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<div class="ChapSects"><a href="chap70.html#X84DBFB8287C3F1B4">70 <span class="Heading">Tables of Marks</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X80883EC17968F442">70.1 <span class="Heading">More about Tables of Marks</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X7D29539F7C14956D">70.2 <span class="Heading">Table of Marks Objects in GAP</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X7B5E4B5F81AF6B00">70.3 <span class="Heading">Constructing Tables of Marks</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X85B262AB7E219C34">70.3-1 TableOfMarks</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7B30FF3A79CCB0DF">70.3-2 TableOfMarksByLattice</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X79ABFA0A833DDCFE">70.3-3 LatticeSubgroupsByTom</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X7AC0FB9685DCBCFD">70.4 <span class="Heading">Printing Tables of Marks</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7DC656517D8335DC">70.4-1 ViewObj</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X86379C0D7D17AD92">70.4-2 PrintObj</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X821F9438839F445D">70.4-3 Display</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X82385925797B5108">70.5 <span class="Heading">Sorting Tables of Marks</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X786A948E82C36F0E">70.5-1 SortedTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7EFD937D804662F6">70.5-2 PermutationTom</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X82271C4F7FD21FAA">70.6 <span class="Heading">Technical Details about Tables of Marks</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X870985C58547FED4">70.6-1 InfoTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7AC1A73D8100C7EC">70.6-2 IsTableOfMarks</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7ACF943D84BDF89E">70.6-3 TableOfMarksFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X87789FD27831B2A2">70.6-4 TableOfMarksComponents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X8491CDBF8543A7D5">70.6-5 ConvertToTableOfMarks</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X838D3B87827D6923">70.7 <span class="Heading">Attributes of Tables of Marks</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X78F486A28561D006">70.7-1 MarksTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X82E5DA217A5D1134">70.7-2 NrSubsTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X781AA1B28178AE9A">70.7-3 LengthsTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7A33C7C38083CC09">70.7-4 ClassTypesTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7A53E923819FE173">70.7-5 ClassNamesTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X86B9891C788D5107">70.7-6 FusionsTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X81E41D3880FA6C4C">70.7-7 UnderlyingGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X817238FB79A3462F">70.7-8 IdempotentsTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X810E53597B5BB4F8">70.7-9 Identifier</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X8463272986781E17">70.7-10 MatTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7D32C8B0786D16C1">70.7-11 MoebiusTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X78525D04849A48EA">70.7-12 WeightsTom</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X78A1B2E4826A9518">70.8 <span class="Heading">Properties of Tables of Marks</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7C93BAEC78B7C2B4">70.8-1 IsAbelianTom</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X7A40D99D7816F126">70.9 <span class="Heading">Other Operations for Tables of Marks</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7D8B4BE08094B137">70.9-1 IsInternallyConsistent</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X8528D9397FFAF477">70.9-2 DerivedSubgroupTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7C29BD438127DFBE">70.9-3 DerivedSubgroupsTomPossible</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7CE6C45881F7F7D4">70.9-4 NormalizerTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7F87B2797827E5DE">70.9-5 ContainedTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7EE050FB87D6F0E7">70.9-6 ContainingTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X838DE06B823C19CA">70.9-7 CyclicExtensionsTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X80890C247EB1E35C">70.9-8 DecomposedFixedPointVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7B1C1A7C867A4082">70.9-9 EulerianFunctionByTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X8224E51382FDB912">70.9-10 IntersectionsTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X859F069C8428B598">70.9-11 FactorGroupTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X8325811586C00ECF">70.9-12 MaximalSubgroupsTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7923B19D7C47BF63">70.9-13 MinimalSupergroupsTom</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X7FE9BE477A90199F">70.10 <span class="Heading">Accessing Subgroups via Tables of Marks</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7B0B6FDD806E9734">70.10-1 GeneratorsSubgroupsTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7898BE7284E47FF3">70.10-2 StraightLineProgramsTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7889DB6D790593B9">70.10-3 IsTableOfMarksWithGens</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7F625AB880B73AC3">70.10-4 RepresentativeTom</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X79ADA60880BE9C49">70.11 <span class="Heading">The Interface between Tables of Marks and Character Tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7A82CB487DBDDC53">70.11-1 FusionCharTableTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X8016499282F0BA37">70.11-2 PermCharsTom</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X7CF66FAE7A8858E4">70.12 <span class="Heading">Generic Construction of Tables of Marks</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7CAA5B6C85CB9A8D">70.12-1 TableOfMarksCyclic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X7AADB47B8079C99E">70.12-2 TableOfMarksDihedral</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap70.html#X78E9DDF885E12687">70.12-3 TableOfMarksFrobenius</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap70.html#X794ABC7187A9285B">70.13 <span class="Heading">The Library of Tables of Marks</span></a>
</span>
</div>
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<h3>70 <span class="Heading">Tables of Marks</span></h3>
<p>The concept of a <em>table of marks</em> was introduced by W. Burnside in his book "Theory of Groups of Finite Order", see <a href="chapBib.html#biBBur55">[Bur55]</a>. Therefore a table of marks is sometimes called a <em>Burnside matrix</em>.</p>
<p>The table of marks of a finite group <span class="SimpleMath">G</span> is a matrix whose rows and columns are labelled by the conjugacy classes of subgroups of <span class="SimpleMath">G</span> and where for two subgroups <span class="SimpleMath">A</span> and <span class="SimpleMath">B</span> the <span class="SimpleMath">(A, B)</span>-entry is the number of fixed points of <span class="SimpleMath">B</span> in the transitive action of <span class="SimpleMath">G</span> on the cosets of <span class="SimpleMath">A</span> in <span class="SimpleMath">G</span>. So the table of marks characterizes the set of all permutation representations of <span class="SimpleMath">G</span>.</p>
<p>Moreover, the table of marks gives a compact description of the subgroup lattice of <span class="SimpleMath">G</span>, since from the numbers of fixed points the numbers of conjugates of a subgroup <span class="SimpleMath">B</span> contained in a subgroup <span class="SimpleMath">A</span> can be derived.</p>
<p>A table of marks of a given group <span class="SimpleMath">G</span> can be constructed from the subgroup lattice of <span class="SimpleMath">G</span> (see <a href="chap70.html#X7B5E4B5F81AF6B00"><span class="RefLink">70.3</span></a>). For several groups, the table of marks can be restored from the <strong class="pkg">GAP</strong> library of tables of marks (see <a href="chap70.html#X794ABC7187A9285B"><span class="RefLink">70.13</span></a>).</p>
<p>Given the table of marks of <span class="SimpleMath">G</span>, one can display it (see <a href="chap70.html#X7AC0FB9685DCBCFD"><span class="RefLink">70.4</span></a>) and derive information about <span class="SimpleMath">G</span> and its Burnside ring from it (see <a href="chap70.html#X838D3B87827D6923"><span class="RefLink">70.7</span></a>, <a href="chap70.html#X78A1B2E4826A9518"><span class="RefLink">70.8</span></a>, <a href="chap70.html#X7A40D99D7816F126"><span class="RefLink">70.9</span></a>). Moreover, tables of marks in <strong class="pkg">GAP</strong> provide an easy access to the classes of subgroups of their underlying groups (see <a href="chap70.html#X7FE9BE477A90199F"><span class="RefLink">70.10</span></a>).</p>
<p><a id="X80883EC17968F442" name="X80883EC17968F442"></a></p>
<h4>70.1 <span class="Heading">More about Tables of Marks</span></h4>
<p>Let <span class="SimpleMath">G</span> be a finite group with <span class="SimpleMath">n</span> conjugacy classes of subgroups <span class="SimpleMath">C_1, C_2, ..., C_n</span> and representatives <span class="SimpleMath">H_i ∈ C_i</span>, <span class="SimpleMath">1 ≤ i ≤ n</span>. The <em>table of marks</em> of <span class="SimpleMath">G</span> is defined to be the <span class="SimpleMath">n × n</span> matrix <span class="SimpleMath">M = (m_ij)</span> where the <em>mark</em> <span class="SimpleMath">m_ij</span> is the number of fixed points of the subgroup <span class="SimpleMath">H_j</span> in the action of <span class="SimpleMath">G</span> on the right cosets of <span class="SimpleMath">H_i</span> in <span class="SimpleMath">G</span>.</p>
<p>Since <span class="SimpleMath">H_j</span> can only have fixed points if it is contained in a point stabilizer the matrix <span class="SimpleMath">M</span> is lower triangular if the classes <span class="SimpleMath">C_i</span> are sorted according to the condition that if <span class="SimpleMath">H_i</span> is contained in a conjugate of <span class="SimpleMath">H_j</span> then <span class="SimpleMath">i ≤ j</span>.</p>
<p>Moreover, the diagonal entries <span class="SimpleMath">m_ii</span> are nonzero since <span class="SimpleMath">m_ii</span> equals the index of <span class="SimpleMath">H_i</span> in its normalizer in <span class="SimpleMath">G</span>. Hence <span class="SimpleMath">M</span> is invertible. Since any transitive action of <span class="SimpleMath">G</span> is equivalent to an action on the cosets of a subgroup of <span class="SimpleMath">G</span>, one sees that the table of marks completely characterizes the set of all permutation representations of <span class="SimpleMath">G</span>.</p>
<p>The marks <span class="SimpleMath">m_ij</span> have further meanings. If <span class="SimpleMath">H_1</span> is the trivial subgroup of <span class="SimpleMath">G</span> then each mark <span class="SimpleMath">m_i1</span> in the first column of <span class="SimpleMath">M</span> is equal to the index of <span class="SimpleMath">H_i</span> in <span class="SimpleMath">G</span> since the trivial subgroup fixes all cosets of <span class="SimpleMath">H_i</span>. If <span class="SimpleMath">H_n = G</span> then each <span class="SimpleMath">m_nj</span> in the last row of <span class="SimpleMath">M</span> is equal to <span class="SimpleMath">1</span> since there is only one coset of <span class="SimpleMath">G</span> in <span class="SimpleMath">G</span>. In general, <span class="SimpleMath">m_ij</span> equals the number of conjugates of <span class="SimpleMath">H_i</span> containing <span class="SimpleMath">H_j</span>, multiplied by the index of <span class="SimpleMath">H_i</span> in its normalizer in <span class="SimpleMath">G</span>. Moreover, the number <span class="SimpleMath">c_ij</span> of conjugates of <span class="SimpleMath">H_j</span> which are contained in <span class="SimpleMath">H_i</span> can be derived from the marks <span class="SimpleMath">m_ij</span> via the formula</p>
