GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
<?xml version="2.1" encoding="ISO-8859-1"?>12<!DOCTYPE Book SYSTEM "gapdoc.dtd" [<!ENTITY ATLAS "<Package>Atlas</Package>">]>34<Book Name="tomlib"> <!-- REQUIRED -->56<!--7The title page8-->9<TitlePage>10<!-- REQUIRED -->11<Title>12<Package>The GAP Table Of Marks Library</Package>13</Title>1415<Subtitle> <!-- OPTIONAL -->16<Br/>1718 19</Subtitle>2021<Version>Version 1.2.6</Version>22<!-- OPTIONAL -->23<Author>24Liam Naughton<Br/> <!-- REQUIRED -->25Goetz Pfeiffer<Br/>26 27<Address>28School of Mathematics, Statistics and Applied Mathematics<Br/>29National University of Ireland Galway,<Br/>30Galway,<Br/>31Ireland.32</Address>33<Email> [email protected], [email protected]</Email>34<Homepage>http://schmidt.nuigalway.ie/tomlib</Homepage>35</Author>3637<Date>November 2016</Date>38<!-- OPTIONAL -->3940<Copyright>41<!-- OPTIONAL -->42©right; 2016 We adopt the copyright regulations of GAP as detailed in the copyright notice in the GAP manual.43</Copyright>4445<Acknowledgements>46<!-- OPTIONAL -->47<P/>484950<P/>This documentation was prepared with the <Package>GAPDoc</Package> package51by Frank Luebeck and Max Neunhoffer.52</Acknowledgements>5354</TitlePage>55565758<TableOfContents/> <!-- OPTIONAL -->59606162<!--63The document64-->65<Body> <!-- REQUIRED -->666768<Chapter>69<Heading>The GAP Table of Marks Library</Heading>707172<Section><Heading>Tables Of Marks</Heading>7374The concept of a <E>Table of Marks</E><Index>table of marks</Index> was introduced by W.Burnside in his75book ``Theory of Groups of Finite Order'' <Cite Key="Bur55"/>. Therefore a table of marks is sometimes called a <E>Burnside matrix</E><Index>Burnside matrix</Index>.7677The table of marks of a finite group <M>G</M> is a matrix whose rows and78columns are labelled by the conjugacy classes of subgroups of <M>G</M>79and where for two subgroups <M>H</M> and <M>K</M> the <M>(H, K)</M>–entry is80the number of fixed points of <M>K</M> in the transitive action of <M>G</M>81on the cosets of <M>H</M> in <M>G</M>.82So the table of marks characterizes the set of all permutation83representations of <M>G</M>.8485Moreover, the table of marks gives a compact description of the subgroup86lattice of <M>G</M>, since from the numbers of fixed points the numbers of87conjugates of a subgroup <M>K</M> contained in a subgroup <M>H</M> can be derived.8889For small groups the table of marks of <M>G</M> can be constructed directly in GAP by first computing the entire subgroup lattice of <M>G</M>. However, for larger groups this method is unfeasible. The GAP Table of Marks library provides access to several hundred table of marks and their maximal subgroups.9091929394</Section>9596<Section>97<Heading>Installing The Table of Marks Library</Heading>9899Download the archives in your preferred format. Unpack the archives inside the pkg dirctory of your GAP installation.100Load the package101<Log>102gap> LoadPackage("tomlib");103true</Log>104</Section>105<Section>106<Heading>Contents</Heading>107TomLib contains several hundred tables of marks. For a complete list of the contents of the library do the following.108<Log>109gap> names:=AllLibTomNames();;110gap> "A5" in names;111true112</Log>113The current version of the tomlib contains the tables of marks of the groups listed below as well as the tables of many of their maximal subgroups114and automorphism groups.115116117118The Alternating groups <M>A_n</M>119<List>120<Item> for <M> n = 5, 6, 7, 8, 9, 10, 11, 12, 13 </M>.</Item>121</List>122The Symmetric groups <M>S_n</M>123<List>124<Item> for <M>n = 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 </M>.</Item>125</List>126The Linear groups127<M>L_{2}(n)</M> for128<List>129<Item> <M>n = 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53</M></Item>130<Item> <M>n = 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125 .</M></Item>131</List>132along with133<List>134<Item><M>L_{3}(4), L_{3}(3), L_{3}(5), L_{3}(7), L_{3}(9)</M></Item>135<Item><M>L_{4}(3), L_{3}(8), L_{3}(11) </M>.</Item>136</List>137138139The Unitary groups140<List>141<Item><M>U_{3}(3), U_{4}(3), U_{3}(5), U_{3}(4), U_{3}(11), U_{3}(7), U_{3}(8)</M></Item>142<Item><M>U_{3}(9), U_{4}(2), U_{5}(2)</M></Item>143</List>144The Sporadic Groups145<List>146<Item> <M>Co_3, HS, McL, He, J_1, J_2, J_3, M_{11}, M_{12}, M_{22}, M_{23}, M_{24} </M> </Item>147</List>148The names given to each subgroup are consistent with those used in Robert Wilson's atlas <Cite Key="AGR"/>149For example if you wish to access the table of marks of the maximal subgroup <M>"5:4 \times A5"</M> of the Higman-Sims group do the following:150<Log>151gap> TableOfMarks("5:4xA5");152TableOfMarks( "5:4xA5" )153</Log>154</Section>155156157<Section>158<Heading>Administrative Functions</Heading>159Here we document some of the administrative facilities for the the &GAP; library of tables of marks.160161<#Include Label="LIBTOMKNOWN">162<#Include Label="IsLibTomRep">163<#Include Label="TableOfMarksFromLibrary">164<#Include Label="ConvertToLibTom">165<#Include Label="SetActualLibFileName">166<#Include Label="LIBTOM">167<#Include Label="AllLibTomNames">168<#Include Label="NamesLibTom">169<#Include Label="NotifiedFusionsOfLibTom">170<#Include Label="NotifiedFusionsToLibTom">171<#Include Label="UnloadTableOfMarksData">172</Section>173<!--174%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->175<Section Label="Standard Generators of Groups">176<Heading>Standard Generators of Groups</Heading>177178<#Include Label="[1]{stdgen}">179<#Include Label="StandardGeneratorsInfo:stdgen">180<!-- %T replace by an example for isom. type as soon as this is181implemented! -->182<#Include Label="HumanReadableDefinition">183<#Include Label="StandardGeneratorsFunctions">184<#Include Label="IsStandardGeneratorsOfGroup">185<#Include Label="StandardGeneratorsOfGroup">186<#Include Label="StandardGeneratorsInfo:tom">187188</Section>189190</Chapter>191192193194</Body>195196<Bibliography Databases="bib.xml"/>197<TheIndex/>198</Book>199200201202203204205