%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A tom.tex GAP documentation Goetz Pfeiffer. %% %A @(#)$Id: tom.tex,v 3.12 1994/06/10 02:49:01 vfelsch Rel $ %% %Y Copyright 1991-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %% This file describes the functions dealing with Tables Of Marks. %% %H $Log: tom.tex,v $ %H Revision 3.12 1994/06/10 02:49:01 vfelsch %H updated examples %H %H Revision 3.11 1993/02/19 10:48:42 gap %H adjustments in line length and spelling %H %H Revision 3.10 1993/02/12 17:02:59 felsch %H examples adjusted to line length 72 %H %H Revision 3.9 1993/02/11 17:46:09 martin %H changed '@' to '&' %H %H Revision 3.8 1993/02/02 12:46:54 felsch %H long lines fixed %H %H Revision 3.7 1993/01/22 19:35:52 martin %H changed |""| to '\"\"' etc. %H %H Revision 3.6 1993/01/22 17:23:17 goetz %H deleted some lines. %H %H Revision 3.5 1993/01/21 17:04:47 goetz %H added references, formatted examples. %H %H Revision 3.4 1993/01/20 15:41:23 goetz %H corrections and additions. %H %H Revision 3.3 1993/01/19 13:40:17 goetz %H extensions and corrections. %H %H Revision 3.2 1992/12/01 14:29:10 goetz %H added formula. %H %H Revision 3.1 1992/12/01 14:23:51 goetz %H formatted displayed material. %H %H Revision 3.0 1992/11/15 15:47:46 goetz %H Initial Revision. %H %% \Chapter{Tables of Marks} The concept of a table of marks was introduced by W.~Burnside in his book \cite{Bur55}. Therefore a table of marks is sometimes called a 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 $A$ and $B$ the $(A, B)$--entry is the number of fixed points of $B$ in the transitive action of $G$ on the cosets of $A$ in $G$. So the table of marks characterizes 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 $B$ contained in a subgroup $A$ can be derived. This chapter describes a function (see "TableOfMarks") which restores a table of marks from the {\GAP} library of tables of marks (see "The Library of Tables of Marks") or which computes the table of marks for a given group from the subgroup lattice of that group. Moreover this package contains a function to display a table of marks (see "DisplayTom"), a function to check the consistency of a table of marks (see "TestTom"), functions which switch between several forms of representation (see "Marks", "NrSubs", "MatTom", and "TomMat"), functions which derive information about the group from the table of marks (see "DecomposedFixedPointVector", "NormalizerTom", "IntersectionsTom", "IsCyclicTom", "FusionCharTableTom", "PermCharsTom", "MoebiusTom", "CyclicExtensionsTom", "IdempotentsTom", "ClassTypesTom", and "ClassNamesTom"), and some functions for the generic construction of a table of marks (see "TomCyclic", "TomDihedral", and "TomFrobenius"). The functions described in this chapter are implemented in the file 'LIBNAME/\"tom.g\"'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{More about Tables of Marks} Let $G$ be a finite group with $n$ conjugacy classes of subgroups $C_1, \ldots, C_n$ and representatives $H_i \in C_i$, $i = 1, \ldots, n$. The *table of marks* of $G$ is defined to be the $n \times n$ matrix $M = (m_{ij})$ where $m_{ij}$ is the number of fixed points of the subgroup $H_j$ in the action of $G$ on the cosets of $H_i$ in $G$. Since $H_j$ can only have fixed points if it is contained in a one point stablizer the matrix $M$ is lower triangular if the classes $C_i$ are sorted according to the following condition; if $H_i$ is contained in a conjugate of $H_j$ then $i \leq j$. Moreover, the diagonal entries $m_{ii}$ are nonzero since $m_{ii}$ equals the index of $H_i$ in its normalizer in $G$. Hence $M$ is invertible. Since any transitive action of $G$ is equivalent to an action on the cosets of a subgroup of $G$, one sees that the table of marks completely characterizes permutation representations of $G$. The entries $m_{ij}$ have further meanings. If $H_1$ is the trivial subgroup of $G$ then each mark $m_{i1}$ in the first column of $M$ is equal to the index of $H_i$ in $G$ since the trivial subgroup fixes all cosets of $H_i$. If $H_n = G$ then each $m_{nj}$ in the last row of $M$ is equal to 1 since there is only one coset of $G$ in $G$. In general, $m_{ij}$ equals the number of conjugates of $H_i$ which contain $H_j$, multiplied by the index of $H_i$ in its normalizer in $G$. Moreover, the number $c_{ij}$ of conjugates of $H_j$ which are contained in $H_i$ can be derived from the marks $m_{ij}$ via the formula \[ c_{ij} = \frac{m_{ij} m_{j1}}{m_{i1} m_{jj}}. \] Both the marks $m_{ij}$ and the numbers of subgroups $c_{ij}$ are needed for the functions described in this chapter. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Table of Marks Records} A table of marks is represented by a record. This record has at least a component 'subs' which is a list where for each conjugacy class of subgroups the class numbers of its subgroups are stored. These are exactly the positions in the corresponding row of the table of marks which have nonzero entries. The marks themselves can be stored in the component 'marks' which is a list that contains for each entry in the component 'subs' the corresponding nonzero value of the table of marks. The same information is, however, given by the three components 'nrSubs', 'length', and 'order', where 'nrSubs' is a list which contains for each entry in the component 'subs' the corresponding number of conjugates which are contained in a subgroup, 'length' is a list which contains for each class of subgroups its length, and 'order' is a list which contains for each class of subgroups their order. So a table of marks consists either of the components 'subs' and 'marks' or of the components 'subs', 'nrSubs', 'length', and 'order'. The functions 'Marks' (see "Marks") and 'NrSubs' (see "NrSubs") will derive one representation from the other when needed. Additional information about a table of marks is needed by some functions. The class numbers of normalizers are stored in the component 'normalizer'. The number of the derived subgroup of the whole group is stored in the component 'derivedSubgroup'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{The Library of Tables of Marks} This package of functions comes together with a library of tables of marks. The library files are stored in a directory 'TOMNAME'. The file 'TOMNAME/\"tmprimar.tom\"' is the primary file of the library of tables of marks. It contains the information where to find a table and the function 'TomLibrary' which restores a table from the library. The secondary files are | tmaltern.tom tmmath24.tom tmsuzuki.tom tmunitar.tom tmlinear.tom tmmisc.tom tmsporad.tom tmsymple.tom | The list 'TOMLIST' contains for each table an entry with its name and the name of the file where it is stored. A table of marks which is restored from the library will be stored as a component of the record 'TOM'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{TableOfMarks} 'TableOfMarks( )' If the argument given to 'TableOfMarks' is a string then 'TableOfMarks' will search the library of tables of marks (see "The Library of Tables of Marks") for a table with name . If such a table is found then 'TableOfMarks' will return a copy of that table. Otherwise 'TableOfMarks' will return 'false'. | gap> a5 := TableOfMarks( "A5" ); rec( derivedSubgroup := 9, normalizer := [ 9, 4, 6, 8, 7, 6, 7, 8, 9 ], nrSubs := [ [ 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 3, 1 ], [ 1, 1 ], [ 1, 3, 1, 1 ], [ 1, 5, 1, 1 ], [ 1, 3, 4, 1, 1 ], [ 1, 15, 10, 5, 6, 10, 6, 5, 1 ] ], order := [ 1, 2, 3, 4, 5, 6, 10, 12, 60 ], subs := [ [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 2, 4 ], [ 1, 5 ], [ 1, 2, 3, 6 ], [ 1, 2, 5, 7 ], [ 1, 2, 3, 4, 8 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ], length := [ 1, 15, 10, 5, 6, 10, 6, 5, 1 ] ) gap> TableOfMarks( "A10" ); &W TableOfMarks: no table of marks A10 found. false | 'TableOfMarks( )' \index{TableOfMarks} If 'TableOfMarks' is called with a group as its argument then the table of marks of that group will be computed and returned in the compressed format. The computation of the table of marks requires the knowledge of the complete subgroup lattice of the group . If the lattice is not yet known then it will be constructed (see "Lattice"). This may take a while if the group is large. Moreover, as the 'Lattice' command is involved