<!DOCTYPE Book SYSTEM "gapdoc.dtd" [<!ENTITY ATLAS "Atlas">]>
<Book Name="tomlib">
<#Include SYSTEM "title.xml">
<TableOfContents/>
<Body>
<Chapter>
<Heading>The GAP Table of Marks Library</Heading>
<Section><Heading>Tables Of Marks</Heading>
The concept of a <E>Table of Marks</E><Index>table of marks</Index> was introduced by W.Burnside in his
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>.
The table of marks of a finite group <M>G</M> is a matrix whose rows and
columns are labelled by the conjugacy classes of subgroups of <M>G</M>
and where for two subgroups <M>H</M> and <M>K</M> the <M>(H, K)</M>–entry is
the number of fixed points of <M>K</M> in the transitive action of <M>G</M>
on the cosets of <M>H</M> in <M>G</M>.
So the table of marks characterizes the set of all permutation
representations of <M>G</M>.
Moreover, the table of marks gives a compact description of the subgroup
lattice of <M>G</M>, since from the numbers of fixed points the numbers of
conjugates of a subgroup <M>K</M> contained in a subgroup <M>H</M> can be derived.
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.
</Section>
<Section>
<Heading>Installing The Table of Marks Library</Heading>
Download the archives in your preferred format. Unpack the archives inside the pkg dirctory of your GAP installation.
Load the package
<Log>
gap> LoadPackage("tomlib");
true</Log>
</Section>
<Section>
<Heading>Contents</Heading>
TomLib contains several hundred tables of marks. For a complete list of the contents of the library do the following.
<Log>
gap> names:=AllLibTomNames();;
gap> "A5" in names;
true
</Log>
The 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
and automorphism groups.
The Alternating groups <M>A_n</M>
<List>
<Item> for <M> n = 5, 6, 7, 8, 9, 10, 11, 12, 13 </M>.</Item>
</List>
The Symmetric groups <M>S_n</M>
<List>
<Item> for <M>n = 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 </M>.</Item>
</List>
The Linear groups
<M>L_{2}(n)</M> for
<List>
<Item> <M>n = 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53</M></Item>
<Item> <M>n = 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125 .</M></Item>
</List>
along with
<List>
<Item><M>L_{3}(4), L_{3}(3), L_{3}(5), L_{3}(7), L_{3}(9)</M></Item>
<Item><M>L_{4}(3), L_{3}(8), L_{3}(11) </M>.</Item>
</List>
The Unitary groups
<List>
<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>
<Item><M>U_{3}(9), U_{4}(2), U_{5}(2)</M></Item>
</List>
The Sporadic Groups
<List>
<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>
</List>
The names given to each subgroup are consistent with those used in Robert Wilson's atlas
For 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:
<Log>
gap> TableOfMarks("5:4xA5");
TableOfMarks( "5:4xA5" )
</Log>
</Section>
<Section>
<Heading>Administrative Functions</Heading>
Here we document some of the administrative facilities for the the &GAP; library of tables of marks.
<#Include Label="[1]{stdgen}">
<#Include Label="StandardGeneratorsInfo:stdgen"> <!-- %T replace by an example for isom. type as soon as this is
implemented! -->
<#Include Label="HumanReadableDefinition">
<#Include Label="StandardGeneratorsFunctions">
<#Include Label="IsStandardGeneratorsOfGroup">
<#Include Label="StandardGeneratorsOfGroup">
<#Include Label="StandardGeneratorsInfo:tom">
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