<p class="pcenter">c_ij = ( m_ij m_j1 ) / ( m_i1 m_jj )</p>
<p>.</p>
<p>Both the marks <span class="SimpleMath">m_ij</span> and the numbers of subgroups <span class="SimpleMath">c_ij</span> are needed for the functions described in this chapter.</p>
<p>A brief survey of properties of tables of marks and a description of algorithms for the interactive construction of tables of marks using <strong class="pkg">GAP</strong> can be found in <a href="chapBib.html#biBPfe97">[Pfe97]</a>.</p>
<p><a id="X7D29539F7C14956D" name="X7D29539F7C14956D"></a></p>
<h4>70.2 <span class="Heading">Table of Marks Objects in GAP</span></h4>
<p>A table of marks of a group <span class="SimpleMath">G</span> in <strong class="pkg">GAP</strong> is represented by an immutable (see <a href="chap12.html#X7F0C119682196D65"><span class="RefLink">12.6</span></a>) object <var class="Arg">tom</var> in the category <code class="func">IsTableOfMarks</code> (<a href="chap70.html#X7AC1A73D8100C7EC"><span class="RefLink">70.6-2</span></a>), with defining attributes <code class="func">SubsTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) and <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>). These two attributes encode the matrix of marks in a compressed form. The <code class="func">SubsTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) value of <var class="Arg">tom</var> is a list where for each conjugacy class of subgroups the class numbers of its subgroups are stored. These are exactly the positions in the corresponding row of the matrix of marks which have nonzero entries. The marks themselves are stored via the <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) value of <var class="Arg">tom</var>, which is a list that contains for each entry in <code class="code">SubsTom( <var class="Arg">tom</var> )</code> the corresponding nonzero value of the table of marks.</p>
<p>It is possible to create table of marks objects that do not store a group, moreover one can create a table of marks object from a matrix of marks (see <code class="func">TableOfMarks</code> (<a href="chap70.html#X85B262AB7E219C34"><span class="RefLink">70.3-1</span></a>)). So it may happen that a table of marks object in <strong class="pkg">GAP</strong> is in fact <em>not</em> the table of marks of a group. To some extent, the consistency of a table of marks object can be checked (see <a href="chap70.html#X7A40D99D7816F126"><span class="RefLink">70.9</span></a>), but <strong class="pkg">GAP</strong> knows no general way to prove or disprove that a given matrix of nonnegative integers is the matrix of marks for a group. Many functions for tables of marks work well without access to the group –this is one of the arguments why tables of marks are so useful–, but for example normalizers (see <code class="func">NormalizerTom</code> (<a href="chap70.html#X7CE6C45881F7F7D4"><span class="RefLink">70.9-4</span></a>)) and derived subgroups (see <code class="func">DerivedSubgroupTom</code> (<a href="chap70.html#X8528D9397FFAF477"><span class="RefLink">70.9-2</span></a>)) of subgroups are in general not uniquely determined by the matrix of marks.</p>
<p><strong class="pkg">GAP</strong> tables of marks are assumed to be in lower triangular form, that is, if a subgroup from the conjugacy class corresponding to the <span class="SimpleMath">i</span>-th row is contained in a subgroup from the class corresponding to the <span class="SimpleMath">j</span>-th row j then <span class="SimpleMath">i ≤ j</span>.</p>
<p>The <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) information can be computed from the values of the attributes <code class="func">NrSubsTom</code> (<a href="chap70.html#X82E5DA217A5D1134"><span class="RefLink">70.7-2</span></a>), <code class="func">LengthsTom</code> (<a href="chap70.html#X781AA1B28178AE9A"><span class="RefLink">70.7-3</span></a>), <code class="func">OrdersTom</code> (<a href="chap70.html#X82E5DA217A5D1134"><span class="RefLink">70.7-2</span></a>), and <code class="func">SubsTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>). <code class="func">NrSubsTom</code> (<a href="chap70.html#X82E5DA217A5D1134"><span class="RefLink">70.7-2</span></a>) stores a list containing for each entry in the <code class="func">SubsTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) value the corresponding number of conjugates that are contained in a subgroup, <code class="func">LengthsTom</code> (<a href="chap70.html#X781AA1B28178AE9A"><span class="RefLink">70.7-3</span></a>) a list containing for each conjugacy class of subgroups its length, and <code class="func">OrdersTom</code> (<a href="chap70.html#X82E5DA217A5D1134"><span class="RefLink">70.7-2</span></a>) a list containing for each class of subgroups their order. So the <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) value of <var class="Arg">tom</var> may be missing provided that the values of <code class="func">NrSubsTom</code> (<a href="chap70.html#X82E5DA217A5D1134"><span class="RefLink">70.7-2</span></a>), <code class="func">LengthsTom</code> (<a href="chap70.html#X781AA1B28178AE9A"><span class="RefLink">70.7-3</span></a>), and <code class="func">OrdersTom</code> (<a href="chap70.html#X82E5DA217A5D1134"><span class="RefLink">70.7-2</span></a>) are stored in <var class="Arg">tom</var>.</p>
<p>Additional information about a table of marks is needed by some functions. The class numbers of normalizers in <span class="SimpleMath">G</span> and the number of the derived subgroup of <span class="SimpleMath">G</span> can be stored via appropriate attributes (see <code class="func">NormalizersTom</code> (<a href="chap70.html#X7CE6C45881F7F7D4"><span class="RefLink">70.9-4</span></a>), <code class="func">DerivedSubgroupTom</code> (<a href="chap70.html#X8528D9397FFAF477"><span class="RefLink">70.9-2</span></a>)).</p>
<p>If <var class="Arg">tom</var> stores its group <span class="SimpleMath">G</span> and a bijection from the rows and columns of the matrix of marks of <var class="Arg">tom</var> to the classes of subgroups of <span class="SimpleMath">G</span> then clearly normalizers, derived subgroup etc. can be computed from this information. But in general a table of marks need not have access to <span class="SimpleMath">G</span>, for example <var class="Arg">tom</var> might have been constructed from a generic table of marks (see <a href="chap70.html#X7CF66FAE7A8858E4"><span class="RefLink">70.12</span></a>), or as table of marks of a factor group from a given table of marks (see <code class="func">FactorGroupTom</code> (<a href="chap70.html#X859F069C8428B598"><span class="RefLink">70.9-11</span></a>)). Access to the group <span class="SimpleMath">G</span> is provided by the attribute <code class="func">UnderlyingGroup</code> (<a href="chap70.html#X81E41D3880FA6C4C"><span class="RefLink">70.7-7</span></a>) if this value is set. Access to the relevant information about conjugacy classes of subgroups of <span class="SimpleMath">G</span> –compatible with the ordering of rows and columns of the marks in <var class="Arg">tom</var>– is signalled by the filter <code class="func">IsTableOfMarksWithGens</code> (<a href="chap70.html#X7889DB6D790593B9"><span class="RefLink">70.10-3</span></a>).</p>
<p>Several examples in this chapter require the <strong class="pkg">GAP</strong> package <strong class="pkg">TomLib</strong> (the <strong class="pkg">GAP</strong> Library of Tables of Marks) to be available. If it is not yet loaded then we load it now.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "tomlib" );</span>
true
</pre></div>
<p><a id="X7B5E4B5F81AF6B00" name="X7B5E4B5F81AF6B00"></a></p>
<h4>70.3 <span class="Heading">Constructing Tables of Marks</span></h4>
<p><a id="X85B262AB7E219C34" name="X85B262AB7E219C34"></a></p>
<h5>70.3-1 TableOfMarks</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarks</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarks</code>( <var class="Arg">string</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarks</code>( <var class="Arg">matrix</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>In the first form, <var class="Arg">G</var> must be a finite group, and <code class="func">TableOfMarks</code> constructs the table of marks of <var class="Arg">G</var>. This computation requires the knowledge of the complete subgroup lattice of <var class="Arg">G</var> (see <code class="func">LatticeSubgroups</code> (<a href="chap39.html#X7B104E2C86166188"><span class="RefLink">39.20-1</span></a>)). If the lattice is not yet stored then it will be constructed. This may take a while if <var class="Arg">G</var> is large. The result has the <code class="func">IsTableOfMarksWithGens</code> (<a href="chap70.html#X7889DB6D790593B9"><span class="RefLink">70.10-3</span></a>) value <code class="keyw">true</code>.</p>
<p>In the second form, <var class="Arg">string</var> must be a string, and <code class="func">TableOfMarks</code> gets the table of marks with name <var class="Arg">string</var> from the <strong class="pkg">GAP</strong> library (see <a href="chap70.html#X794ABC7187A9285B"><span class="RefLink">70.13</span></a>). If no table of marks with this name is contained in the library then <code class="keyw">fail</code> is returned.</p>
<p>In the third form, <var class="Arg">matrix</var> must be a matrix or a list of rows describing a lower triangular matrix where the part above the diagonal is omitted. For such an argument <var class="Arg">matrix</var>, <code class="func">TableOfMarks</code> returns a table of marks object (see <a href="chap70.html#X7D29539F7C14956D"><span class="RefLink">70.2</span></a>) for which <var class="Arg">matrix</var> is the matrix of marks. Note that not every matrix (containing only nonnegative integers and having lower triangular shape) describes a table of marks of a group. Necessary conditions are checked with <code class="func">IsInternallyConsistent</code> (<a href="chap70.html#X7D8B4BE08094B137"><span class="RefLink">70.9-1</span></a>) (see <a href="chap70.html#X7A40D99D7816F126"><span class="RefLink">70.9</span></a>), and <code class="keyw">fail</code> is returned if <var class="Arg">matrix</var> is proved not to describe a matrix of marks; but if <code class="func">TableOfMarks</code> returns a table of marks object created from a matrix then it may still happen that this object does not describe the table of marks of a group.</p>
<p>For an overview of operations for table of marks objects, see the introduction to Chapter <a href="chap70.html#X84DBFB8287C3F1B4"><span class="RefLink">70</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tom:= TableOfMarks( AlternatingGroup( 5 ) );</span>
TableOfMarks( Alt( [ 1 .. 5 ] ) )
<span class="GAPprompt">gap></span> <span class="GAPinput">TableOfMarks( "J5" );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">a5:= TableOfMarks( "A5" );</span>
TableOfMarks( "A5" )
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ [ 60, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 30, 2, 0, 0, 0, 0, 0, 0, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 20, 0, 2, 0, 0, 0, 0, 0, 0 ], [ 15, 3, 0, 3, 0, 0, 0, 0, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 12, 0, 0, 0, 2, 0, 0, 0, 0 ], [ 10, 2, 1, 0, 0, 1, 0, 0, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 6, 2, 0, 0, 1, 0, 1, 0, 0 ], [ 5, 1, 2, 1, 0, 0, 0, 1, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TableOfMarks( mat );</span>
TableOfMarks( <9 classes> )
</pre></div>
<p>The following <code class="func">TableOfMarks</code> methods for a group are installed.</p>
<ul>
<li><p>If the group is known to be cyclic then <code class="func">TableOfMarks</code> constructs the table of marks essentially without the group, instead the knowledge about the structure of cyclic groups is used.</p>
</li>
<li><p>If the lattice of subgroups is already stored in the group then <code class="func">TableOfMarks</code> computes the table of marks from the lattice (see <code class="func">TableOfMarksByLattice</code> (<a href="chap70.html#X7B30FF3A79CCB0DF"><span class="RefLink">70.3-2</span></a>)).</p>