the applicability of 'TableOfMarks' underlies the same restrictions with respect to the soluble residuum of as described in section "Lattice". The result of 'TableOfMarks' is assigned to the component 'tableOfMarks' of the group record , so that the next call to 'TableOfMarks' with the same argument can just return this component 'tableOfMarks'. *Warning*\:\ Note that 'TableOfMarks' has changed with the release {\GAP} 3.2. It now returns the table of marks in compressed form. However, you can apply the 'MatTom' command (see "MatTom") to convert it into the square matrix which was returned by 'TableOfMarks' in {\GAP} version 3.1. | gap> alt5 := AlternatingPermGroup( 5 );; gap> TableOfMarks( alt5 ); rec( subs := [ [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 2, 4 ], [ 1, 5 ], [ 1, 2, 3, 6 ], [ 1, 2, 5, 7 ], [ 1, 2, 3, 4, 8 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ], marks := [ [ 60 ], [ 30, 2 ], [ 20, 2 ], [ 15, 3, 3 ], [ 12, 2 ], [ 10, 2, 1, 1 ], [ 6, 2, 1, 1 ], [ 5, 1, 2, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ] ) gap> last = alt5.tableOfMarks; true | For a pretty print display of a table of marks see "DisplayTom". %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Marks} 'Marks( )' 'Marks' returns the list of lists of marks of the table of marks . If these are not yet stored in the component 'marks' of then they will be computed and assigned to the component 'marks'. | gap> Marks( a5 ); [ [ 60 ], [ 30, 2 ], [ 20, 2 ], [ 15, 3, 3 ], [ 12, 2 ], [ 10, 2, 1, 1 ], [ 6, 2, 1, 1 ], [ 5, 1, 2, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{NrSubs} 'NrSubs( )' 'NrSubs' returns the list of lists of numbers of subgroups of the table of marks . If these are not yet stored in the component 'nrSubs' of then they will be computed and assigned to the component 'nrSubs'. 'NrSubs' also has to compute the orders and lengths from the marks. | gap> NrSubs( a5 ); [ [ 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 3, 1 ], [ 1, 1 ], [ 1, 3, 1, 1 ], [ 1, 5, 1, 1 ], [ 1, 3, 4, 1, 1 ], [ 1, 15, 10, 5, 6, 10, 6, 5, 1 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{WeightsTom} 'WeightsTom( )' 'WeightsTom' extracts the weights from a table of marks , i.e., the diagonal entries, indicating the index of a subgroup in its normalizer. | gap> wt := WeightsTom( a5 ); [ 60, 2, 2, 3, 2, 1, 1, 1, 1 ] | This information may be used to obtain the numbers of conjugate supergroups from the marks. | gap> marks := Marks( a5 );; gap> List( [ 1 .. 9 ], x -> marks[x] / wt[x] ); [ [ 1 ], [ 15, 1 ], [ 10, 1 ], [ 5, 1, 1 ], [ 6, 1 ], [ 10, 2, 1, 1 ], [ 6, 2, 1, 1 ], [ 5, 1, 2, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ]| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{MatTom} 'MatTom( )' 'MatTom' produces a square matrix corresponding to the table of marks in compressed form. For large tables this may need a lot of space. | gap> MatTom( a5 ); [ [ 60, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 30, 2, 0, 0, 0, 0, 0, 0, 0 ], [ 20, 0, 2, 0, 0, 0, 0, 0, 0 ], [ 15, 3, 0, 3, 0, 0, 0, 0, 0 ], [ 12, 0, 0, 0, 2, 0, 0, 0, 0 ], [ 10, 2, 1, 0, 0, 1, 0, 0, 0 ], [ 6, 2, 0, 0, 1, 0, 1, 0, 0 ], [ 5, 1, 2, 1, 0, 0, 0, 1, 0 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{TomMat} 'TomMat( )' Given a matrix which contains the marks of a group as its entries, 'TomMat' will produce the corresponding table of marks record. | gap> mat:= > [ [ 60, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 30, 2, 0, 0, 0, 0, 0, 0, 0 ], > [ 20, 0, 2, 0, 0, 0, 0, 0, 0 ], [ 15, 3, 0, 3, 0, 0, 0, 0, 0 ], > [ 12, 0, 0, 0, 2, 0, 0, 0, 0 ], [ 10, 2, 1, 0, 0, 1, 0, 0, 0 ], > [ 6, 2, 0, 0, 1, 0, 1, 0, 0 ], [ 5, 1, 2, 1, 0, 0, 0, 1, 0 ], > [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ];; gap> TomMat( mat ); rec( subs := [ [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 2, 4 ], [ 1, 5 ], [ 1, 2, 3, 6 ], [ 1, 2, 5, 7 ], [ 1, 2, 3, 4, 8 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ], marks := [ [ 60 ], [ 30, 2 ], [ 20, 2 ], [ 15, 3, 3 ], [ 12, 2 ], [ 10, 2, 1, 1 ], [ 6, 2, 1, 1 ], [ 5, 1, 2, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ] ) gap> TomMat( IdentityMat( 7 ) ); rec( subs := [ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ] ], marks := [ [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ] ] ) | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{DecomposedFixedPointVector} 'DecomposedFixedPointVector( , )' Let the group with table of marks act as a permutation group on its conjugacy classes of subgroups, then is assumed to be a vector of fixed point numbers, i.e., a vector which contains for each class of subgroups the number of fixed points under that action. 