The concept of a table of marks was introduced by W. Burnside in his book Theory of Groups of Finite Order
, see [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 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 B contained in a subgroup A can be derived.
A table of marks of a given group G can be constructed from the subgroup lattice of G (see 70.3). For several groups, the table of marks can be restored from the GAP library of tables of marks (see 70.13).
Given the table of marks of G, one can display it (see 70.4) and derive information about G and its Burnside ring from it (see 70.7, 70.8, 70.9). Moreover, tables of marks in GAP provide an easy access to the classes of subgroups of their underlying groups (see 70.10).
Let G be a finite group with n conjugacy classes of subgroups C_1, C_2, ..., C_n and representatives H_i ∈ C_i, 1 ≤ i ≤ n. The table of marks of G is defined to be the n × n matrix M = (m_ij) where the mark m_ij is the number of fixed points of the subgroup H_j in the action of G on the right cosets of H_i in G.
Since H_j can only have fixed points if it is contained in a point stabilizer the matrix M is lower triangular if the classes C_i are sorted according to the condition that if H_i is contained in a conjugate of H_j then i ≤ 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 the set of all permutation representations of G.
The marks 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 containing 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 = ( 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.
A brief survey of properties of tables of marks and a description of algorithms for the interactive construction of tables of marks using GAP can be found in [Pfe97].
A table of marks of a group G in GAP is represented by an immutable (see 12.6) object tom in the category IsTableOfMarks (70.6-2), with defining attributes SubsTom (70.7-1) and MarksTom (70.7-1). These two attributes encode the matrix of marks in a compressed form. The SubsTom (70.7-1) value of tom 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 matrix of marks which have nonzero entries. The marks themselves are stored via the MarksTom (70.7-1) value of tom, which is a list that contains for each entry in SubsTom( tom ) the corresponding nonzero value of the table of marks.
It is possible to create table of marks objects that do not store a group, moreover one can create a table of marks object from a matrix of marks (see TableOfMarks (70.3-1)). So it may happen that a table of marks object in GAP is in fact not the table of marks of a group. To some extent, the consistency of a table of marks object can be checked (see 70.9), but GAP knows no general way to prove or disprove that a given matrix of nonnegative integers is the matrix of marks for a group. Many functions for tables of marks work well without access to the group –this is one of the arguments why tables of marks are so useful–, but for example normalizers (see NormalizerTom (70.9-4)) and derived subgroups (see DerivedSubgroupTom (70.9-2)) of subgroups are in general not uniquely determined by the matrix of marks.
GAP tables of marks are assumed to be in lower triangular form, that is, if a subgroup from the conjugacy class corresponding to the i-th row is contained in a subgroup from the class corresponding to the j-th row j then i ≤ j.
The MarksTom (70.7-1) information can be computed from the values of the attributes NrSubsTom (70.7-2), LengthsTom (70.7-3), OrdersTom (70.7-2), and SubsTom (70.7-1). NrSubsTom (70.7-2) stores a list containing for each entry in the SubsTom (70.7-1) value the corresponding number of conjugates that are contained in a subgroup, LengthsTom (70.7-3) a list containing for each conjugacy class of subgroups its length, and OrdersTom (70.7-2) a list containing for each class of subgroups their order. So the MarksTom (70.7-1) value of tom may be missing provided that the values of NrSubsTom (70.7-2), LengthsTom (70.7-3), and OrdersTom (70.7-2) are stored in tom.
Additional information about a table of marks is needed by some functions. The class numbers of normalizers in G and the number of the derived subgroup of G can be stored via appropriate attributes (see NormalizersTom (70.9-4), DerivedSubgroupTom (70.9-2)).
