integers [<i>a</i><sub>1</sub>,…, <i>a</i><sub><i>k</i></sub> is interpreted as <b>Z</b>. <p> <pre> 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 ] </pre> <p> <a name = "SSEC002.5"></a> <li><code>FirstCohomologyPPPPcps( </code><var>G</var><code>[, </code><var>p</var><code>] ) F</code> <p> computes the first-cohomology groups <i>H</i><sup>1</sup>(<i>G</i>,<i>R</i>) of the <var>p</var>-power-poly-pcp-groups <var>G</var> with coefficients in <i>R</i>, where <i>R</i> ≅ <i>GF</i>(<i>p</i> the prime <i>p</i> is given or <i>R</i> ≅ <b>Z</b> otherwise. The action of <i>G</i> on <i>R</i> is taken to be trivial. The function returns a list of integers [<i>a</i><sub>1</sub>,…, <i>a</i><sub><i>k</i></sub> is interpreted as <b>Z</b>. <p> <pre> 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 ); [ ] </pre> <p> <a name = "SSEC002.6"></a> <li><code>SecondCohomologyPPPPcps( </code><var>G</var><code>[, </code><var>p</var><code>] ) F</code> <p> computes the second-cohomology groups <i>H</i><sup>2</sup>(<i>G</i>,<i>R</i>) of the <var>p</var>-power-poly-pcp-groups <var>G</var> with coefficients in <i>R</i>, where <i>R</i> ≅ <i>GF</i>(<i>p</i> the prime <i>p</i> is given or <i>R</i> ≅ <b>Z</b> otherwise. The action of <i>G</i> on <i>R</i> is taken to be trivial. The function returns a list of integers [<i>a</i><sub>1</sub>,…, <i>a</i><sub><i>k</i></sub> is interpreted as <b>Z</b>. <p> <pre> 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 ] </pre> <p> <p> <h2><a name="SECT003">5.3 Info classes for the computation of the Schur extension</a></h2> <p><p> The following info classes are available <p> <a name = "SSEC003.1"></a> <li><code>InfoConsistencyRelPPowerPoly V</code> <p> <p> <dl compact> <dt><code>level 1</code> <dd> shows which consistency relations are computed and gives the result; </dl> <p> the default value is 0. <p> <a name = "SSEC003.2"></a> <li><code>InfoCollectingPPowerPoly V</code> <p> <p> <dl compact> <dt><code>level 1</code> <dd> shows what is done during collecting; </dl> <p> the default value is 0. <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href = "theindex.htm">In <P> <address>SymbCompCC manual<br>February 2022 </address></body></html> ¤ Dauer der Verarbeitung: 0.16 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28