</li>
<li><p>If the group is known to be solvable then <code class="func">TableOfMarks</code> takes the lattice of subgroups (see <code class="func">LatticeSubgroups</code> (<a href="chap39.html#X7B104E2C86166188"><span class="RefLink">39.20-1</span></a>)) of the group –which means that the lattice is computed if it is not yet stored– and then computes the table of marks from it. This method is also accessible via the global function <code class="func">TableOfMarksByLattice</code> (<a href="chap70.html#X7B30FF3A79CCB0DF"><span class="RefLink">70.3-2</span></a>).</p>
</li>
<li><p>If the group doesn't know its lattice of subgroups or its conjugacy classes of subgroups then the table of marks and the conjugacy classes of subgroups are computed at the same time by the cyclic extension method. Only the table of marks is stored because the conjugacy classes of subgroups or the lattice of subgroups can be easily read off (see <code class="func">LatticeSubgroupsByTom</code> (<a href="chap70.html#X79ABFA0A833DDCFE"><span class="RefLink">70.3-3</span></a>)).</p>
</li>
</ul>
<p>Conversely, the lattice of subgroups of a group with known table of marks can be computed using the table of marks, via the function <code class="func">LatticeSubgroupsByTom</code> (<a href="chap70.html#X79ABFA0A833DDCFE"><span class="RefLink">70.3-3</span></a>). This is also installed as a method for <code class="func">LatticeSubgroups</code> (<a href="chap39.html#X7B104E2C86166188"><span class="RefLink">39.20-1</span></a>).</p>
<p><a id="X7B30FF3A79CCB0DF" name="X7B30FF3A79CCB0DF"></a></p>
<h5>70.3-2 TableOfMarksByLattice</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarksByLattice</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">TableOfMarksByLattice</code> computes the table of marks of the group <var class="Arg">G</var> from the lattice of subgroups of <var class="Arg">G</var>. This lattice is computed via <code class="func">LatticeSubgroups</code> (<a href="chap39.html#X7B104E2C86166188"><span class="RefLink">39.20-1</span></a>) if it is not yet stored in <var class="Arg">G</var>. The function <code class="func">TableOfMarksByLattice</code> is installed as a method for <code class="func">TableOfMarks</code> (<a href="chap70.html#X85B262AB7E219C34"><span class="RefLink">70.3-1</span></a>) for solvable groups and groups with stored subgroup lattice, and is available as a global variable only in order to provide explicit access to this method.</p>
<p><a id="X79ABFA0A833DDCFE" name="X79ABFA0A833DDCFE"></a></p>
<h5>70.3-3 LatticeSubgroupsByTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LatticeSubgroupsByTom</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">LatticeSubgroupsByTom</code> computes the lattice of subgroups of <var class="Arg">G</var> from the table of marks of <var class="Arg">G</var>, using <code class="func">RepresentativeTom</code> (<a href="chap70.html#X7F625AB880B73AC3"><span class="RefLink">70.10-4</span></a>).</p>
<p><a id="X7AC0FB9685DCBCFD" name="X7AC0FB9685DCBCFD"></a></p>
<h4>70.4 <span class="Heading">Printing Tables of Marks</span></h4>
<p><a id="X7DC656517D8335DC" name="X7DC656517D8335DC"></a></p>
<h5>70.4-1 ViewObj</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ViewObj</code>( <var class="Arg">tom</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>The default <code class="func">ViewObj</code> (<a href="chap6.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) method for tables of marks prints the string <code class="code">"TableOfMarks"</code>, followed by –if known– the identifier (see <code class="func">Identifier</code> (<a href="chap70.html#X810E53597B5BB4F8"><span class="RefLink">70.7-9</span></a>)) or the group of the table of marks enclosed in brackets; if neither group nor identifier are known then just the number of conjugacy classes of subgroups is printed instead.</p>
<p><a id="X86379C0D7D17AD92" name="X86379C0D7D17AD92"></a></p>
<h5>70.4-2 PrintObj</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrintObj</code>( <var class="Arg">tom</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>The default <code class="func">PrintObj</code> (<a href="chap6.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) method for tables of marks does the same as <code class="func">ViewObj</code> (<a href="chap6.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>), except that <code class="func">PrintObj</code> (<a href="chap6.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) is used for the group instead of <code class="func">ViewObj</code> (<a href="chap6.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>).</p>
<p><a id="X821F9438839F445D" name="X821F9438839F445D"></a></p>
<h5>70.4-3 Display</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Display</code>( <var class="Arg">tom</var>[, <var class="Arg">arec</var>] )</td><td class="tdright">( method )</td></tr></table></div>
<p>The default <code class="func">Display</code> (<a href="chap6.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>) method for a table of marks <var class="Arg">tom</var> produces a formatted output of the marks in <var class="Arg">tom</var>. Each line of output begins with the number of the corresponding class of subgroups. This number is repeated if the output spreads over several pages. The number of columns printed at one time depends on the actual line length, which can be accessed and changed by the function <code class="func">SizeScreen</code> (<a href="chap6.html#X8723E0A1837894F3"><span class="RefLink">6.12-1</span></a>).</p>
<p>An interactive variant of <code class="func">Display</code> (<a href="chap6.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>) is the <code class="func">Browse</code> (<span class="RefLink">???</span>) method for tables of marks that is provided by the <strong class="pkg">GAP</strong> package <strong class="pkg">Browse</strong>, see <code class="func">Browse</code> (<span class="RefLink">???</span>).</p>
<p>The optional second argument <var class="Arg">arec</var> of <code class="func">Display</code> (<a href="chap6.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>) can be used to change the default style for displaying a table of marks. <var class="Arg">arec</var> must be a record, its relevant components are the following.</p>
<dl>
<dt><strong class="Mark"><code class="code">classes</code></strong></dt>
<dd><p>a list of class numbers to select only the rows and columns of the matrix that correspond to this list for printing,</p>
</dd>
<dt><strong class="Mark"><code class="code">form</code></strong></dt>
<dd><p>one of the strings <code class="code">"subgroups"</code>, <code class="code">"supergroups"</code>; in the former case, at position <span class="SimpleMath">(i,j)</span> of the matrix the number of conjugates of <span class="SimpleMath">H_j</span> contained in <span class="SimpleMath">H_i</span> is printed, and in the latter case, at position <span class="SimpleMath">(i,j)</span> the number of conjugates of <span class="SimpleMath">H_i</span> which contain <span class="SimpleMath">H_j</span> is printed.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tom:= TableOfMarks( "A5" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tom );</span>
1: 60
2: 30 2
3: 20 . 2
4: 15 3 . 3
5: 12 . . . 2
6: 10 2 1 . . 1
7: 6 2 . . 1 . 1
8: 5 1 2 1 . . . 1
9: 1 1 1 1 1 1 1 1 1
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tom, rec( classes:= [ 1, 2, 3, 4, 8 ] ) );</span>
1: 60
2: 30 2
3: 20 . 2
4: 15 3 . 3
8: 5 1 2 1 1
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tom, rec( form:= "subgroups" ) );</span>
1: 1
2: 1 1
3: 1 . 1
4: 1 3 . 1
5: 1 . . . 1
6: 1 3 1 . . 1
7: 1 5 . . 1 . 1
8: 1 3 4 1 . . . 1
9: 1 15 10 5 6 10 6 5 1
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tom, rec( form:= "supergroups" ) );</span>
1: 1
2: 15 1
3: 10 . 1
4: 5 1 . 1
5: 6 . . . 1
6: 10 2 1 . . 1
7: 6 2 . . 1 . 1
8: 5 1 2 1 . . . 1
9: 1 1 1 1 1 1 1 1 1
</pre></div>
<p><a id="X82385925797B5108" name="X82385925797B5108"></a></p>
<h4>70.5 <span class="Heading">Sorting Tables of Marks</span></h4>
<p><a id="X786A948E82C36F0E" name="X786A948E82C36F0E"></a></p>
<h5>70.5-1 SortedTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SortedTom</code>( <var class="Arg">tom</var>, <var class="Arg">perm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">SortedTom</code> returns a table of marks where the rows and columns of the table of marks <var class="Arg">tom</var> are reordered according to the permutation <var class="Arg">perm</var>.</p>
<p><em>Note</em> that in each table of marks in <strong class="pkg">GAP</strong>, the matrix of marks is assumed to have lower triangular shape (see <a href="chap70.html#X7D29539F7C14956D"><span class="RefLink">70.2</span></a>). If the permutation <var class="Arg">perm</var> does <em>not</em> have this property then the functions for tables of marks might return wrong results when applied to the output of <code class="func">SortedTom</code>.</p>
<p>The returned table of marks has only those attribute values stored that are known for <var class="Arg">tom</var> and listed in <code class="func">TableOfMarksComponents</code> (<a href="chap70.html#X87789FD27831B2A2"><span class="RefLink">70.6-4</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tom:= TableOfMarksCyclic( 6 );; Display( tom );</span>
1: 6
2: 3 3
3: 2 . 2
4: 1 1 1 1
<span class="GAPprompt">gap></span> <span class="GAPinput">sorted:= SortedTom( tom, (2,3) );; Display( sorted );</span>
1: 6
2: 2 2
3: 3 . 3
4: 1 1 1 1
<span class="GAPprompt">gap></span> <span class="GAPinput">wrong:= SortedTom( tom, (1,2) );; Display( wrong );</span>
1: 3
2: . 6
3: . 2 2
4: 1 1 1 1
</pre></div>
<p><a id="X7EFD937D804662F6" name="X7EFD937D804662F6"></a></p>
<h5>70.5-2 PermutationTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PermutationTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For the table of marks <var class="Arg">tom</var> of the group <span class="SimpleMath">G</span> stored as <code class="func">UnderlyingGroup</code> (<a href="chap70.html#X81E41D3880FA6C4C"><span class="RefLink">70.7-7</span></a>) value of <var class="Arg">tom</var>, <code class="func">PermutationTom</code> is a permutation <span class="SimpleMath">π</span> such that the <span class="SimpleMath">i</span>-th conjugacy class of subgroups of <span class="SimpleMath">G</span> belongs to the <span class="SimpleMath">i^π</span>-th column and row of marks in <var class="Arg">tom</var>.</p>
<p>This attribute value is bound only if <var class="Arg">tom</var> was obtained from another table of marks by permuting with <code class="func">SortedTom</code> (<a href="chap70.html#X786A948E82C36F0E"><span class="RefLink">70.5-1</span></a>), and there is no default method to compute its value.</p>
<p>The attribute is necessary because the original and the sorted table of marks have the same identifier and the same group, and information computed from the group may depend on the ordering of marks, for example the fusion from the ordinary character table of <span class="SimpleMath">G</span> into <var class="Arg">tom</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MarksTom( tom )[2];</span>
[ 3, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MarksTom( sorted )[2];</span>
[ 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">HasPermutationTom( sorted );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">PermutationTom( sorted );</span>
(2,3)
</pre></div>
<p><a id="X82271C4F7FD21FAA" name="X82271C4F7FD21FAA"></a></p>
<h4>70.6 <span class="Heading">Technical Details about Tables of Marks</span></h4>
<p><a id="X870985C58547FED4" name="X870985C58547FED4"></a></p>
<h5>70.6-1 InfoTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoTom</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>is the info class for computations concerning tables of marks.</p>
<p><a id="X7AC1A73D8100C7EC" name="X7AC1A73D8100C7EC"></a></p>