If tom stores its group G and a bijection from the rows and columns of the matrix of marks of tom to the classes of subgroups of G then clearly normalizers, derived subgroup etc. can be computed from this information. But in general a table of marks need not have access to G, for example tom might have been constructed from a generic table of marks (see 70.12), or as table of marks of a factor group from a given table of marks (see FactorGroupTom (70.9-11)). Access to the group G is provided by the attribute UnderlyingGroup (70.7-7) if this value is set. Access to the relevant information about conjugacy classes of subgroups of G –compatible with the ordering of rows and columns of the marks in tom– is signalled by the filter IsTableOfMarksWithGens (70.10-3).
Several examples in this chapter require the GAP package TomLib (the GAP Library of Tables of Marks) to be available. If it is not yet loaded then we load it now.
gap> LoadPackage( "tomlib" ); true
‣ TableOfMarks( G ) | ( attribute ) |
‣ TableOfMarks( string ) | ( attribute ) |
‣ TableOfMarks( matrix ) | ( attribute ) |
In the first form, G must be a finite group, and TableOfMarks constructs the table of marks of G. This computation requires the knowledge of the complete subgroup lattice of G (see LatticeSubgroups (39.20-1)). If the lattice is not yet stored then it will be constructed. This may take a while if G is large. The result has the IsTableOfMarksWithGens (70.10-3) value true.
In the second form, string must be a string, and TableOfMarks gets the table of marks with name string from the GAP library (see 70.13). If no table of marks with this name is contained in the library then fail is returned.
In the third form, matrix must be a matrix or a list of rows describing a lower triangular matrix where the part above the diagonal is omitted. For such an argument matrix, TableOfMarks returns a table of marks object (see 70.2) for which matrix is the matrix of marks. Note that not every matrix (containing only nonnegative integers and having lower triangular shape) describes a table of marks of a group. Necessary conditions are checked with IsInternallyConsistent (70.9-1) (see 70.9), and fail is returned if matrix is proved not to describe a matrix of marks; but if TableOfMarks returns a table of marks object created from a matrix then it may still happen that this object does not describe the table of marks of a group.
For an overview of operations for table of marks objects, see the introduction to Chapter 70.
gap> tom:= TableOfMarks( AlternatingGroup( 5 ) ); TableOfMarks( Alt( [ 1 .. 5 ] ) ) gap> TableOfMarks( "J5" ); fail gap> a5:= TableOfMarks( "A5" ); TableOfMarks( "A5" ) 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> TableOfMarks( mat ); TableOfMarks( <9 classes> )
The following TableOfMarks methods for a group are installed.
If the group is known to be cyclic then TableOfMarks constructs the table of marks essentially without the group, instead the knowledge about the structure of cyclic groups is used.
If the lattice of subgroups is already stored in the group then TableOfMarks computes the table of marks from the lattice (see TableOfMarksByLattice (70.3-2)).
If the group is known to be solvable then TableOfMarks takes the lattice of subgroups (see LatticeSubgroups (39.20-1)) of the group –which means that the lattice is computed if it is not yet stored– and then computes the table of marks from it. This method is also accessible via the global function TableOfMarksByLattice (70.3-2).
If the group doesn't know its lattice of subgroups or its conjugacy classes of subgroups then the table of marks and the conjugacy classes of subgroups are computed at the same time by the cyclic extension method. Only the table of marks is stored because the conjugacy classes of subgroups or the lattice of subgroups can be easily read off (see LatticeSubgroupsByTom (70.3-3)).
Conversely, the lattice of subgroups of a group with known table of marks can be computed using the table of marks, via the function LatticeSubgroupsByTom (70.3-3). This is also installed as a method for LatticeSubgroups (39.20-1).
‣ TableOfMarksByLattice( G ) | ( function ) |
TableOfMarksByLattice computes the table of marks of the group G from the lattice of subgroups of G. This lattice is computed via LatticeSubgroups (39.20-1) if it is not yet stored in G. The function TableOfMarksByLattice is installed as a method for TableOfMarks (70.3-1) for solvable groups and groups with stored subgroup lattice, and is available as a global variable only in order to provide explicit access to this method.
‣ LatticeSubgroupsByTom( G ) | ( function ) |
LatticeSubgroupsByTom computes the lattice of subgroups of G from the table of marks of G, using RepresentativeTom (70.10-4).
