## `The GAP Table Of Marks Library`

Version 1.2.2
Liam Naughton
Goetz Pfeiffer

    				School of Mathematics, Statistics and Applied Mathematics

    				National University of Ireland Galway,

    				Galway,

    				Ireland.

liam.naughton@nuigalway.ie, goetz.pfeiffer@nuigalway.ie http://schmidt.nuigalway.ie/tomlib
September 2011
&copyright; 2011 We adopt the copyright regulations of GAP as detailed in the copyright notice in the GAP manual.

This documentation was prepared with the `GAPDoc` package by Frank Luebeck and Max Neunhoffer.
The GAP Table of Marks Library
Tables Of Marks

The concept of a Table of Marks table of marks was introduced by W.Burnside in his book ``Theory of Groups of Finite Order'' . Therefore a table of marks is sometimes called a Burnside matrix Burnside matrix.

The table of marks of a finite group G is a matrix whose rows and columns are labelled by the conjugacy classes of subgroups of G and where for two subgroups H and K the (H, K)–entry is the number of fixed points of K in the transitive action of G on the cosets of H in G. So the table of marks characterizes the set of all permutation representations of G.

Moreover, the table of marks gives a compact description of the subgroup lattice of G, since from the numbers of fixed points the numbers of conjugates of a subgroup K contained in a subgroup H can be derived.

For small groups the table of marks of G can be constructed directly in GAP by first computing the entire subgroup lattice of G. 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.

Installing The Table of Marks Library

Download the archives in your preferred format. Unpack the archives inside the pkg dirctory of your GAP installation. Load the package gap> LoadPackage("tomlib"); true
Contents
TomLib contains several hundred tables of marks. For a complete list of the contents of the library do the following. gap> names:=AllLibTomNames();; gap> "A5" in names; true 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 A_n for n = 5, 6, 7, 8, 9, 10, 11, 12, 13 . The Symmetric groups S_n for n = 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 . The Linear groups L_{2}(n) for n = 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53 n = 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125 . along with L_{3}(4), L_{3}(3), L_{3}(5), L_{3}(7), L_{3}(9) L_{4}(3), L_{3}(8), L_{3}(11) .

The Unitary groups U_{3}(3), U_{4}(3), U_{3}(5), U_{3}(4), U_{3}(11), U_{3}(7), U_{3}(8) U_{3}(9), U_{4}(2), U_{5}(2) The Sporadic Groups HS, McL, He, J_1, J_2, J_3, M_{11}, M_{12}, M_{22}, M_{23}, M_{24} The names given to each subgroup are consistent with those used in Robert Wilon's atlas For example if you wish to access the table of marks of the maximal subgroup "5:4 \times A5" of the Higman-Sims group do the following: gap> TableOfMarks("5:4 x A5"); TableOfMarks( "5:4 x A5" )
Administrative Functions
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">
Standard Generators of Groups
<#Include Label="[1]{stdgen}"> <#Include Label="StandardGeneratorsInfo:stdgen"> <#Include Label="HumanReadableDefinition"> <#Include Label="StandardGeneratorsFunctions"> <#Include Label="IsStandardGeneratorsOfGroup"> <#Include Label="StandardGeneratorsOfGroup"> <#Include Label="StandardGeneratorsInfo:tom">