<h5>70.6-2 IsTableOfMarks</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTableOfMarks</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Each table of marks belongs to this category.</p>
<p><a id="X7ACF943D84BDF89E" name="X7ACF943D84BDF89E"></a></p>
<h5>70.6-3 TableOfMarksFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarksFamily</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Each table of marks belongs to this family.</p>
<p><a id="X87789FD27831B2A2" name="X87789FD27831B2A2"></a></p>
<h5>70.6-4 TableOfMarksComponents</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarksComponents</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>The list <code class="func">TableOfMarksComponents</code> is used when a table of marks object is created from a record via <code class="func">ConvertToTableOfMarks</code> (<a href="chap70.html#X8491CDBF8543A7D5"><span class="RefLink">70.6-5</span></a>). <code class="func">TableOfMarksComponents</code> contains at position <span class="SimpleMath">2i-1</span> a name of an attribute and at position <span class="SimpleMath">2i</span> the corresponding attribute getter function.</p>
<p><a id="X8491CDBF8543A7D5" name="X8491CDBF8543A7D5"></a></p>
<h5>70.6-5 ConvertToTableOfMarks</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConvertToTableOfMarks</code>( <var class="Arg">record</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">ConvertToTableOfMarks</code> converts a record with components from <code class="func">TableOfMarksComponents</code> (<a href="chap70.html#X87789FD27831B2A2"><span class="RefLink">70.6-4</span></a>) into a table of marks object with the corresponding attributes.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">record:= rec( MarksTom:= [ [ 4 ], [ 2, 2 ], [ 1, 1, 1 ] ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> SubsTom:= [ [ 1 ], [ 1, 2 ], [ 1, 2, 3 ] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ConvertToTableOfMarks( record );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">record;</span>
TableOfMarks( <3 classes> )
</pre></div>
<p><a id="X838D3B87827D6923" name="X838D3B87827D6923"></a></p>
<h4>70.7 <span class="Heading">Attributes of Tables of Marks</span></h4>
<p><a id="X78F486A28561D006" name="X78F486A28561D006"></a></p>
<h5>70.7-1 MarksTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MarksTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubsTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The matrix of marks (see <a href="chap70.html#X80883EC17968F442"><span class="RefLink">70.1</span></a>) of the table of marks <var class="Arg">tom</var> is stored in a compressed form where zeros are omitted, using the attributes <code class="func">MarksTom</code> and <code class="func">SubsTom</code>. If <span class="SimpleMath">M</span> is the square matrix of marks of <var class="Arg">tom</var> (see <code class="func">MatTom</code> (<a href="chap70.html#X8463272986781E17"><span class="RefLink">70.7-10</span></a>)) then the <code class="func">SubsTom</code> value of <var class="Arg">tom</var> is a list that contains at position <span class="SimpleMath">i</span> the list of all positions of nonzero entries of the <span class="SimpleMath">i</span>-th row of <span class="SimpleMath">M</span>, and the <code class="func">MarksTom</code> value of <var class="Arg">tom</var> is a list that contains at position <span class="SimpleMath">i</span> the list of the corresponding marks.</p>
<p><code class="func">MarksTom</code> and <code class="func">SubsTom</code> are defining attributes of tables of marks (see <a href="chap70.html#X7D29539F7C14956D"><span class="RefLink">70.2</span></a>). There is no default method for computing the <code class="func">SubsTom</code> value, and the default <code class="func">MarksTom</code> method needs the values of <code class="func">NrSubsTom</code> (<a href="chap70.html#X82E5DA217A5D1134"><span class="RefLink">70.7-2</span></a>) and <code class="func">OrdersTom</code> (<a href="chap70.html#X82E5DA217A5D1134"><span class="RefLink">70.7-2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a5:= TableOfMarks( "A5" );</span>
TableOfMarks( "A5" )
<span class="GAPprompt">gap></span> <span class="GAPinput">MarksTom( a5 );</span>
[ [ 60 ], [ 30, 2 ], [ 20, 2 ], [ 15, 3, 3 ], [ 12, 2 ],
[ 10, 2, 1, 1 ], [ 6, 2, 1, 1 ], [ 5, 1, 2, 1, 1 ],
[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SubsTom( a5 );</span>
[ [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 2, 4 ], [ 1, 5 ], [ 1, 2, 3, 6 ],
[ 1, 2, 5, 7 ], [ 1, 2, 3, 4, 8 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ]
</pre></div>
<p><a id="X82E5DA217A5D1134" name="X82E5DA217A5D1134"></a></p>
<h5>70.7-2 NrSubsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrSubsTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrdersTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Instead of storing the marks (see <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>)) of the table of marks <var class="Arg">tom</var> one can use a matrix which contains at position <span class="SimpleMath">(i,j)</span> the number of subgroups of conjugacy class <span class="SimpleMath">j</span> that are contained in one member of the conjugacy class <span class="SimpleMath">i</span>. These values are stored in the <code class="func">NrSubsTom</code> value in the same way as the marks in the <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) value.</p>
<p><code class="func">OrdersTom</code> returns a list that contains at position <span class="SimpleMath">i</span> the order of a representative of the <span class="SimpleMath">i</span>-th conjugacy class of subgroups of <var class="Arg">tom</var>.</p>
<p>One can compute the <code class="func">NrSubsTom</code> and <code class="func">OrdersTom</code> values from the <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) value of <var class="Arg">tom</var> and vice versa.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrSubsTom( a5 );</span>
[ [ 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 3, 1 ], [ 1, 1 ], [ 1, 3, 1, 1 ],
[ 1, 5, 1, 1 ], [ 1, 3, 4, 1, 1 ], [ 1, 15, 10, 5, 6, 10, 6, 5, 1 ]
]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersTom( a5 );</span>
[ 1, 2, 3, 4, 5, 6, 10, 12, 60 ]
</pre></div>
<p><a id="X781AA1B28178AE9A" name="X781AA1B28178AE9A"></a></p>
<h5>70.7-3 LengthsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LengthsTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a table of marks <var class="Arg">tom</var>, <code class="func">LengthsTom</code> returns a list of the lengths of the conjugacy classes of subgroups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LengthsTom( a5 );</span>
[ 1, 15, 10, 5, 6, 10, 6, 5, 1 ]
</pre></div>
<p><a id="X7A33C7C38083CC09" name="X7A33C7C38083CC09"></a></p>
<h5>70.7-4 ClassTypesTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassTypesTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">ClassTypesTom</code> distinguishes isomorphism types of the classes of subgroups of the table of marks <var class="Arg">tom</var> as far as this is possible from the <code class="func">SubsTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) and <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) values of <var class="Arg">tom</var>.</p>
<p>Two subgroups are clearly not isomorphic if they have different orders. Moreover, isomorphic subgroups must contain the same number of subgroups of each type.</p>
<p>Each type is represented by a positive integer. <code class="func">ClassTypesTom</code> returns the list which contains for each class of subgroups its corresponding type.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a6:= TableOfMarks( "A6" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassTypesTom( a6 );</span>
[ 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15,
15, 16 ]
</pre></div>
<p><a id="X7A53E923819FE173" name="X7A53E923819FE173"></a></p>
<h5>70.7-5 ClassNamesTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassNamesTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">ClassNamesTom</code> constructs generic names for the conjugacy classes of subgroups of the table of marks <var class="Arg">tom</var>. In general, the generic name of a class of non-cyclic subgroups consists of three parts and has the form <code class="code">"(</code><var class="Arg">o</var><code class="code">)_{</code><var class="Arg">t</var><code class="code">}</code><var class="Arg">l</var><code class="code">"</code>, where <var class="Arg">o</var> indicates the order of the subgroup, <var class="Arg">t</var> is a number that distinguishes different types of subgroups of the same order, and <var class="Arg">l</var> is a letter that distinguishes classes of subgroups of the same type and order. The type of a subgroup is determined by the numbers of its subgroups of other types (see <code class="func">ClassTypesTom</code> (<a href="chap70.html#X7A33C7C38083CC09"><span class="RefLink">70.7-4</span></a>)). This is slightly weaker than isomorphism.</p>
<p>The letter is omitted if there is only one class of subgroups of that order and type, and the type is omitted if there is only one class of that order. Moreover, the braces <code class="code">{}</code> around the type are omitted if the type number has only one digit.</p>
<p>For classes of cyclic subgroups, the parentheses round the order and the type are omitted. Hence the most general form of their generic names is <code class="code">"<var class="Arg">o</var>,<var class="Arg">l</var>"</code>. Again, the letter is omitted if there is only one class of cyclic subgroups of that order.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassNamesTom( a6 );</span>
[ "1", "2", "3a", "3b", "5", "4", "(4)_2a", "(4)_2b", "(6)a", "(6)b",
"(8)", "(9)", "(10)", "(12)a", "(12)b", "(18)", "(24)a", "(24)b",
"(36)", "(60)a", "(60)b", "(360)" ]
</pre></div>
<p><a id="X86B9891C788D5107" name="X86B9891C788D5107"></a></p>
<h5>70.7-6 FusionsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionsTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a table of marks <var class="Arg">tom</var>, <code class="func">FusionsTom</code> is a list of fusions into other tables of marks. Each fusion is a list of length two, the first entry being the <code class="func">Identifier</code> (<a href="chap70.html#X810E53597B5BB4F8"><span class="RefLink">70.7-9</span></a>)) value of the image table, the second entry being the list of images of the class positions of <var class="Arg">tom</var> in the image table.</p>
<p>This attribute is mainly used for tables of marks in the <strong class="pkg">GAP</strong> library (see <a href="chap70.html#X794ABC7187A9285B"><span class="RefLink">70.13</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= FusionsTom( a6 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus[1];</span>
[ "L3(4)",
[ 1, 2, 3, 3, 14, 5, 9, 7, 15, 15, 24, 26, 27, 32, 33, 50, 57, 55,
63, 73, 77, 90 ] ]
</pre></div>
<p><a id="X81E41D3880FA6C4C" name="X81E41D3880FA6C4C"></a></p>
<h5>70.7-7 UnderlyingGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingGroup</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">UnderlyingGroup</code> is used to access an underlying group that is stored on the table of marks <var class="Arg">tom</var>. There is no default method to compute an underlying group if it is not stored.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingGroup( a6 );</span>
Group([ (1,2)(3,4), (1,2,4,5)(3,6) ])
</pre></div>
<p><a id="X817238FB79A3462F" name="X817238FB79A3462F"></a></p>
<h5>70.7-8 IdempotentsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdempotentsTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdempotentsTomInfo</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">IdempotentsTom</code> encodes the idempotents of the integral Burnside ring described by the table of marks <var class="Arg">tom</var>. The return value is a list <span class="SimpleMath">l</span> of positive integers such that each row vector describing a primitive idempotent has value <span class="SimpleMath">1</span> at all positions with the same entry in <span class="SimpleMath">l</span>, and <span class="SimpleMath">0</span> at all other positions.</p>