‣ ViewObj( tom ) | ( method ) |
The default ViewObj (6.3-5) method for tables of marks prints the string "TableOfMarks", followed by –if known– the identifier (see Identifier (70.7-9)) or the group of the table of marks enclosed in brackets; if neither group nor identifier are known then just the number of conjugacy classes of subgroups is printed instead.
‣ PrintObj( tom ) | ( method ) |
The default PrintObj (6.3-5) method for tables of marks does the same as ViewObj (6.3-5), except that PrintObj (6.3-5) is used for the group instead of ViewObj (6.3-5).
‣ Display( tom[, arec] ) | ( method ) |
The default Display (6.3-6) method for a table of marks tom produces a formatted output of the marks in tom. 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. The number of columns printed at one time depends on the actual line length, which can be accessed and changed by the function SizeScreen (6.12-1).
An interactive variant of Display (6.3-6) is the Browse (Browse: Browse) method for tables of marks that is provided by the GAP package Browse, see Browse (Browse: Browse for tables of marks).
The optional second argument arec of Display (6.3-6) can be used to change the default style for displaying a table of marks. arec must be a record, its relevant components are the following.
classesa list of class numbers to select only the rows and columns of the matrix that correspond to this list for printing,
formone of the strings "subgroups", "supergroups"; in the former case, at position (i,j) of the matrix the number of conjugates of H_j contained in H_i is printed, and in the latter case, at position (i,j) the number of conjugates of H_i which contain H_j is printed.
gap> tom:= TableOfMarks( "A5" );; gap> Display( tom ); 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 gap> Display( tom, 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 gap> Display( tom, 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> Display( tom, 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
‣ SortedTom( tom, perm ) | ( operation ) |
SortedTom returns a table of marks where the rows and columns of the table of marks tom are reordered according to the permutation perm.
Note that in each table of marks in GAP, the matrix of marks is assumed to have lower triangular shape (see 70.2). If the permutation perm does not have this property then the functions for tables of marks might return wrong results when applied to the output of SortedTom.
The returned table of marks has only those attribute values stored that are known for tom and listed in TableOfMarksComponents (70.6-4).
gap> tom:= TableOfMarksCyclic( 6 );; Display( tom ); 1: 6 2: 3 3 3: 2 . 2 4: 1 1 1 1 gap> sorted:= SortedTom( tom, (2,3) );; Display( sorted ); 1: 6 2: 2 2 3: 3 . 3 4: 1 1 1 1 gap> wrong:= SortedTom( tom, (1,2) );; Display( wrong ); 1: 3 2: . 6 3: . 2 2 4: 1 1 1 1
‣ PermutationTom( tom ) | ( attribute ) |
For the table of marks tom of the group G stored as UnderlyingGroup (70.7-7) value of tom, PermutationTom is a permutation π such that the i-th conjugacy class of subgroups of G belongs to the i^π-th column and row of marks in tom.
This attribute value is bound only if tom was obtained from another table of marks by permuting with SortedTom (70.5-1), and there is no default method to compute its value.
The attribute is necessary because the original and the sorted table of marks have the same identifier and the same group, and information computed from the group may depend on the ordering of marks, for example the fusion from the ordinary character table of G into tom.
gap> MarksTom( tom )[2]; [ 3, 3 ] gap> MarksTom( sorted )[2]; [ 2, 2 ] gap> HasPermutationTom( sorted ); true gap> PermutationTom( sorted ); (2,3)
‣ InfoTom | ( info class ) |
is the info class for computations concerning tables of marks.
‣ IsTableOfMarks( obj ) | ( category ) |
Each table of marks belongs to this category.
‣ TableOfMarksFamily | ( family ) |
Each table of marks belongs to this family.
‣ TableOfMarksComponents | ( global variable ) |
The list TableOfMarksComponents is used when a table of marks object is created from a record via ConvertToTableOfMarks (70.6-5). TableOfMarksComponents contains at position 2i-1 a name of an attribute and at position 2i the corresponding attribute getter function.