'DecomposedFixedPointVector' returns the decomposition of into rows of the table of marks. This decomposition corresponds to a decomposition of the action into transitive constituents. Trailing zeros in may be omitted. | gap> DecomposedFixedPointVector( a5, [ 16, 4, 1, 0, 1, 1, 1 ] ); [ ,,,,, 1, 1 ] | The vector may be any vector of integers. The resulting decomposition, however, will not be integral, in general. | gap> DecomposedFixedPointVector( a5, [ 0, 0, 0, 0, 1, 1 ] ); [ 2/5, -1, -1/2,, 1/2, 1 ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{TestTom} 'TestTom( )' 'TestTom' decomposes all tensor products of rows of the table of marks . It returns 'true' if all decomposition numbers are nonnegative integers and 'false' otherwise. This provides a strong consistency check for a table of marks. | gap> TestTom( a5 ); true | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{DisplayTom} 'DisplayTom( )' 'DisplayTom' produces a formatted output for the table of marks . Each line of output begins with the number of the corresponding class of subgroups. This number is repeated if the output spreads over several pages. | gap> DisplayTom( a5 ); 1: 60 2: 30 2 3: 20 . 2 4: 15 3 . 3 5: 12 . . . 2 6: 10 2 1 . . 1 7: 6 2 . . 1 . 1 8: 5 1 2 1 . . . 1 9: 1 1 1 1 1 1 1 1 1 | 'DisplayTom( , )' In this form 'DisplayTom' takes a record as an additional parameter. If this record has a component 'classes' which contains a list of class numbers then only the rows and columns of the matrix corresponding to this list are printed. | gap> DisplayTom( a5, rec( classes := [ 1, 2, 3, 4, 8 ] ) ); 1: 60 2: 30 2 3: 20 . 2 4: 15 3 . 3 8: 5 1 2 1 1 | The record may also have a component 'form' which enables the printing of tables of numbers of subgroups. If .'form' has the value '\"subgroups\"' then at position $(i,j)$ the number of conjugates of $H_j$ contained in $H_i$ will be printed. If it has the value '\"supergroups\"' then at position $(i,j)$ the number of conjugates of $H_i$ which contain $H_j$ will be printed. | gap> DisplayTom( a5, rec( form := "subgroups" ) ); 1: 1 2: 1 1 3: 1 . 1 4: 1 3 . 1 5: 1 . . . 1 6: 1 3 1 . . 1 7: 1 5 . . 1 . 1 8: 1 3 4 1 . . . 1 9: 1 15 10 5 6 10 6 5 1 gap> DisplayTom( a5, rec( form := "supergroups" ) ); 1: 1 2: 15 1 3: 10 . 1 4: 5 1 . 1 5: 6 . . . 1 6: 10 2 1 . . 1 7: 6 2 . . 1 . 1 8: 5 1 2 1 . . . 1 9: 1 1 1 1 1 1 1 1 1 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{NormalizerTom} 'NormalizerTom( , )' 'NormalizerTom' tries to find conjugacy class of the normalizer of a subgroup with class number . It will return the list of class numbers of those subgroups which have the right size and contain the subgroup and all subgroups which clearly contain it as a normal subgroup. If the normalizer is uniquely determined by these conditions then only its class number will be returned. 'NormalizerTom' should never return an empty list. | gap> NormalizerTom( a5, 4 ); 8 | The example shows that a subgroup with class number 4 in $A_5$ (which is a Kleinan four group) is normalized by a subgroup in class 8. This class contains the subgroups of $A_5$ which are isomorphic to $A_4$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IntersectionsTom} 'IntersectionsTom( , , )' The intersections of the two conjugacy classes of subgroups with class numbers and , respectively, are determined by the decomposition of the tensor product of their rows of marks. 