<p>According to A. Dress <a href="chapBib.html#biBDre69">[Dre69]</a> (see also <a href="chapBib.html#biBPfe97">[Pfe97]</a>), these idempotents correspond to the classes of perfect subgroups, and each such idempotent is the characteristic function of all those subgroups that arise by cyclic extension from the corresponding perfect subgroup (see <code class="func">CyclicExtensionsTom</code> (<a href="chap70.html#X838DE06B823C19CA"><span class="RefLink">70.9-7</span></a>)).</p>
<p><code class="func">IdempotentsTomInfo</code> returns a record with components <code class="code">fixpointvectors</code> and <code class="code">primidems</code>, both bound to lists. The <span class="SimpleMath">i</span>-th entry of the <code class="code">fixpointvectors</code> list is the <span class="SimpleMath">0-1</span>-vector describing the <span class="SimpleMath">i</span>-th primitive idempotent, and the <span class="SimpleMath">i</span>-th entry of <code class="code">primidems</code> is the decomposition of this idempotent in the rows of <var class="Arg">tom</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdempotentsTom( a5 );</span>
[ 1, 1, 1, 1, 1, 1, 1, 1, 9 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdempotentsTomInfo( a5 );</span>
rec(
fixpointvectors := [ [ 1, 1, 1, 1, 1, 1, 1, 1, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ],
primidems := [ [ 1, -2, -1, 0, 0, 1, 1, 1 ],
[ -1, 2, 1, 0, 0, -1, -1, -1, 1 ] ] )
</pre></div>
<p><a id="X810E53597B5BB4F8" name="X810E53597B5BB4F8"></a></p>
<h5>70.7-9 Identifier</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Identifier</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The identifier of a table of marks <var class="Arg">tom</var> is a string. It is used for printing the table of marks (see <a href="chap70.html#X7AC0FB9685DCBCFD"><span class="RefLink">70.4</span></a>) and in fusions between tables of marks (see <code class="func">FusionsTom</code> (<a href="chap70.html#X86B9891C788D5107"><span class="RefLink">70.7-6</span></a>)).</p>
<p>If <var class="Arg">tom</var> is a table of marks from the <strong class="pkg">GAP</strong> library of tables of marks (see <a href="chap70.html#X794ABC7187A9285B"><span class="RefLink">70.13</span></a>) then it has an identifier, and if <var class="Arg">tom</var> was constructed from a group with <code class="func">Name</code> (<a href="chap12.html#X7F14EF9D81432113"><span class="RefLink">12.8-2</span></a>) then this name is chosen as <code class="func">Identifier</code> value. There is no default method to compute an identifier in all other cases.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Identifier( a5 );</span>
"A5"
</pre></div>
<p><a id="X8463272986781E17" name="X8463272986781E17"></a></p>
<h5>70.7-10 MatTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">MatTom</code> returns the square matrix of marks (see <a href="chap70.html#X80883EC17968F442"><span class="RefLink">70.1</span></a>) of the table of marks <var class="Arg">tom</var> which is stored in a compressed form using the attributes <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) and <code class="func">SubsTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) This may need substantially more space than the values of <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>) and <code class="func">SubsTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MatTom( a5 );</span>
[ [ 60, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 30, 2, 0, 0, 0, 0, 0, 0, 0 ],
[ 20, 0, 2, 0, 0, 0, 0, 0, 0 ], [ 15, 3, 0, 3, 0, 0, 0, 0, 0 ],
[ 12, 0, 0, 0, 2, 0, 0, 0, 0 ], [ 10, 2, 1, 0, 0, 1, 0, 0, 0 ],
[ 6, 2, 0, 0, 1, 0, 1, 0, 0 ], [ 5, 1, 2, 1, 0, 0, 0, 1, 0 ],
[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ]
</pre></div>
<p><a id="X7D32C8B0786D16C1" name="X7D32C8B0786D16C1"></a></p>
<h5>70.7-11 MoebiusTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MoebiusTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">MoebiusTom</code> computes the Möbius values both of the subgroup lattice of the group <span class="SimpleMath">G</span> with table of marks <var class="Arg">tom</var> and of the poset of conjugacy classes of subgroups of <span class="SimpleMath">G</span>. It returns a record where the component <code class="code">mu</code> contains the Möbius values of the subgroup lattice, and the component <code class="code">nu</code> contains the Möbius values of the poset.</p>
<p>Moreover, according to an observation of Isaacs et al. (see <a href="chapBib.html#biBHIO89">[HIÖ89]</a>, <a href="chapBib.html#biBPah93">[Pah93]</a>), the values on the subgroup lattice often can be derived from those of the poset of conjugacy classes. These "expected values" are returned in the component <code class="code">ex</code>, and the list of numbers of those subgroups where the expected value does not coincide with the actual value are returned in the component <code class="code">hyp</code>. For the computation of these values, the position of the derived subgroup of <span class="SimpleMath">G</span> is needed (see <code class="func">DerivedSubgroupTom</code> (<a href="chap70.html#X8528D9397FFAF477"><span class="RefLink">70.9-2</span></a>)). If it is not uniquely determined then the result does not have the components <code class="code">ex</code> and <code class="code">hyp</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MoebiusTom( a5 );</span>
rec( ex := [ -60, 4, 2,,, -1, -1, -1, 1 ], hyp := [ ],
mu := [ -60, 4, 2,,, -1, -1, -1, 1 ],
nu := [ -1, 2, 1,,, -1, -1, -1, 1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">tom:= TableOfMarks( "M12" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">moebius:= MoebiusTom( tom );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">moebius.hyp;</span>
[ 1, 2, 4, 16, 39, 45, 105 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">moebius.mu[1]; moebius.ex[1];</span>
95040
190080
</pre></div>
<p><a id="X78525D04849A48EA" name="X78525D04849A48EA"></a></p>
<h5>70.7-12 WeightsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WeightsTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">WeightsTom</code> extracts the <em>weights</em> from the table of marks <var class="Arg">tom</var>, i.e., the diagonal entries of the matrix of marks (see <code class="func">MarksTom</code> (<a href="chap70.html#X78F486A28561D006"><span class="RefLink">70.7-1</span></a>)), indicating the index of a subgroup in its normalizer.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">wt:= WeightsTom( a5 );</span>
[ 60, 2, 2, 3, 2, 1, 1, 1, 1 ]
</pre></div>
<p>This information may be used to obtain the numbers of conjugate supergroups from the marks.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">marks:= MarksTom( a5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( [ 1 .. 9 ], x -> marks[x] / wt[x] );</span>
[ [ 1 ], [ 15, 1 ], [ 10, 1 ], [ 5, 1, 1 ], [ 6, 1 ], [ 10, 2, 1, 1 ],
[ 6, 2, 1, 1 ], [ 5, 1, 2, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ]
</pre></div>
<p><a id="X78A1B2E4826A9518" name="X78A1B2E4826A9518"></a></p>
<h4>70.8 <span class="Heading">Properties of Tables of Marks</span></h4>
<p>For a table of marks <var class="Arg">tom</var> of a group <span class="SimpleMath">G</span>, the following properties have the same meaning as the corresponding properties for <span class="SimpleMath">G</span>. Additionally, if a positive integer <var class="Arg">sub</var> is given as the second argument then the value of the corresponding property for the <var class="Arg">sub</var>-th class of subgroups of <var class="Arg">tom</var> is returned.</p>
<p><a id="X7C93BAEC78B7C2B4" name="X7C93BAEC78B7C2B4"></a></p>
<h5>70.8-1 IsAbelianTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAbelianTom</code>( <var class="Arg">tom</var>[, <var class="Arg">sub</var>] )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCyclicTom</code>( <var class="Arg">tom</var>[, <var class="Arg">sub</var>] )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNilpotentTom</code>( <var class="Arg">tom</var>[, <var class="Arg">sub</var>] )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPerfectTom</code>( <var class="Arg">tom</var>[, <var class="Arg">sub</var>] )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSolvableTom</code>( <var class="Arg">tom</var>[, <var class="Arg">sub</var>] )</td><td class="tdright">( property )</td></tr></table></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tom:= TableOfMarks( "A5" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAbelianTom( tom ); IsPerfectTom( tom );</span>
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAbelianTom( tom, 3 ); IsNilpotentTom( tom, 7 );</span>
true
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPerfectTom( tom, 7 ); IsSolvableTom( tom, 7 );</span>
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [ 1 .. 6 ] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print( i, ": ", IsCyclicTom(a5, i), " " );</span>
<span class="GAPprompt">></span> <span class="GAPinput">od; Print( "\n" );</span>
1: true 2: true 3: true 4: false 5: true 6: false
</pre></div>
<p><a id="X7A40D99D7816F126" name="X7A40D99D7816F126"></a></p>
<h4>70.9 <span class="Heading">Other Operations for Tables of Marks</span></h4>
<p><a id="X7D8B4BE08094B137" name="X7D8B4BE08094B137"></a></p>
<h5>70.9-1 IsInternallyConsistent</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInternallyConsistent</code>( <var class="Arg">tom</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>For a table of marks <var class="Arg">tom</var>, <code class="func">IsInternallyConsistent</code> decomposes all tensor products of rows of <var class="Arg">tom</var>. It returns <code class="keyw">true</code> if all decomposition numbers are nonnegative integers, and <code class="keyw">false</code> otherwise. This provides a strong consistency check for a table of marks.</p>
<p><a id="X8528D9397FFAF477" name="X8528D9397FFAF477"></a></p>
<h5>70.9-2 DerivedSubgroupTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivedSubgroupTom</code>( <var class="Arg">tom</var>, <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivedSubgroupsTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a table of marks <var class="Arg">tom</var> and a positive integer <var class="Arg">sub</var>, <code class="func">DerivedSubgroupTom</code> returns either a positive integer <span class="SimpleMath">i</span> or a list <span class="SimpleMath">l</span> of positive integers. In the former case, the result means that the derived subgroups of the subgroups in the <var class="Arg">sub</var>-th class of <var class="Arg">tom</var> lie in the <span class="SimpleMath">i</span>-th class. In the latter case, the class of the derived subgroups could not be uniquely determined, and the position of the class of derived subgroups is an entry of <span class="SimpleMath">l</span>.</p>
<p>Values computed with <code class="func">DerivedSubgroupTom</code> are stored using the attribute <code class="func">DerivedSubgroupsTomPossible</code> (<a href="chap70.html#X7C29BD438127DFBE"><span class="RefLink">70.9-3</span></a>).</p>
<p><code class="func">DerivedSubgroupsTom</code> is just the list of <code class="func">DerivedSubgroupTom</code> values for all values of <var class="Arg">sub</var>.</p>
<p><a id="X7C29BD438127DFBE" name="X7C29BD438127DFBE"></a></p>
<h5>70.9-3 DerivedSubgroupsTomPossible</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivedSubgroupsTomPossible</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivedSubgroupsTomUnique</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">tom</var> be a table of marks. The value of the attribute <code class="func">DerivedSubgroupsTomPossible</code> is a list in which the value at position <span class="SimpleMath">i</span> –if bound– is a positive integer or a list; the meaning of the entry is the same as in <code class="func">DerivedSubgroupTom</code> (<a href="chap70.html#X8528D9397FFAF477"><span class="RefLink">70.9-2</span></a>).</p>