‣ ConvertToTableOfMarks( record ) | ( function ) |
ConvertToTableOfMarks converts a record with components from TableOfMarksComponents (70.6-4) into a table of marks object with the corresponding attributes.
gap> record:= rec( MarksTom:= [ [ 4 ], [ 2, 2 ], [ 1, 1, 1 ] ], > SubsTom:= [ [ 1 ], [ 1, 2 ], [ 1, 2, 3 ] ] );; gap> ConvertToTableOfMarks( record );; gap> record; TableOfMarks( <3 classes> )
‣ MarksTom( tom ) | ( attribute ) |
‣ SubsTom( tom ) | ( attribute ) |
The matrix of marks (see 70.1) of the table of marks tom is stored in a compressed form where zeros are omitted, using the attributes MarksTom and SubsTom. If M is the square matrix of marks of tom (see MatTom (70.7-10)) then the SubsTom value of tom is a list that contains at position i the list of all positions of nonzero entries of the i-th row of M, and the MarksTom value of tom is a list that contains at position i the list of the corresponding marks.
MarksTom and SubsTom are defining attributes of tables of marks (see 70.2). There is no default method for computing the SubsTom value, and the default MarksTom method needs the values of NrSubsTom (70.7-2) and OrdersTom (70.7-2).
gap> a5:= TableOfMarks( "A5" ); TableOfMarks( "A5" ) gap> MarksTom( 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 ] ] gap> SubsTom( a5 ); [ [ 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 ] ]
‣ NrSubsTom( tom ) | ( attribute ) |
‣ OrdersTom( tom ) | ( attribute ) |
Instead of storing the marks (see MarksTom (70.7-1)) of the table of marks tom one can use a matrix which contains at position (i,j) the number of subgroups of conjugacy class j that are contained in one member of the conjugacy class i. These values are stored in the NrSubsTom value in the same way as the marks in the MarksTom (70.7-1) value.
OrdersTom returns a list that contains at position i the order of a representative of the i-th conjugacy class of subgroups of tom.
One can compute the NrSubsTom and OrdersTom values from the MarksTom (70.7-1) value of tom and vice versa.
gap> NrSubsTom( 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 ] ] gap> OrdersTom( a5 ); [ 1, 2, 3, 4, 5, 6, 10, 12, 60 ]
‣ LengthsTom( tom ) | ( attribute ) |
For a table of marks tom, LengthsTom returns a list of the lengths of the conjugacy classes of subgroups.
gap> LengthsTom( a5 ); [ 1, 15, 10, 5, 6, 10, 6, 5, 1 ]
‣ ClassTypesTom( tom ) | ( attribute ) |
ClassTypesTom distinguishes isomorphism types of the classes of subgroups of the table of marks tom as far as this is possible from the SubsTom (70.7-1) and MarksTom (70.7-1) values of tom.
Two subgroups are clearly not isomorphic if they have different orders. Moreover, isomorphic subgroups must contain the same number of subgroups of each type.
Each type is represented by a positive integer. ClassTypesTom returns the list which contains for each class of subgroups its corresponding type.
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 ]
‣ ClassNamesTom( tom ) | ( attribute ) |
ClassNamesTom constructs generic names for the conjugacy classes of subgroups of the table of marks tom. In general, the generic name of a class of non-cyclic subgroups consists of three parts and has the form "(o)_{t}l", where o indicates the order of the subgroup, t is a number that distinguishes different types of subgroups of the same order, and l is a letter that 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 (70.7-4)). 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 {} around the type are omitted if the type number has only one digit.
For classes of cyclic subgroups, the parentheses round the order and the type are omitted. Hence the most general form of their generic names is "o,l". 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", "(8)", "(9)", "(10)", "(12)a", "(12)b", "(18)", "(24)a", "(24)b", "(36)", "(60)a", "(60)b", "(360)" ]
‣ FusionsTom( tom ) | ( attribute ) |
For a table of marks tom, FusionsTom is a list of fusions into other tables of marks. Each fusion is a list of length two, the first entry being the Identifier (70.7-9)) value of the image table, the second entry being the list of images of the class positions of tom in the image table.