'IntersectionsTom' returns this decomposition. | gap> IntersectionsTom( a5, 8, 8 ); [ ,, 1,,,,, 1 ] | Any two subgroups of class number 8 ($A_4$) of $A_5$ are either equal and their intersection has again class number 8, or their intersection has class number $3$, and is a cyclic subgroup of order 3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsCyclicTom} 'IsCyclicTom( , )' A subgroup is cyclic if and only if the sum over the corresponding row of the inverse table of marks is nonzero (see \cite{Ker91}, page 125). Thus we only have to decompose the corresponding idempotent. | gap> for i in [ 1 .. 6 ] do > Print( i, ": ", IsCyclicTom(a5, i), " " ); > od; Print( "\n" ); 1: true 2: true 3: true 4: false 5: true 6: false | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{FusionCharTableTom} 'FusionCharTableTom( , )' 'FusionCharTableTom' determines the fusion of the classes of elements from the character table into classes of cyclic subgroups on the table of marks . | gap> a5c := CharTable( "A5" );; gap> fus := FusionCharTableTom( a5c, a5 ); [ 1, 2, 3, 5, 5 ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PermCharsTom} 'PermCharsTom( , )' 'PermCharsTom' reads the list of permutation characters from the table of marks . It therefore has to know the fusion map which sends each conjugacy class of elements of the group to the conjugacy class of subgroups they generate. | gap> PermCharsTom( a5, fus ); [ [ 60, 0, 0, 0, 0 ], [ 30, 2, 0, 0, 0 ], [ 20, 0, 2, 0, 0 ], [ 15, 3, 0, 0, 0 ], [ 12, 0, 0, 2, 2 ], [ 10, 2, 1, 0, 0 ], [ 6, 2, 0, 1, 1 ], [ 5, 1, 2, 0, 0 ], [ 1, 1, 1, 1, 1 ] ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{MoebiusTom} 'MoebiusTom( )' 'MoebiusTom' computes the M{\accent127 o}bius values both of the subgroup lattice of the group with table of marks and of the poset of conjugacy classes of subgroups. It returns a record where the component 'mu' contains the M{\accent127 o}bius values of the subgroup lattice, and the component 'nu' contains the M{\accent127 o}bius values of the poset. Moreover, according to a conjecture of Isaacs et al. (see \cite{HIO89}, \cite{Pah93}), the values on the poset of conjugacy classes are derived from those of the subgroup lattice. These theoretical values are returned in the component 'ex'. For that computation, the derived subgroup must be known in the component 'derivedSubgroup' of . The numbers of those subgroups where the theoretical value does not coincide with the actual value are returned in the component 'hyp'. | gap> MoebiusTom( a5 ); rec( mu := [ -60, 4, 2,,, -1, -1, -1, 1 ], nu := [ -1, 2, 1,,, -1, -1, -1, 1 ], ex := [ -60, 4, 2,,, -1, -1, -1, 1 ], hyp := [ ] ) | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CyclicExtensionsTom} 'CyclicExtensionsTom( ,

)' According to A.~Dress \cite{Dre69}, two columns of the table of marks are equal modulo the prime

if and only if the corresponding subgroups are connected by a chain of normal extensions of order

. 'CyclicExtensionsTom' returns the classes of this equivalence relation. This information is not used by 'NormalizerTom' although it might give additional restrictions in the search of normalizers. | gap> CyclicExtensionsTom( a5, 2 ); [ [ 1, 2, 4 ], [ 3, 6 ], [ 5, 7 ], [ 8 ], [ 9 ] ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IdempotentsTom} 'IdempotentsTom( )' 'IdempotentsTom' returns the list of idempotents of the integral Burnside ring described by the table of marks . According to A.~Dress \cite{Dre69}, these idempotents correspond to the classes of perfect subgroups, and each such idempotent is the characteristic function of all those subgroups which arise by cyclic extension from the corresponding perfect subgroup. | gap> IdempotentsTom( a5 ); [ 1, 1, 1, 1, 1, 1, 1, 1, 9 ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ClassTypesTom} 'ClassTypesTom( )' 'ClassTypesTom' distinguishes isomorphism types of the classes of subgroups of the table of marks as far as this is possible. Two subgroups are clearly not isomorphic if they have different orders. Moreover, isomorphic subgroups must contain the same number of subgroups of each type. The types are represented by numbers. 