<p>If the value of the attribute <code class="func">DerivedSubgroupsTomUnique</code> is known for <var class="Arg">tom</var> then it is a list of positive integers, the value at position <span class="SimpleMath">i</span> being the position of the class of derived subgroups of the <span class="SimpleMath">i</span>-th class of subgroups in <var class="Arg">tom</var>.</p>
<p>The derived subgroups are in general not uniquely determined by the table of marks if no <code class="func">UnderlyingGroup</code> (<a href="chap70.html#X81E41D3880FA6C4C"><span class="RefLink">70.7-7</span></a>) value is stored, so there is no default method for <code class="func">DerivedSubgroupsTomUnique</code>. But in some cases the derived subgroups are explicitly set when the table of marks is constructed. In this case, <code class="func">DerivedSubgroupTom</code> (<a href="chap70.html#X8528D9397FFAF477"><span class="RefLink">70.9-2</span></a>) does not set values in the <code class="func">DerivedSubgroupsTomPossible</code> list.</p>
<p>The <code class="func">DerivedSubgroupsTomUnique</code> value is automatically set when the last missing unique value is entered in the <code class="func">DerivedSubgroupsTomPossible</code> list by <code class="func">DerivedSubgroupTom</code> (<a href="chap70.html#X8528D9397FFAF477"><span class="RefLink">70.9-2</span></a>).</p>
<p><a id="X7CE6C45881F7F7D4" name="X7CE6C45881F7F7D4"></a></p>
<h5>70.9-4 NormalizerTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizerTom</code>( <var class="Arg">tom</var>, <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizersTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">tom</var> be the table of marks of a group <span class="SimpleMath">G</span>, say. <code class="func">NormalizerTom</code> tries to find the conjugacy class of the normalizer <span class="SimpleMath">N</span> in <span class="SimpleMath">G</span> of a subgroup <span class="SimpleMath">U</span> in the <var class="Arg">sub</var>-th class of <var class="Arg">tom</var>. The return value is either the list of class numbers of those subgroups that have the right size and contain the subgroup and all subgroups that clearly contain it as a normal subgroup, or the class number of the normalizer if it is uniquely determined by these conditions. If <var class="Arg">tom</var> knows the subgroup lattice of <span class="SimpleMath">G</span> (see <code class="func">IsTableOfMarksWithGens</code> (<a href="chap70.html#X7889DB6D790593B9"><span class="RefLink">70.10-3</span></a>)) then all normalizers are uniquely determined. <code class="func">NormalizerTom</code> should never return an empty list.</p>
<p><code class="func">NormalizersTom</code> returns the list of positions of the classes of normalizers of subgroups in <var class="Arg">tom</var>. In addition to the criteria for a single class of subgroup used by <code class="func">NormalizerTom</code>, the approximations of normalizers for several classes are used and thus <code class="func">NormalizersTom</code> may return better approximations than <code class="func">NormalizerTom</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalizerTom( a5, 4 );</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalizersTom( a5 );</span>
[ 9, 4, 6, 8, 7, 6, 7, 8, 9 ]
</pre></div>
<p>The example shows that a subgroup with class number 4 in <span class="SimpleMath">A_5</span> (which is a Kleinian four group) is normalized by a subgroup in class 8. This class contains the subgroups of <span class="SimpleMath">A_5</span> which are isomorphic to <span class="SimpleMath">A_4</span>.</p>
<p><a id="X7F87B2797827E5DE" name="X7F87B2797827E5DE"></a></p>
<h5>70.9-5 ContainedTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ContainedTom</code>( <var class="Arg">tom</var>, <var class="Arg">sub1</var>, <var class="Arg">sub2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">ContainedTom</code> returns the number of subgroups in class <var class="Arg">sub1</var> of the table of marks <var class="Arg">tom</var> that are contained in one fixed member of the class <var class="Arg">sub2</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ContainedTom( a5, 3, 5 ); ContainedTom( a5, 3, 8 );</span>
0
4
</pre></div>
<p><a id="X7EE050FB87D6F0E7" name="X7EE050FB87D6F0E7"></a></p>
<h5>70.9-6 ContainingTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ContainingTom</code>( <var class="Arg">tom</var>, <var class="Arg">sub1</var>, <var class="Arg">sub2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">ContainingTom</code> returns the number of subgroups in class <var class="Arg">sub2</var> of the table of marks <var class="Arg">tom</var> that contain one fixed member of the class <var class="Arg">sub1</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ContainingTom( a5, 3, 5 ); ContainingTom( a5, 3, 8 );</span>
0
2
</pre></div>
<p><a id="X838DE06B823C19CA" name="X838DE06B823C19CA"></a></p>
<h5>70.9-7 CyclicExtensionsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclicExtensionsTom</code>( <var class="Arg">tom</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclicExtensionsTom</code>( <var class="Arg">tom</var>[, <var class="Arg">list</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>According to A. Dress <a href="chapBib.html#biBDre69">[Dre69]</a>, two columns of the table of marks <var class="Arg">tom</var> are equal modulo the prime <var class="Arg">p</var> if and only if the corresponding subgroups are connected by a chain of normal extensions of order <var class="Arg">p</var>.</p>
<p>Called with <var class="Arg">tom</var> and <var class="Arg">p</var>, <code class="func">CyclicExtensionsTom</code> returns the classes of this equivalence relation.</p>
<p>In the second form, <var class="Arg">list</var> must be a list of primes, and the return value is the list of classes of the relation obtained by considering chains of normal extensions of prime order where all primes are in <var class="Arg">list</var>. The default value for <var class="Arg">list</var> is the set of prime divisors of the order of the group of <var class="Arg">tom</var>.</p>
<p>(This information is <em>not</em> used by <code class="func">NormalizerTom</code> (<a href="chap70.html#X7CE6C45881F7F7D4"><span class="RefLink">70.9-4</span></a>) although it might give additional restrictions in the search of normalizers.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclicExtensionsTom( a5, 2 );</span>
[ [ 1, 2, 4 ], [ 3, 6 ], [ 5, 7 ], [ 8 ], [ 9 ] ]
</pre></div>
<p><a id="X80890C247EB1E35C" name="X80890C247EB1E35C"></a></p>
<h5>70.9-8 DecomposedFixedPointVector</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DecomposedFixedPointVector</code>( <var class="Arg">tom</var>, <var class="Arg">fix</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">tom</var> be the table of marks of the group <span class="SimpleMath">G</span>, say, and let <var class="Arg">fix</var> be a vector of fixed point numbers w.r.t. an action of <span class="SimpleMath">G</span>, i.e., a vector which contains for each class of subgroups the number of fixed points under the given action. <code class="func">DecomposedFixedPointVector</code> returns the decomposition of <var class="Arg">fix</var> into rows of the table of marks. This decomposition corresponds to a decomposition of the action into transitive constituents. Trailing zeros in <var class="Arg">fix</var> may be omitted.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DecomposedFixedPointVector( a5, [ 16, 4, 1, 0, 1, 1, 1 ] );</span>
[ 0, 0, 0, 0, 0, 1, 1 ]
</pre></div>
<p>The vector <var class="Arg">fix</var> may be any vector of integers. The resulting decomposition, however, will not be integral, in general.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DecomposedFixedPointVector( a5, [ 0, 0, 0, 0, 1, 1 ] );</span>
[ 2/5, -1, -1/2, 0, 1/2, 1 ]
</pre></div>
<p><a id="X7B1C1A7C867A4082" name="X7B1C1A7C867A4082"></a></p>
<h5>70.9-9 EulerianFunctionByTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EulerianFunctionByTom</code>( <var class="Arg">tom</var>, <var class="Arg">n</var>[, <var class="Arg">sub</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Called with two arguments, <code class="func">EulerianFunctionByTom</code> computes the Eulerian function (see <code class="func">EulerianFunction</code> (<a href="chap39.html#X843E0CCA8351FDF4"><span class="RefLink">39.16-3</span></a>)) of the underlying group <span class="SimpleMath">G</span> of the table of marks <var class="Arg">tom</var>, that is, the number of <var class="Arg">n</var>-tuples of elements in <span class="SimpleMath">G</span> that generate <span class="SimpleMath">G</span>. If the optional argument <var class="Arg">sub</var> is given then <code class="func">EulerianFunctionByTom</code> computes the Eulerian function of each subgroup in the <var class="Arg">sub</var>-th class of subgroups of <var class="Arg">tom</var>.</p>
<p>For a group <span class="SimpleMath">G</span> whose table of marks is known, <code class="func">EulerianFunctionByTom</code> is installed as a method for <code class="func">EulerianFunction</code> (<a href="chap39.html#X843E0CCA8351FDF4"><span class="RefLink">39.16-3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">EulerianFunctionByTom( a5, 2 );</span>
2280
<span class="GAPprompt">gap></span> <span class="GAPinput">EulerianFunctionByTom( a5, 3 );</span>
200160
<span class="GAPprompt">gap></span> <span class="GAPinput">EulerianFunctionByTom( a5, 2, 3 );</span>
8
</pre></div>
<p><a id="X8224E51382FDB912" name="X8224E51382FDB912"></a></p>
<h5>70.9-10 IntersectionsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntersectionsTom</code>( <var class="Arg">tom</var>, <var class="Arg">sub1</var>, <var class="Arg">sub2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The intersections of the groups in the <var class="Arg">sub1</var>-th conjugacy class of subgroups of the table of marks <var class="Arg">tom</var> with the groups in the <var class="Arg">sub2</var>-th conjugacy classes of subgroups of <var class="Arg">tom</var> are determined up to conjugacy by the decomposition of the tensor product of their rows of marks. <code class="func">IntersectionsTom</code> returns a list <span class="SimpleMath">l</span> that describes this decomposition. The <span class="SimpleMath">i</span>-th entry in <span class="SimpleMath">l</span> is the multiplicity of groups in the <span class="SimpleMath">i</span>-th conjugacy class as an intersection.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IntersectionsTom( a5, 8, 8 );</span>
[ 0, 0, 1, 0, 0, 0, 0, 1 ]
</pre></div>
<p>Any two subgroups of class number 8 (<span class="SimpleMath">A_4</span>) of <span class="SimpleMath">A_5</span> are either equal and their intersection has again class number 8, or their intersection has class number <span class="SimpleMath">3</span>, and is a cyclic subgroup of order 3.</p>
<p><a id="X859F069C8428B598" name="X859F069C8428B598"></a></p>
<h5>70.9-11 FactorGroupTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FactorGroupTom</code>( <var class="Arg">tom</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a table of marks <var class="Arg">tom</var> of the group <span class="SimpleMath">G</span>, say, and the normal subgroup <span class="SimpleMath">N</span> of <span class="SimpleMath">G</span> corresponding to the <var class="Arg">n</var>-th class of subgroups of <var class="Arg">tom</var>, <code class="func">FactorGroupTom</code> returns the table of marks of the factor group <span class="SimpleMath">G / N</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s4:= TableOfMarks( SymmetricGroup( 4 ) );</span>
TableOfMarks( Sym( [ 1 .. 4 ] ) )
<span class="GAPprompt">gap></span> <span class="GAPinput">LengthsTom( s4 );</span>
[ 1, 3, 6, 4, 1, 3, 3, 4, 3, 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersTom( s4 );</span>
[ 1, 2, 2, 3, 4, 4, 4, 6, 8, 12, 24 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">s3:= FactorGroupTom( s4, 5 );</span>