This attribute is mainly used for tables of marks in the GAP library (see 70.13).
gap> fus:= FusionsTom( a6 );; gap> fus[1]; [ "L3(4)", [ 1, 2, 3, 3, 14, 5, 9, 7, 15, 15, 24, 26, 27, 32, 33, 50, 57, 55, 63, 73, 77, 90 ] ]
‣ UnderlyingGroup( tom ) | ( attribute ) |
UnderlyingGroup is used to access an underlying group that is stored on the table of marks tom. There is no default method to compute an underlying group if it is not stored.
gap> UnderlyingGroup( a6 ); Group([ (1,2)(3,4), (1,2,4,5)(3,6) ])
‣ IdempotentsTom( tom ) | ( attribute ) |
‣ IdempotentsTomInfo( tom ) | ( attribute ) |
IdempotentsTom encodes the idempotents of the integral Burnside ring described by the table of marks tom. The return value is a list l of positive integers such that each row vector describing a primitive idempotent has value 1 at all positions with the same entry in l, and 0 at all other positions.
According to A. Dress [Dre69] (see also [Pfe97]), these idempotents correspond to the classes of perfect subgroups, and each such idempotent is the characteristic function of all those subgroups that arise by cyclic extension from the corresponding perfect subgroup (see CyclicExtensionsTom (70.9-7)).
IdempotentsTomInfo returns a record with components fixpointvectors and primidems, both bound to lists. The i-th entry of the fixpointvectors list is the 0-1-vector describing the i-th primitive idempotent, and the i-th entry of primidems is the decomposition of this idempotent in the rows of tom.
gap> IdempotentsTom( a5 ); [ 1, 1, 1, 1, 1, 1, 1, 1, 9 ] gap> IdempotentsTomInfo( a5 ); rec( fixpointvectors := [ [ 1, 1, 1, 1, 1, 1, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ], primidems := [ [ 1, -2, -1, 0, 0, 1, 1, 1 ], [ -1, 2, 1, 0, 0, -1, -1, -1, 1 ] ] )
‣ Identifier( tom ) | ( attribute ) |
The identifier of a table of marks tom is a string. It is used for printing the table of marks (see 70.4) and in fusions between tables of marks (see FusionsTom (70.7-6)).
If tom is a table of marks from the GAP library of tables of marks (see 70.13) then it has an identifier, and if tom was constructed from a group with Name (12.8-2) then this name is chosen as Identifier value. There is no default method to compute an identifier in all other cases.
gap> Identifier( a5 ); "A5"
‣ MatTom( tom ) | ( attribute ) |
MatTom returns the square matrix of marks (see 70.1) of the table of marks tom which is stored in a compressed form using the attributes MarksTom (70.7-1) and SubsTom (70.7-1) This may need substantially more space than the values of MarksTom (70.7-1) and SubsTom (70.7-1).
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 ] ]
‣ MoebiusTom( tom ) | ( attribute ) |
MoebiusTom computes the Möbius values both of the subgroup lattice of the group G with table of marks tom and of the poset of conjugacy classes of subgroups of G. It returns a record where the component mu contains the Möbius values of the subgroup lattice, and the component nu contains the Möbius values of the poset.
