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<html><head><title>[SymbCompCC] 3 p-power-poly-pcp-groups</title></head>
<body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Nex <h1>3 p-power-poly-pcp-groups</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP003.htm#SECT001">Example</a> <li> <A HREF="CHAP003.htm#SECT002">Obtaining p-power-poly-pcp-groups</a> <li> <A HREF="CHAP003.htm#SECT003">Operations and functions for p-power-poly-pcp-group elem <li> <A HREF="CHAP003.htm#SECT004">Operations and functions for p-power-poly-pcp-groups</a> <li> <A HREF="CHAP003.htm#SECT005">Info classes for the p-power-poly-pcp-groups</a> <li> <A HREF="CHAP003.htm#SECT006">Global variables for the p-power-poly-pcp-groups</a> </ol><p> <p> Eick and Leedham-Green <a href="biblio.htm#ELG08"><cite>ELG08</cite></a> defined for a prime <var>p< coclass <var>r</var> infinite coclass sequences. These sequences consist of finite <var>p</var>-groups of coclass <var>r</var>. For each infinite coclass sequence there exists a consistent pp-presentation (see Section <a href="CHAP002.htm#SECT002">Background on (polycyclic) parametrised presentati such that if we choose a natural number for the parameter and possibly reduce the exponents modulo the relative orders, we obtain a consistent polycyclic presentation for a group in the sequence; and for each group in the sequence there exists a natural number such that using this as a value for the parameter, we obtain a polycyclic presentation for the group. <p> We use these consistent pp-presentations to compute parametrised groups, which we call <var>p</var>-power-poly-pcp-groups. Furthermore, methods for these are presented. Without specifying the parameter we compute certain properties and using the <var>p</var>-power-poly-pcp-groups we do this for all groups they represent at once. <p> The <var>p</var>-power-poly-pcp-groups have a consistent pp-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>d</i></sub> and <i>c</i><sub>1</ non-negative integers <var>n</var>, <var>d</var> and <var>m</var>, and relations of the form, where <i>rel</i>[<i>i</i>,<i>j</i>] stores the right hand sides of the relations (see Section <a href="CHAP002.htm#SECT002">Background on (polycyclic) parametrised presentati information on pp-presentations), <p> <br clear="all" /><table border="0" width="100%"><tr><td> <table border="0" cellspacing="0" cellp where the <i>t</i><sub><i>i</i></sub>'s commute modulo 〈c1,…, cm〉 and the <i>c</i><sub><i>i</i></sub>'s are central. So rel (see Section Obtaining p-power-poly-pcp-gro are the right hand sides of the relations, where some depend on the parameter. The relative orders <var>expo</var> and <var>expo_vec[i]</var> of the generators <i>t</i><sub><i>j</i></sub> a <i>c</i><sub><i>i</i></sub> depend on the parameter. <p> <p> <h2><a name="SECT001">3.1 Example</a></h2> <p><p> In this section we present the well-known example of quaternion groups <i>Q</i><sub>2<sup><i>x</i>+3</sup></sub>. They have a pp-presentation of the following form: <p> <br clear="all" /><table border="0" width="100%"><tr><td> <table border="0" cellspacing="0" cellp <p> <p> <h2><a name="SECT002">3.2 Obtaining p-power-poly-pcp-groups</a></h2> <p><p> To obtain <var>p</var>-power-poly-pcp-groups: <p> <a name = "SSEC002.1"></a> <li><code>PPPPcpGroups( </code><var>rel</var><code>, </code><var>n</var><code>, </code><var>d</var><code>, </code>< <li><code>PPPPcpGroups( </code><var>rec</var><code> ) F</code> <p> returns the p-power-poly-pcp-groups described by the consistent pp-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> <i>c</i><sub>1</sub>, …, <i>c</i><sub><i>m</i></sub>, for