TableOfMarks( Group([ f1, f2 ]) )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( s3 );</span>
1: 6
2: 3 1
3: 2 . 2
4: 1 1 1 1
</pre></div>
<p><a id="X8325811586C00ECF" name="X8325811586C00ECF"></a></p>
<h5>70.9-12 MaximalSubgroupsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalSubgroupsTom</code>( <var class="Arg">tom</var>[, <var class="Arg">sub</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Called with a table of marks <var class="Arg">tom</var>, <code class="func">MaximalSubgroupsTom</code> returns a list of length two, the first entry being the list of positions of the classes of maximal subgroups of the whole group of <var class="Arg">tom</var>, the second entry being the list of class lengths of these groups.</p>
<p>Called with a table of marks <var class="Arg">tom</var> and a position <var class="Arg">sub</var>, the same information for the <var class="Arg">sub</var>-th class of subgroups is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalSubgroupsTom( s4 );</span>
[ [ 10, 9, 8 ], [ 1, 3, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalSubgroupsTom( s4, 10 );</span>
[ [ 5, 4 ], [ 1, 4 ] ]
</pre></div>
<p><a id="X7923B19D7C47BF63" name="X7923B19D7C47BF63"></a></p>
<h5>70.9-13 MinimalSupergroupsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalSupergroupsTom</code>( <var class="Arg">tom</var>, <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a table of marks <var class="Arg">tom</var>, <code class="func">MinimalSupergroupsTom</code> returns a list of length two, the first entry being the list of positions of the classes containing the minimal supergroups of the groups in the <var class="Arg">sub</var>-th class of subgroups of <var class="Arg">tom</var>, the second entry being the list of class lengths of these groups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalSupergroupsTom( s4, 5 );</span>
[ [ 9, 10 ], [ 3, 1 ] ]
</pre></div>
<p><a id="X7FE9BE477A90199F" name="X7FE9BE477A90199F"></a></p>
<h4>70.10 <span class="Heading">Accessing Subgroups via Tables of Marks</span></h4>
<p>Let <var class="Arg">tom</var> be the table of marks of the group <span class="SimpleMath">G</span>, and assume that <var class="Arg">tom</var> has access to <span class="SimpleMath">G</span> via the <code class="func">UnderlyingGroup</code> (<a href="chap70.html#X81E41D3880FA6C4C"><span class="RefLink">70.7-7</span></a>) value. Then it makes sense to use <var class="Arg">tom</var> and its ordering of conjugacy classes of subgroups of <span class="SimpleMath">G</span> for storing information for constructing representatives of these classes. The group <span class="SimpleMath">G</span> is in general not sufficient for this, <var class="Arg">tom</var> needs more information; this is available if and only if the <code class="func">IsTableOfMarksWithGens</code> (<a href="chap70.html#X7889DB6D790593B9"><span class="RefLink">70.10-3</span></a>) value of <var class="Arg">tom</var> is <code class="keyw">true</code>. In this case, <code class="func">RepresentativeTom</code> (<a href="chap70.html#X7F625AB880B73AC3"><span class="RefLink">70.10-4</span></a>) can be used to get a subgroup of the <span class="SimpleMath">i</span>-th class, for all <span class="SimpleMath">i</span>.</p>
<p><strong class="pkg">GAP</strong> provides two different possibilities to store generators of the representatives of classes of subgroups. The first is implemented by the attribute <code class="func">GeneratorsSubgroupsTom</code> (<a href="chap70.html#X7B0B6FDD806E9734"><span class="RefLink">70.10-1</span></a>), which uses explicit generators of the subgroups. The second, more general, possibility is implemented by the attribute <code class="func">StraightLineProgramsTom</code> (<a href="chap70.html#X7898BE7284E47FF3"><span class="RefLink">70.10-2</span></a>), which encodes the generators as straight line programs (see <a href="chap37.html#X7DC99E4284093FBB"><span class="RefLink">37.8</span></a>) that evaluate to the generators in question when applied to <em>standard generators</em> of <span class="SimpleMath">G</span>. This means that on the one hand, standard generators of <span class="SimpleMath">G</span> must be known in order to use <code class="func">StraightLineProgramsTom</code> (<a href="chap70.html#X7898BE7284E47FF3"><span class="RefLink">70.10-2</span></a>). On the other hand, the straight line programs allow one to compute easily generators not only of a subgroup <span class="SimpleMath">U</span> of <span class="SimpleMath">G</span> but also generators of the image of <span class="SimpleMath">U</span> in any representation of <span class="SimpleMath">G</span>, provided that one knows standard generators of the image of <span class="SimpleMath">G</span> under this representation. See the manual of the package <strong class="pkg">TomLib</strong> for details and an example.</p>
<p><a id="X7B0B6FDD806E9734" name="X7B0B6FDD806E9734"></a></p>
<h5>70.10-1 GeneratorsSubgroupsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsSubgroupsTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">tom</var> be a table of marks with <code class="func">IsTableOfMarksWithGens</code> (<a href="chap70.html#X7889DB6D790593B9"><span class="RefLink">70.10-3</span></a>) value <code class="keyw">true</code>. Then <code class="func">GeneratorsSubgroupsTom</code> returns a list of length two, the first entry being a list <span class="SimpleMath">l</span> of elements of the group stored as <code class="func">UnderlyingGroup</code> (<a href="chap70.html#X81E41D3880FA6C4C"><span class="RefLink">70.7-7</span></a>) value of <var class="Arg">tom</var>, the second entry being a list that contains at position <span class="SimpleMath">i</span> a list of positions in <span class="SimpleMath">l</span> of generators of a representative of a subgroup in class <span class="SimpleMath">i</span>.</p>
<p>The <code class="func">GeneratorsSubgroupsTom</code> value is known for all tables of marks that have been computed with <code class="func">TableOfMarks</code> (<a href="chap70.html#X85B262AB7E219C34"><span class="RefLink">70.3-1</span></a>) from a group, and there is a method to compute the value for a table of marks that admits <code class="func">RepresentativeTom</code> (<a href="chap70.html#X7F625AB880B73AC3"><span class="RefLink">70.10-4</span></a>).</p>
<p><a id="X7898BE7284E47FF3" name="X7898BE7284E47FF3"></a></p>
<h5>70.10-2 StraightLineProgramsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StraightLineProgramsTom</code>( <var class="Arg">tom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a table of marks <var class="Arg">tom</var> with <code class="func">IsTableOfMarksWithGens</code> (<a href="chap70.html#X7889DB6D790593B9"><span class="RefLink">70.10-3</span></a>) value <code class="keyw">true</code>, <code class="func">StraightLineProgramsTom</code> returns a list that contains at position <span class="SimpleMath">i</span> either a list of straight line programs or a straight line program (see <a href="chap37.html#X7DC99E4284093FBB"><span class="RefLink">37.8</span></a>), encoding the generators of a representative of the <span class="SimpleMath">i</span>-th conjugacy class of subgroups of <code class="code">UnderlyingGroup( <var class="Arg">tom</var> )</code>; in the former case, each straight line program returns a generator, in the latter case, the program returns the list of generators.</p>
<p>There is no default method to compute the <code class="func">StraightLineProgramsTom</code> value of a table of marks if they are not yet stored. The value is known for all tables of marks that belong to the <strong class="pkg">GAP</strong> library of tables of marks (see <a href="chap70.html#X794ABC7187A9285B"><span class="RefLink">70.13</span></a>).</p>
<p><a id="X7889DB6D790593B9" name="X7889DB6D790593B9"></a></p>
<h5>70.10-3 IsTableOfMarksWithGens</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTableOfMarksWithGens</code>( <var class="Arg">tom</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This filter shall express the union of the filters <code class="code">IsTableOfMarks and HasStraightLineProgramsTom</code> and <code class="code">IsTableOfMarks and HasGeneratorsSubgroupsTom</code>. If a table of marks <var class="Arg">tom</var> has this filter set then <var class="Arg">tom</var> can be asked to compute information that is in general not uniquely determined by a table of marks, for example the positions of derived subgroups or normalizers of subgroups (see <code class="func">DerivedSubgroupTom</code> (<a href="chap70.html#X8528D9397FFAF477"><span class="RefLink">70.9-2</span></a>), <code class="func">NormalizerTom</code> (<a href="chap70.html#X7CE6C45881F7F7D4"><span class="RefLink">70.9-4</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a5:= TableOfMarks( "A5" );; IsTableOfMarksWithGens( a5 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">HasGeneratorsSubgroupsTom( a5 ); HasStraightLineProgramsTom( a5 );</span>
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">alt5:= TableOfMarks( AlternatingGroup( 5 ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTableOfMarksWithGens( alt5 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">HasGeneratorsSubgroupsTom(alt5); HasStraightLineProgramsTom(alt5);</span>
true
false
<span class="GAPprompt">gap></span> <span class="GAPinput">progs:= StraightLineProgramsTom( a5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersTom( a5 );</span>
[ 1, 2, 3, 4, 5, 6, 10, 12, 60 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclicTom( a5, 4 );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( progs[4] );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">progs[4][1];</span>
<straight line program>
<span class="GAPprompt">gap></span> <span class="GAPinput"># first generator of an el. ab group of order 4:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( progs[4][1] );</span>
# input:
r:= [ g1, g2 ];
# program:
r[3]:= r[2]*r[1];
r[4]:= r[3]*r[2]^-1*r[1]*r[3]*r[2]^-1*r[1]*r[2];
# return value:
r[4]
<span class="GAPprompt">gap></span> <span class="GAPinput">x:= [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">y:= [ [ Z(2^2), Z(2)^0 ], [ 0*Z(2), Z(2^2)^2 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">res1:= ResultOfStraightLineProgram( progs[4][1], [ x, y ] );</span>
[ [ Z(2)^0, 0*Z(2) ], [ Z(2^2)^2, Z(2)^0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">res2:= ResultOfStraightLineProgram( progs[4][2], [ x, y ] );</span>
[ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">w:= y*x;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">res1 = w*y^-1*x*w*y^-1*x*y;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">subgrp:= Group( res1, res2 );; Size( subgrp ); IsCyclic( subgrp );</span>
4
false
</pre></div>
<p><a id="X7F625AB880B73AC3" name="X7F625AB880B73AC3"></a></p>
<h5>70.10-4 RepresentativeTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativeTom</code>( <var class="Arg">tom</var>, <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativeTomByGenerators</code>( <var class="Arg">tom</var>, <var class="Arg">sub</var>, <var class="Arg">gens</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativeTomByGeneratorsNC</code>( <var class="Arg">tom</var>, <var class="Arg">sub</var>, <var class="Arg">gens</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">tom</var> be a table of marks with <code class="func">IsTableOfMarksWithGens</code> (<a href="chap70.html#X7889DB6D790593B9"><span class="RefLink">70.10-3</span></a>) value <code class="keyw">true</code>, and <var class="Arg">sub</var> a positive integer. <code class="func">RepresentativeTom</code> returns a representative of the <var class="Arg">sub</var>-th conjugacy class of subgroups of <var class="Arg">tom</var>.</p>