Moreover, according to an observation of Isaacs et al. (see [HIÖ89], [Pah93]), the values on the subgroup lattice often can be derived from those of the poset of conjugacy classes. These expected values
are returned in the component ex, and the list of numbers of those subgroups where the expected value does not coincide with the actual value are returned in the component hyp. For the computation of these values, the position of the derived subgroup of G is needed (see DerivedSubgroupTom (70.9-2)). If it is not uniquely determined then the result does not have the components ex and hyp.
gap> MoebiusTom( a5 ); rec( ex := [ -60, 4, 2,,, -1, -1, -1, 1 ], hyp := [ ], mu := [ -60, 4, 2,,, -1, -1, -1, 1 ], nu := [ -1, 2, 1,,, -1, -1, -1, 1 ] ) gap> tom:= TableOfMarks( "M12" );; gap> moebius:= MoebiusTom( tom );; gap> moebius.hyp; [ 1, 2, 4, 16, 39, 45, 105 ] gap> moebius.mu[1]; moebius.ex[1]; 95040 190080
‣ WeightsTom( tom ) | ( attribute ) |
WeightsTom extracts the weights from the table of marks tom, i.e., the diagonal entries of the matrix of marks (see MarksTom (70.7-1)), 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:= MarksTom( 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 ] ]
For a table of marks tom of a group G, the following properties have the same meaning as the corresponding properties for G. Additionally, if a positive integer sub is given as the second argument then the value of the corresponding property for the sub-th class of subgroups of tom is returned.
‣ IsAbelianTom( tom[, sub] ) | ( property ) |
‣ IsCyclicTom( tom[, sub] ) | ( property ) |
‣ IsNilpotentTom( tom[, sub] ) | ( property ) |
‣ IsPerfectTom( tom[, sub] ) | ( property ) |
‣ IsSolvableTom( tom[, sub] ) | ( property ) |
gap> tom:= TableOfMarks( "A5" );; gap> IsAbelianTom( tom ); IsPerfectTom( tom ); false true gap> IsAbelianTom( tom, 3 ); IsNilpotentTom( tom, 7 ); true false gap> IsPerfectTom( tom, 7 ); IsSolvableTom( tom, 7 ); false true 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
‣ IsInternallyConsistent( tom ) | ( method ) |
For a table of marks tom, IsInternallyConsistent decomposes all tensor products of rows of tom. It returns true if all decomposition numbers are nonnegative integers, and false otherwise. This provides a strong consistency check for a table of marks.
‣ DerivedSubgroupTom( tom, sub ) | ( operation ) |
‣ DerivedSubgroupsTom( tom ) | ( function ) |
For a table of marks tom and a positive integer sub, DerivedSubgroupTom returns either a positive integer i or a list l of positive integers. In the former case, the result means that the derived subgroups of the subgroups in the sub-th class of tom lie in the i-th class. In the latter case, the class of the derived subgroups could not be uniquely determined, and the position of the class of derived subgroups is an entry of l.
Values computed with DerivedSubgroupTom are stored using the attribute DerivedSubgroupsTomPossible (70.9-3).
DerivedSubgroupsTom is just the list of DerivedSubgroupTom values for all values of sub.
‣ DerivedSubgroupsTomPossible( tom ) | ( attribute ) |
‣ DerivedSubgroupsTomUnique( tom ) | ( attribute ) |
Let tom be a table of marks. The value of the attribute DerivedSubgroupsTomPossible is a list in which the value at position i –if bound– is a positive integer or a list; the meaning of the entry is the same as in DerivedSubgroupTom (70.9-2).
If the value of the attribute DerivedSubgroupsTomUnique is known for tom then it is a list of positive integers, the value at position i being the position of the class of derived subgroups of the i-th class of subgroups in tom.
The derived subgroups are in general not uniquely determined by the table of marks if no UnderlyingGroup (70.7-7) value is stored, so there is no default method for DerivedSubgroupsTomUnique. But in some cases the derived subgroups are explicitly set when the table of marks is constructed. In this case, DerivedSubgroupTom (70.9-2) does not set values in the DerivedSubgroupsTomPossible list.
The DerivedSubgroupsTomUnique value is automatically set when the last missing unique value is entered in the DerivedSubgroupsTomPossible list by DerivedSubgroupTom (70.9-2).
