1<?xml version="2.1" encoding="utf-8"?> 2 3<!DOCTYPE Book SYSTEM "gapdoc.dtd" [<!ENTITY ATLAS "<Package>Atlas</Package>">]> 4 5<Book Name="tomlib"> 6 <#Include SYSTEM "title.xml"> 7 <TableOfContents/> 8 <Body> 9 10 <Chapter> 11 <Heading>The GAP Table of Marks Library</Heading> 12 13 14 <Section><Heading>Tables Of Marks</Heading> 15 16 The concept of a <E>Table of Marks</E><Index>table of marks</Index> was introduced by W.Burnside in his 17 book ``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>. 18 19 The table of marks of a finite group <M>G</M> is a matrix whose rows and 20 columns are labelled by the conjugacy classes of subgroups of <M>G</M> 21 and where for two subgroups <M>H</M> and <M>K</M> the <M>(H, K)</M>–entry is 22 the number of fixed points of <M>K</M> in the transitive action of <M>G</M> 23 on the cosets of <M>H</M> in <M>G</M>. 24 So the table of marks characterizes the set of all permutation 25 representations of <M>G</M>. 26 27 Moreover, the table of marks gives a compact description of the subgroup 28 lattice of <M>G</M>, since from the numbers of fixed points the numbers of 29 conjugates of a subgroup <M>K</M> contained in a subgroup <M>H</M> can be derived. 30 31 For 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. 32 33 34 35 36 </Section> 37 38 <Section> 39 <Heading>Installing The Table of Marks Library</Heading> 40 41 Download the archives in your preferred format. Unpack the archives inside the pkg dirctory of your GAP installation. 42 Load the package 43 <Log> 44gap> LoadPackage("tomlib"); 45true</Log> 46 </Section> 47 <Section> 48 <Heading>Contents</Heading> 49 TomLib contains several hundred tables of marks. For a complete list of the contents of the library do the following. 50<Log> 51gap> names:=AllLibTomNames();; 52gap> "A5" in names; 53true 54</Log> 55The 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 subgroups 56and automorphism groups. 57 58 59 60The Alternating groups <M>A_n</M> 61<List> 62<Item> for <M> n = 5, 6, 7, 8, 9, 10, 11, 12, 13 </M>.</Item> 63</List> 64The Symmetric groups <M>S_n</M> 65<List> 66<Item> for <M>n = 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 </M>.</Item> 67</List> 68The Linear groups 69<M>L_{2}(n)</M> for 70<List> 71<Item> <M>n = 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53</M></Item> 72<Item> <M>n = 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125 .</M></Item> 73</List> 74along with 75<List> 76<Item><M>L_{3}(4), L_{3}(3), L_{3}(5), L_{3}(7), L_{3}(9)</M></Item> 77<Item><M>L_{4}(3), L_{3}(8), L_{3}(11) </M>.</Item> 78</List> 79 80 81The Unitary groups 82<List> 83<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> 84<Item><M>U_{3}(9), U_{4}(2), U_{5}(2)</M></Item> 85</List> 86The Sporadic Groups 87<List> 88<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> 89</List> 90The names given to each subgroup are consistent with those used in Robert Wilson's atlas <Cite Key="AGR"/> 91For 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: 92<Log> 93gap> TableOfMarks("5:4xA5"); 94TableOfMarks( "5:4xA5" ) 95</Log> 96</Section> 97 98 99<Section> 100<Heading>Administrative Functions</Heading> 101Here we document some of the administrative facilities for the the &GAP; library of tables of marks. 102 103<#Include Label="LIBTOMKNOWN"> 104<#Include Label="IsLibTomRep"> 105<#Include Label="TableOfMarksFromLibrary"> 106<#Include Label="ConvertToLibTom"> 107<#Include Label="SetActualLibFileName"> 108<#Include Label="LIBTOM"> 109<#Include Label="AllLibTomNames"> 110<#Include Label="NamesLibTom"> 111<#Include Label="NotifiedFusionsOfLibTom"> 112<#Include Label="NotifiedFusionsToLibTom"> 113<#Include Label="UnloadTableOfMarksData"> 114</Section> 115<!-- 116%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> 117<Section Label="Standard Generators of Groups"> 118<Heading>Standard Generators of Groups</Heading> 119 120<#Include Label="[1]{stdgen}"> 121<#Include Label="StandardGeneratorsInfo:stdgen"> 122<!-- %T replace by an example for isom. type as soon as this is 123implemented! --> 124<#Include Label="HumanReadableDefinition"> 125<#Include Label="StandardGeneratorsFunctions"> 126<#Include Label="IsStandardGeneratorsOfGroup"> 127<#Include Label="StandardGeneratorsOfGroup"> 128<#Include Label="StandardGeneratorsInfo:tom"> 129 130</Section> 131 132 </Chapter> 133 134 135 136</Body> 137 138<Bibliography Databases="bib.xml"/> 139<TheIndex/> 140</Book> 141 142 143 144 145