<p>If the attribute <code class="func">StraightLineProgramsTom</code> (<a href="chap70.html#X7898BE7284E47FF3"><span class="RefLink">70.10-2</span></a>) is set in <var class="Arg">tom</var> then methods for the operations <code class="func">RepresentativeTomByGenerators</code> and <code class="func">RepresentativeTomByGeneratorsNC</code> are available, which return a representative of the <var class="Arg">sub</var>-th conjugacy class of subgroups of <var class="Arg">tom</var>, as a subgroup of the group generated by <var class="Arg">gens</var>. This means that the standard generators of <var class="Arg">tom</var> are replaced by <var class="Arg">gens</var>.</p>
<p><code class="func">RepresentativeTomByGenerators</code> checks whether mapping the standard generators of <var class="Arg">tom</var> to <var class="Arg">gens</var> extends to a group isomorphism, and returns <code class="keyw">fail</code> if not. <code class="func">RepresentativeTomByGeneratorsNC</code> omits all checks. So <code class="func">RepresentativeTomByGenerators</code> is thought mainly for debugging purposes; note that when several representatives are constructed, it is cheaper to construct (and check) the isomorphism once, and to map the groups returned by <code class="func">RepresentativeTom</code> under this isomorphism. The idea behind <code class="func">RepresentativeTomByGeneratorsNC</code>, however, is to avoid the overhead of using isomorphisms when <var class="Arg">gens</var> are known to be standard generators. In order to proceed like this, the attribute <code class="func">StraightLineProgramsTom</code> (<a href="chap70.html#X7898BE7284E47FF3"><span class="RefLink">70.10-2</span></a>) is needed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeTom( a5, 4 );</span>
Group([ (2,3)(4,5), (2,4)(3,5) ])
</pre></div>
<p><a id="X79ADA60880BE9C49" name="X79ADA60880BE9C49"></a></p>
<h4>70.11 <span class="Heading">The Interface between Tables of Marks and Character Tables</span></h4>
<p>The following examples require the <strong class="pkg">GAP</strong> Character Table Library to be available. If it is not yet loaded then we load it now.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "ctbllib" );</span>
true
</pre></div>
<p><a id="X7A82CB487DBDDC53" name="X7A82CB487DBDDC53"></a></p>
<h5>70.11-1 FusionCharTableTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionCharTableTom</code>( <var class="Arg">tbl</var>, <var class="Arg">tom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PossibleFusionsCharTableTom</code>( <var class="Arg">tbl</var>, <var class="Arg">tom</var>[, <var class="Arg">options</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">tbl</var> be the ordinary character table of the group <span class="SimpleMath">G</span>, say, and <var class="Arg">tom</var> the table of marks of <span class="SimpleMath">G</span>. <code class="func">FusionCharTableTom</code> determines the fusion of the classes of elements from <var class="Arg">tbl</var> to the classes of cyclic subgroups on <var class="Arg">tom</var>, that is, a list that contains at position <span class="SimpleMath">i</span> the position of the class of cyclic subgroups in <var class="Arg">tom</var> that are generated by elements in the <span class="SimpleMath">i</span>-th conjugacy class of elements in <var class="Arg">tbl</var>.</p>
<p>Three cases are handled differently.</p>
<ol>
<li><p>The fusion is explicitly stored on <var class="Arg">tbl</var>. Then nothing has to be done. This happens only if both <var class="Arg">tbl</var> and <var class="Arg">tom</var> are tables from the <strong class="pkg">GAP</strong> library (see <a href="chap70.html#X794ABC7187A9285B"><span class="RefLink">70.13</span></a> and the manual of the <strong class="pkg">GAP</strong> Character Table Library).</p>
</li>
<li><p>The <code class="func">UnderlyingGroup</code> (<a href="chap70.html#X81E41D3880FA6C4C"><span class="RefLink">70.7-7</span></a>) values of <var class="Arg">tbl</var> and <var class="Arg">tom</var> are known and equal. Then the group is used to compute the fusion.</p>
</li>
<li><p>There is neither fusion nor group information available. In this case only necessary conditions can be checked, and if they are not sufficient to determine the fusion uniquely then <code class="keyw">fail</code> is returned by <code class="func">FusionCharTableTom</code>.</p>
</li>
</ol>
<p><code class="func">PossibleFusionsCharTableTom</code> computes the list of possible fusions from <var class="Arg">tbl</var> to <var class="Arg">tom</var>, according to the criteria that have been checked. So if <code class="func">FusionCharTableTom</code> returns a unique fusion then the list returned by <code class="func">PossibleFusionsCharTableTom</code> for the same arguments contains exactly this fusion, and if <code class="func">FusionCharTableTom</code> returns <code class="keyw">fail</code> then the length of this list is different from <span class="SimpleMath">1</span>.</p>
<p>The optional argument <var class="Arg">options</var> must be a record that may have the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">fusionmap</code></strong></dt>
<dd><p>a parametrized map which is an approximation of the desired map,</p>
</dd>
<dt><strong class="Mark"><code class="code">quick</code></strong></dt>
<dd><p>a Boolean; if <code class="keyw">true</code> then as soon as only one possibility remains this possibility is returned immediately; the default value is <code class="keyw">false</code>.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a5c:= CharacterTable( "A5" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= FusionCharTableTom( a5c, a5 );</span>
[ 1, 2, 3, 5, 5 ]
</pre></div>
<p><a id="X8016499282F0BA37" name="X8016499282F0BA37"></a></p>
<h5>70.11-2 PermCharsTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PermCharsTom</code>( <var class="Arg">fus</var>, <var class="Arg">tom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PermCharsTom</code>( <var class="Arg">tbl</var>, <var class="Arg">tom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">PermCharsTom</code> returns the list of transitive permutation characters from the table of marks <var class="Arg">tom</var>. In the first form, <var class="Arg">fus</var> must be the fusion map from the ordinary character table of the group of <var class="Arg">tom</var> to <var class="Arg">tom</var> (see <code class="func">FusionCharTableTom</code> (<a href="chap70.html#X7A82CB487DBDDC53"><span class="RefLink">70.11-1</span></a>)). In the second form, <var class="Arg">tbl</var> must be the character table of the group of which <var class="Arg">tom</var> is the table of marks. If the fusion map is not uniquely determined (see <code class="func">FusionCharTableTom</code> (<a href="chap70.html#X7A82CB487DBDDC53"><span class="RefLink">70.11-1</span></a>)) then <code class="keyw">fail</code> is returned.</p>
<p>If the fusion map <var class="Arg">fus</var> is given as first argument then each transitive permutation character is represented by its values list. If the character table <var class="Arg">tbl</var> is given then the permutation characters are class function objects (see Chapter <a href="chap72.html#X7C91D0D17850E564"><span class="RefLink">72</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PermCharsTom( a5c, a5 );</span>
[ Character( CharacterTable( "A5" ), [ 60, 0, 0, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 30, 2, 0, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 15, 3, 0, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 12, 0, 0, 2, 2 ] ),
Character( CharacterTable( "A5" ), [ 10, 2, 1, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 6, 2, 0, 1, 1 ] ),
Character( CharacterTable( "A5" ), [ 5, 1, 2, 0, 0 ] ),
Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PermCharsTom( fus, a5 )[1];</span>
[ 60, 0, 0, 0, 0 ]
</pre></div>
<p><a id="X7CF66FAE7A8858E4" name="X7CF66FAE7A8858E4"></a></p>
<h4>70.12 <span class="Heading">Generic Construction of Tables of Marks</span></h4>
<p>The following three operations construct a table of marks only from the data given, i.e., without underlying group.</p>
<p><a id="X7CAA5B6C85CB9A8D" name="X7CAA5B6C85CB9A8D"></a></p>
<h5>70.12-1 TableOfMarksCyclic</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarksCyclic</code>( <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">TableOfMarksCyclic</code> returns the table of marks of the cyclic group of order <var class="Arg">n</var>.</p>
<p>A cyclic group of order <var class="Arg">n</var> has as its subgroups for each divisor <span class="SimpleMath">d</span> of <var class="Arg">n</var> a cyclic subgroup of order <span class="SimpleMath">d</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( TableOfMarksCyclic( 6 ) );</span>
1: 6
2: 3 3
3: 2 . 2
4: 1 1 1 1
</pre></div>
<p><a id="X7AADB47B8079C99E" name="X7AADB47B8079C99E"></a></p>
<h5>70.12-2 TableOfMarksDihedral</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarksDihedral</code>( <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">TableOfMarksDihedral</code> returns the table of marks of the dihedral group of order <var class="Arg">m</var>.</p>
<p>For each divisor <span class="SimpleMath">d</span> of <var class="Arg">m</var>, a dihedral group of order <span class="SimpleMath">m = 2n</span> contains subgroups of order <span class="SimpleMath">d</span> according to the following rule. If <span class="SimpleMath">d</span> is odd and divides <span class="SimpleMath">n</span> then there is only one cyclic subgroup of order <span class="SimpleMath">d</span>. If <span class="SimpleMath">d</span> is even and divides <span class="SimpleMath">n</span> then there are a cyclic subgroup of order <span class="SimpleMath">d</span> and two classes of dihedral subgroups of order <span class="SimpleMath">d</span> (which are cyclic, too, in the case <span class="SimpleMath">d = 2</span>, see the example below). Otherwise (i.e., if <span class="SimpleMath">d</span> does not divide <span class="SimpleMath">n</span>) there is just one class of dihedral subgroups of order <span class="SimpleMath">d</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( TableOfMarksDihedral( 12 ) );</span>
1: 12
2: 6 6
3: 6 . 2
4: 6 . . 2
5: 4 . . . 4
6: 3 3 1 1 . 1
7: 2 2 . . 2 . 2
8: 2 . 2 . 2 . . 2
9: 2 . . 2 2 . . . 2
10: 1 1 1 1 1 1 1 1 1 1
</pre></div>
<p><a id="X78E9DDF885E12687" name="X78E9DDF885E12687"></a></p>
<h5>70.12-3 TableOfMarksFrobenius</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TableOfMarksFrobenius</code>( <var class="Arg">p</var>, <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">TableOfMarksFrobenius</code> computes the table of marks of a Frobenius group of order <span class="SimpleMath">p q</span>, where <span class="SimpleMath">p</span> is a prime and <span class="SimpleMath">q</span> divides <span class="SimpleMath">p-1</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( TableOfMarksFrobenius( 5, 4 ) );</span>
1: 20
2: 10 2
3: 5 1 1
4: 4 . . 4
5: 2 2 . 2 2
6: 1 1 1 1 1 1
</pre></div>
<p><a id="X794ABC7187A9285B" name="X794ABC7187A9285B"></a></p>
<h4>70.13 <span class="Heading">The Library of Tables of Marks</span></h4>
<p>The <strong class="pkg">GAP</strong> package <strong class="pkg">TomLib</strong> provides access to several hundred tables of marks of almost simple groups and their maximal subgroups. If this package is installed then the tables from this database can be accessed via <code class="func">TableOfMarks</code> (<a href="chap70.html#X85B262AB7E219C34"><span class="RefLink">70.3-1</span></a>) with argument a string. If also the <strong class="pkg">GAP</strong> Character Table Library is installed and contains the ordinary character table of the group for which one wants to fetch the table of marks then one can also call <code class="func">TableOfMarks</code> (<a href="chap70.html#X85B262AB7E219C34"><span class="RefLink">70.3-1</span></a>) with argument the character table.</p>
<p>A list of all names of tables of marks that are provided by the <strong class="pkg">TomLib</strong> package can be obtained via <code class="func">AllLibTomNames</code> (<span class="RefLink">???</span>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">names:= AllLibTomNames();;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">"A5" in names;</span>
true
</pre></div>
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