some non-negative integers <var>n</var>, <var>d</var> and <var> relations of the form <p> <br clear="all" /><table border="0" width="100%"><tr><td> <table border="0" cellspacing="0" cellp <p> The input consists of the following: <p> <p> <dl compact> <dt><code></code><var>rel</var><code></code> <dd> is the list of relations, where each relation is presente list consisting of tuples; the first entry <var>i</var> of a tuple is the index of the generator (if <i>i</i> ≤ <i>n</i>, then it represents generator <i>g</i><sub><i>i</i></sub>, if <i>n</i> < <i>i</i> ≤ <i>d</i>, then it represents generator <i>t</i><sub><i>i</i>−<i>n</i></sub> and otherwise it represents generator <i>c</i><sub><i>i</i>−<i>n</i>−<i>d</i></sub>) and the second entry of the tuple is the corresponding exponent. Note that the exponents of the <i>g</i><sub><i>i</i></sub>'s are saved as integers and all other exponents as lists, representing elements depending on the parameter. <dt><code></code><var>n</var><code></code> <dd> is the number of generators <i>g</i><sub><i>i</i></sub>, <dt><code></code><var>d</var><code></code> <dd> is the number of generators <i>t</i><sub><i>i</i></sub>, <dt><code></code><var>m</var><code></code> <dd> is the number of generators <i>c</i><sub><i>i</i></sub>, <dt><code></code><var>expo</var><code></code> <dd> is the relative order of all generators <i>t</i><sub><i>i</i></su given as a list to represent an element depending on the parameter, <dt><code></code><var>expo_vec</var><code></code> <dd> is the list of relative orders, where the <var>i</var>t list gives the relative order of the generator <i>c</i><sub><i>i</i></sub>; note that each relative order is given as a list to represent an element depending on the parameter, <dt><code></code><var>prime</var><code></code> <dd> is the underlying prime <var>p</var>, <dt><code></code><var>cc</var><code></code> <dd> if the <var>p</var>-power-poly-pcp-groups represent an inf sequence of <var>p</var>-groups of coclass <var>r</var>, then <var>cc</var> = <var>r</var>. If they represen Schur extensions of groups in an infinite coclass sequence, then <var>cc</var> is the coclass of the groups in this infinite coclass sequence. <dt><code></code><var>name</var><code></code> <dd> a string to name the <var>p</var>-power-poly-pcp-groups. <dt><code></code><var>rec</var><code></code> <dd> is a record of the form <var>rec( rel, expo, n, d, m, prime, cc, expo_vec, name )</var>. </dl> <p> The pp-presentation is described at the beginning of Chapter <a href="badlink:SymbCompCC:p-power-poly-pcp-group">p-power-poly-pcp-group</a>. Note th checked and that the presentation has to be consistent. <p> <pre> gap> ParPresGlobalVar_2_1[1]; rec( rel := [ [ [ [ 1, 0 ] ] ], [ [ [ 2, 1 ], [ 3, -1+2*2^x ] ], [ [ 3, 1 ] ] ], [ [ [ 3, -1+2*2^x ] ], [ [ 3, 1 ] ], [ [ 3, 0 ] ] ] ], expo := 2*2^x, n := 2, d := 1, m := 0, prime := 2, cc := 1, expo_vec := [ ], name := "D" ) gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] ); < P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] > </pre> <p> <a name = "SSEC002.2"></a> <li><code>PPPPcpGroupsElement( </code><var>G</var><code>, </code><var>word</var><code> ) F</code> <p> constructs an element in <var>p</var>-power-poly-pcp-groups, where <var>G</var> is a <var>p</var>-power-poly-pcp-group (thus representing an infinite coclass sequence through a pp-presentation) with generators <i>g</i><sub>1</sub>, …, <i>g</i><sub><i>n</i></sub>, <i>t</i><sub>1</sub> entry <var>i</var> in the tuple gives the index of the generator (if <i>i</i> ≤ <i>n</i>, then it represents generator <i>g</i><sub><i>i</i></sub>, if <i>n</i> < <i>i</i> ≤ <i>d</i>, then it represents generator <i>t</i><sub><i>i</i>−<i>n</i></sub> and otherwise it represents generator <i>c</i><sub><i>i</i>−<i>n</i>−<i>d</i></sub>) and t entry of the tuple is the corresponding exponent. Note that the exponents of the <i>g</i><sub><i>i</i></sub>'s must be integers, while all other exponents can be intege or lists, representing an element depending on the parameter. <p> <pre> gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[3] ); < P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] > gap> g1 := PPPPcpGroupsElement( G , [[1,1]] ); g1 gap> g := PPPPcpGroupsElement( G , [[1,1],[2,1],[3,1]] ); g1*g2*t1 gap> h := PPPPcpGroupsElement( G , [[1,1],[2,1],[3,G!.expo-1]] ); g1*g2*t1^(-1+2*2^x) </pre> <p> <p> <h2><a name="SECT003">3.3 Operations and functions for p-power-poly-pcp-group elements</a></h2> <p><p> The typical operations for group elements can be carried out for <var>p</var>-power-poly-pcp-group elements, like <code>*</code>, <code>/</code>, Inverse, One, equa ShallowCopy. <p> <a name = "SSEC003.1"></a> <li><code>CollectPPPPcp( </code><var>a</var><code> ) F</code> <p> collects the <var>p</var>-power-poly-pcp-group element <var>a</var> so that after reducing to integers for every specific value for the parameter <var>x</var>, the element is collected in the polycyclic group, represented by <var>x</var> in the underlying pp-presentation. <p> Note that the global variable <code>COLLECT_PPOWERPOLY_PCP</code> determines whether every element will be collected immediately, when created, or not, see <a href="CHAP003.htm#SSEC006.1">COLLECT_PPOWERPOLY_PCP</a>. <p> <p> <h2><a name="SECT004">3.4 Operations and functions for p-power-poly-pcp-groups</a></h2> <p><p> For <var>p</var>-power-poly-pcp-groups: <p> <a name = "SSEC004.1"></a> <li><code>GeneratorsOfGroup( </code><var>G</var><code> )</code> <p> returns a set of generators for the <var>p</var>-power-poly-pcp-groups <var>G</var>. <p> <a name = "SSEC004.2"></a> <li><code>One( </code><var>G</var><code> )</code> <p> obtains the identity element of the <var>p</var>-power-poly-pcp-groups <var>G</var>. <p> <a name = "SSEC004.3"></a> <li><code>IsConsistentPPPPcp( </code><var>G</var><code> ) F</code> <li><code>IsConsistentPPPPcp( </code><var>ParPres</var><code> ) F</code> <p> checks if the underlying pp-presentation of the <var>p</var>-power-poly-pcp-groups <var>G</var> is consistent or if the pp-presenta-tion <var>ParPres</var> is consistent. <p> <a name = "SSEC004.4"></a> <li><code>GetPcGroupPPowerPoly( </code><var>ParPres</var><code>, </code><var>n</var><code> ) F</code> <li><code>GetPcGroupPPowerPoly( </code><var>G</var><code>, </code><var>n</var><code> ) F</code> <p> takes the pp-presentation given by the record <var>ParPres</var> as in <a href="CHAP003.htm#SSEC002.1">PPPPcpGroups</a> or the <var>p</var>-power-poly-pcp-groups <va non-negative integer, as a value for the parameter to obtain a pc-presentation for the corresponding finite <var>p</var>-group. <p> <a name = "SSEC004.5"></a> <li><code>GetPcpGroupPPowerPoly( </code><var>ParPres</var><code>, </code><var>n</var><code> ) F</code> <li><code>GetPcpGroupPPowerPoly( </code><var>G</var><code>, </code><var>n</var><code> ) F</code> <p> takes pp-presentation given by the record <var>ParPres</var> as in <a href="CHAP003.htm#SSEC002.1">PPPPcpGroups</a> or the <var>p</var>-power-poly-pcp-groups <va non-negative integer, as the parameter to obtain a pcp-presentation for the corresponding finite <var>p</var>-group, for further information we refer to the polycyclic package. <p> <a name = "SSEC004.6"></a> <li><code>GAPInputPPPPcpGroups( </code><var>file</var><code>, </code><var>G</var><code> ) F</code> <li><code>GAPInputPPPPcpGroups( </code><var>file</var><code>, </code><var>ParPres</var><code> ) F</code> <p> prints the <var>p</var>-power-poly-pcp-groups <var>G</var> defined by <var>ParPres</var> in the file <var>file</var> as a record that could be used as input to <a href="CHAP003.htm#SSEC002.1">PPPPcpGroups</a> to create <var>p</var>-power-poly-pcp-group <p> <a name = "SSEC004.7"></a> <li><code>GAPInputPPPPcpGroupsAppend( </code><var>file</var><code>, </code><var>G</var><code> ) F</code> <li><code>GAPInputPPPPcpGroupsAppend( </code><var>file</var><code>, </code><var>ParPres</var><code> ) F</ <p> appends the pp-presentation of the <var>p</var>-power-poly-pcp-groups <var>G</var> defined by <var>ParPres</var> to the file <var>file</var> as a record that could be used as input to <a href="CHAP003.htm#SSEC002.1">PPPPcpGroups</a> to create <var>p</var>-power-poly-pcp-group <p> <a name = "SSEC004.8"></a> <li><code>LatexInputPPPPcpGroups( </code><var>file</var><code>, </code><var>G</var><code> ) F</code> <li><code>LatexInputPPPPcpGroups( </code><var>file</var><code>, </code><var>ParPres</var><code> ) F</code> <p> prints the pp-presentation of <var>G</var> as given by <var>ParPres</var> in latex-code to the file <var>file</var>. Note that only non-trivial relations are printed. <p> <a name = "SSEC004.9"></a> <li><code>LatexInputPPPPcpGroupsAppend( </code><var>file</var><code>, </code><var>G</var><code> ) F</code> <li><code>LatexInputPPPPcpGroupsAppend( </code><var>file</var><code>, </code><var>ParPres</var><code> ) <p> appends the pp-presentation of <var>G</var> as given by <var>ParPres</var> in latex-code to the file <var>file</var>. Note that only non-trivial relations are appended. <p> <a name = "SSEC004.10"></a> <li><code>LatexInputPPPPcpGroupsAllAppend( </code><var>file</var><code>, </code><var>G</var><code> ) F</c <li><code>LatexInputPPPPcpGroupsAllAppend( </code><var>file</var><code>, </code><var>ParPres</var><co <p> appends the pp-presentation of <var>G</var> as given by <var>ParPres</var> in latex-code to the file <var>file</var>. Note that all relations are appended. <p> <p> <h2><a name="SECT005">3.5 Info classes for the p-power-poly-pcp-groups</a></h2> <p><p> The following info classes are available: <p> <a name = "SSEC005.1"></a> <li><code>InfoConsistencyPPPPcp V</code> <p> is an InfoClass with the following levels. <p> <p> <dl compact> <dt><code>level 1</code> <dd> displays the first consistency relation that fails during the consiste <dt><code>level 2</code> <dd> displays which family of consistency relations have been checked durin </dl> <p> the default value is 1. <p> <a name = "SSEC005.2"></a> <li><code>InfoCollectingPPPPcp V</code> <p> is an InfoClass with the following levels. <p> <p> <dl compact> <dt><code>level 1</code> <dd> displays some information during collecting; </dl> <p> the default value is 0. <p> <p> <h2><a name="SECT006">3.6 Global variables for the p-power-poly-pcp-groups</a></h2> <p><p> The following global variables are available with default value: <p> <a name = "SSEC006.1"></a> <li><code>COLLECT_PPOWERPOLY_PCP V</code> <p> is a global variable determining if every <var>p</var>-power-poly-pcp-group element is collected, when created, the default value is true. <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Nex <P> <address>SymbCompCC manual<br>February 2022 </address></body></html> ¤ Dauer der Verarbeitung: 0.2 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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