‣ NormalizerTom( tom, sub ) | ( operation ) |
‣ NormalizersTom( tom ) | ( attribute ) |
Let tom be the table of marks of a group G. NormalizerTom tries to find the conjugacy class of the normalizer N in G of a subgroup U in the sub-th class of tom. The return value is either the list of class numbers of those subgroups that have the right size and contain the subgroup and all subgroups that clearly contain it as a normal subgroup, or the class number of the normalizer if it is uniquely determined by these conditions. If tom knows the subgroup lattice of G (see IsTableOfMarksWithGens (70.10-3)) then all normalizers are uniquely determined. NormalizerTom should never return an empty list.
NormalizersTom returns the list of positions of the classes of normalizers of subgroups in tom. In addition to the criteria for a single class of subgroup used by NormalizerTom, the approximations of normalizers for several classes are used and thus NormalizersTom may return better approximations than NormalizerTom.
gap> NormalizerTom( a5, 4 ); 8 gap> NormalizersTom( a5 ); [ 9, 4, 6, 8, 7, 6, 7, 8, 9 ]
The example shows that a subgroup with class number 4 in A_5 (which is a Kleinian four group) is normalized by a subgroup in class 8. This class contains the subgroups of A_5 which are isomorphic to A_4.
‣ ContainedTom( tom, sub1, sub2 ) | ( operation ) |
ContainedTom returns the number of subgroups in class sub1 of the table of marks tom that are contained in one fixed member of the class sub2.
gap> ContainedTom( a5, 3, 5 ); ContainedTom( a5, 3, 8 ); 0 4
‣ ContainingTom( tom, sub1, sub2 ) | ( operation ) |
ContainingTom returns the number of subgroups in class sub2 of the table of marks tom that contain one fixed member of the class sub1.
gap> ContainingTom( a5, 3, 5 ); ContainingTom( a5, 3, 8 ); 0 2
‣ CyclicExtensionsTom( tom, p ) | ( operation ) |
‣ CyclicExtensionsTom( tom[, list] ) | ( attribute ) |
According to A. Dress [Dre69], two columns of the table of marks tom are equal modulo the prime p if and only if the corresponding subgroups are connected by a chain of normal extensions of order p.
Called with tom and p, CyclicExtensionsTom returns the classes of this equivalence relation.
In the second form, list must be a list of primes, and the return value is the list of classes of the relation obtained by considering chains of normal extensions of prime order where all primes are in list. The default value for list is the set of prime divisors of the order of the group of tom.
(This information is not used by NormalizerTom (70.9-4) although it might give additional restrictions in the search of normalizers.)
gap> CyclicExtensionsTom( a5, 2 ); [ [ 1, 2, 4 ], [ 3, 6 ], [ 5, 7 ], [ 8 ], [ 9 ] ]
‣ DecomposedFixedPointVector( tom, fix ) | ( operation ) |
Let tom be the table of marks of a group G and let fix be a vector of fixed point numbers w.r.t. an action of G, i.e., a vector which contains for each class of subgroups the number of fixed points under the given action. DecomposedFixedPointVector returns the decomposition of fix into rows of the table of marks. This decomposition corresponds to a decomposition of the action into transitive constituents. Trailing zeros in fix may be omitted.
gap> DecomposedFixedPointVector( a5, [ 16, 4, 1, 0, 1, 1, 1 ] ); [ 0, 0, 0, 0, 0, 1, 1 ]
The vector fix 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, 0, 1/2, 1 ]
‣ EulerianFunctionByTom( tom, n[, sub] ) | ( operation ) |
Called with two arguments, EulerianFunctionByTom computes the Eulerian function (see EulerianFunction (39.16-3)) of the underlying group G of the table of marks tom, that is, the number of n-tuples of elements in G that generate G. If the optional argument sub is given then EulerianFunctionByTom computes the Eulerian function of each subgroup in the sub-th class of subgroups of tom.