'ClassTypesTom' returns a list which contains for each class of subgroups its corresponding number. | gap> a6 := TableOfMarks( "A6" );; gap> ClassTypesTom( a6 ); [ 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16 ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ClassNamesTom} 'ClassNamesTom( )' 'ClassNamesTom' constructs generic names for the conjugacy classes of subgroups of the table of marks . In general, the generic name of a class of non--cyclic subgroups consists of three parts, '\"()\"', '\"\_\{\}\"', and '\"\"', and hence has the form '\"()\_\{\}\"', where indicates the order of the subgroups, is a number that distinguishes different types of subgroups of the same order, and is a letter which distinguishes classes of subgroups of the same type and order. The type of a subgroup is determined by the numbers of its subgroups of other types (see "ClassTypesTom"). This is slightly weaker than isomorphism. The letter is omitted if there is only one class of subgroups of that order and type, and the type is omitted if there is only one class of that order. Moreover, the braces round the type are omitted if the type number has only one digit. For classes of cyclic subgoups, the parentheses round the order and the type are omitted. Hence the most general form of their generic names is '\"\,\"'. Again, the letter is omitted if there is only one class of cyclic subgroups of that order. | gap> ClassNamesTom( a6 ); [ "1", "2", "3a", "3b", "5", "4", "(4)_2a", "(4)_2b", "(6)a", "(6)b", "(9)", "(10)", "(8)", "(12)a", "(12)b", "(18)", "(24)a", "(24)b", "(36)", "(60)a", "(60)b", "(360)" ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{TomCyclic} 'TomCyclic( )' 'TomCyclic' constructs the table of marks of the cyclic group of order . A cyclic group of order has as its subgroups for each divisor $d$ of a cyclic subgroup of order $d$. The record which is returned has an additional component 'name' where for each subgroup its order is given as a string. | gap> c6 := TomCyclic( 6 ); rec( name := [ "1", "2", "3", "6" ], subs := [ [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 2, 3, 4 ] ], marks := [ [ 6 ], [ 3, 3 ], [ 2, 2 ], [ 1, 1, 1, 1 ] ] ) gap> DisplayTom( c6 ); 1: 6 2: 3 3 3: 2 . 2 4: 1 1 1 1 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{TomDihedral} 'TomDihedral( )' 'TomDihedral' constructs the table of marks of the dihedral group of order . For each divisor $d$ of , a dihedral group of order $m = 2n$ contains subgroups of order $d$ according to the following rule. If $d$ is odd and divides $n$ then there is only one cyclic subgroup of order $d$. If $d$ is even and divides $n$ then there are a cyclic subgroup of order $d$ and two classes of dihedral subgroups of order $d$ which are cyclic, too, in the case $d = 2$, see example below). Otherwise, (i.e. if $d$ does not divide $n$, there is just one class of dihedral subgroups of order $d$. | gap> d12 := TomDihedral( 12 ); rec( name := [ "1", "2", "D_{2}a", "D_{2}b", "3", "D_{4}", "6", "D_{6}a", "D_{6}b", "D_{12}" ], subs := [ [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 2, 3, 4, 6 ], [ 1, 2, 5, 7 ], [ 1, 3, 5, 8 ], [ 1, 4, 5, 9 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] ], marks := [ [ 12 ], [ 6, 6 ], [ 6, 2 ], [ 6, 2 ], [ 4, 4 ], [ 3, 3, 1, 1, 1 ], [ 2, 2, 2, 2 ], [ 2, 2, 2, 2 ], [ 2, 2, 2, 2 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ] ) gap> DisplayTom( d12 ); 1: 12 2: 6 6 3: 6 . 2 4: 6 . . 2 5: 4 . . . 4 6: 3 3 1 1 . 1 7: 2 2 . . 2 . 2 8: 2 . 2 . 2 . . 2 9: 2 . . 2 2 . . . 2 10: 1 1 1 1 1 1 1 1 1 1 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{TomFrobenius} 'TomFrobenius(

, )' 'TomFrobenius' computes the table of marks of a Frobenius group of order $p q$, where $p$ is a prime and $q$ divides $p-1$. | gap> f20 := TomFrobenius( 5, 4 ); rec( name := [ "1", "2", "4", "5:1", "5:2", "5:4" ], subs := [ [ 1 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 4 ], [ 1, 2, 4, 5 ], [ 1, 2, 3, 4, 5, 6 ] ], marks := [ [ 20 ], [ 10, 2 ], [ 5, 1, 1 ], [ 4, 4 ], [ 2, 2, 2, 2 ], [ 1, 1, 1, 1, 1, 1 ] ] ) gap> DisplayTom( f20 ); 1: 20 2: 10 2 3: 5 1 1 4: 4 . . 4 5: 2 2 . 2 2 6: 1 1 1 1 1 1 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E Emacs . . . . . . . . . . . . . . . . . . . . . . . local emacs variables %% %% Local Variables: %% mode: outline %% outline-regexp: "\\\\Chapter\\|\\\\Section" %% fill-column: 73 %% eval: (hide-body) %% End: %%