/usr/share/gap/pkg/tomlib/doc/tomlib.xml

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<!DOCTYPE Book SYSTEM "gapdoc.dtd" [<!ENTITY ATLAS "<Package>Atlas</Package>">]>

<Book Name="tomlib">                                        <!-- REQUIRED -->
	
	<!--
	    The title page
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	<TitlePage>    
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		<Title>                                             
			<Package>The GAP Table Of Marks Library</Package>
		</Title>
		
		<Subtitle>     	                                     <!-- OPTIONAL -->
			<Br/>
			
			&nbsp;
		</Subtitle>
		
		<Version>Version 1.2.6</Version> 
		<!-- OPTIONAL -->
		<Author> 
			Liam Naughton<Br/>                <!-- REQUIRED -->
			Goetz Pfeiffer<Br/>
			&nbsp;
			<Address>
				School of Mathematics, Statistics and Applied Mathematics<Br/>
				National University of Ireland Galway,<Br/>
				Galway,<Br/>
				Ireland.
			</Address>
			<Email> l.naughton@wlv.ac.uk, goetz.pfeiffer@nuigalway.ie</Email>
			<Homepage>http://schmidt.nuigalway.ie/tomlib</Homepage>
		</Author> 
		
		<Date>November 2016</Date> 
		<!-- OPTIONAL -->
		
		<Copyright> 
			<!-- OPTIONAL -->
			&copyright; 2016 We adopt the copyright regulations of GAP as detailed in the copyright notice in the GAP manual.
		</Copyright> 
		
		<Acknowledgements> 
			<!-- OPTIONAL -->
			<P/>
			
			
			<P/>This documentation was prepared with the <Package>GAPDoc</Package> package 
			by Frank Luebeck and Max Neunhoffer. 
		</Acknowledgements>
		
	</TitlePage>
	
	
	
	<TableOfContents/>                                    <!-- OPTIONAL -->
	
	
	
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	<Body>                                                <!-- REQUIRED -->
		
		
  <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>&ndash;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&gt; 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&gt; names:=AllLibTomNames();;
gap&gt; "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 <Cite Key="AGR"/>
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&gt; 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="LIBTOMKNOWN">
<#Include Label="IsLibTomRep">
<#Include Label="TableOfMarksFromLibrary">
<#Include Label="ConvertToLibTom">
<#Include Label="SetActualLibFileName">
<#Include Label="LIBTOM">
<#Include Label="AllLibTomNames">
<#Include Label="NamesLibTom">
<#Include Label="NotifiedFusionsOfLibTom">
<#Include Label="NotifiedFusionsToLibTom">
<#Include Label="UnloadTableOfMarksData">
</Section>
<!--
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Standard Generators of Groups">
<Heading>Standard Generators of Groups</Heading>

<#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">

</Section>

 </Chapter>
 
  
  
</Body>

<Bibliography Databases="bib.xml"/>
<TheIndex/>  
</Book>
gap-table-of-marks 1r2p6-1 / usr / share / gap / pkg / tomlib / doc / tomlib.xml