For a group G whose table of marks is known, EulerianFunctionByTom is installed as a method for EulerianFunction (39.16-3).
gap> EulerianFunctionByTom( a5, 2 ); 2280 gap> EulerianFunctionByTom( a5, 3 ); 200160 gap> EulerianFunctionByTom( a5, 2, 3 ); 8
‣ IntersectionsTom( tom, sub1, sub2 ) | ( operation ) |
The intersections of the groups in the sub1-th conjugacy class of subgroups of the table of marks tom with the groups in the sub2-th conjugacy classes of subgroups of tom are determined up to conjugacy by the decomposition of the tensor product of their rows of marks. IntersectionsTom returns a list l that describes this decomposition. The i-th entry in l is the multiplicity of groups in the i-th conjugacy class as an intersection.
gap> IntersectionsTom( a5, 8, 8 ); [ 0, 0, 1, 0, 0, 0, 0, 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.
‣ FactorGroupTom( tom, n ) | ( operation ) |
For a table of marks tom of a group G and the normal subgroup N of G corresponding to the n-th class of subgroups of tom, FactorGroupTom returns the table of marks of the factor group G / N.
gap> s4:= TableOfMarks( SymmetricGroup( 4 ) ); TableOfMarks( Sym( [ 1 .. 4 ] ) ) gap> LengthsTom( s4 ); [ 1, 3, 6, 4, 1, 3, 3, 4, 3, 1, 1 ] gap> OrdersTom( s4 ); [ 1, 2, 2, 3, 4, 4, 4, 6, 8, 12, 24 ] gap> s3:= FactorGroupTom( s4, 5 ); TableOfMarks( Group([ f1, f2 ]) ) gap> Display( s3 ); 1: 6 2: 3 1 3: 2 . 2 4: 1 1 1 1
‣ MaximalSubgroupsTom( tom[, sub] ) | ( attribute ) |
Called with a table of marks tom, MaximalSubgroupsTom returns a list of length two, the first entry being the list of positions of the classes of maximal subgroups of the whole group of tom, the second entry being the list of class lengths of these groups.
Called with a table of marks tom and a position sub, the same information for the sub-th class of subgroups is returned.
gap> MaximalSubgroupsTom( s4 ); [ [ 10, 9, 8 ], [ 1, 3, 4 ] ] gap> MaximalSubgroupsTom( s4, 10 ); [ [ 5, 4 ], [ 1, 4 ] ]
‣ MinimalSupergroupsTom( tom, sub ) | ( operation ) |
For a table of marks tom, MinimalSupergroupsTom returns a list of length two, the first entry being the list of positions of the classes containing the minimal supergroups of the groups in the sub-th class of subgroups of tom, the second entry being the list of class lengths of these groups.
gap> MinimalSupergroupsTom( s4, 5 ); [ [ 9, 10 ], [ 3, 1 ] ]
Let tom be the table of marks of the group G, and assume that tom has access to G via the UnderlyingGroup (70.7-7) value. Then it makes sense to use tom and its ordering of conjugacy classes of subgroups of G for storing information for constructing representatives of these classes. The group G is in general not sufficient for this, tom needs more information; this is available if and only if the IsTableOfMarksWithGens (70.10-3) value of tom is true. In this case, RepresentativeTom (70.10-4) can be used to get a subgroup of the i-th class, for all i.
GAP provides two different possibilities to store generators of the representatives of classes of subgroups. The first is implemented by the attribute GeneratorsSubgroupsTom (70.10-1), which uses explicit generators of the subgroups. The second, more general, possibility is implemented by the attribute StraightLineProgramsTom (70.10-2), which encodes the generators as straight line programs (see 37.8) that evaluate to the generators in question when applied to standard generators of G. This means that on the one hand, standard generators of G must be known in order to use StraightLineProgramsTom (70.10-2). On the other hand, the straight line programs allow one to compute easily generators not only of a subgroup U of G but also generators of the image of U in any representation of G, provided that one knows standard generators of the image of G under this representation. See the manual of the package TomLib for details and an example.
‣ GeneratorsSubgroupsTom( tom ) | ( attribute ) |
Let tom be a table of marks with IsTableOfMarksWithGens (70.10-3) value true. Then GeneratorsSubgroupsTom returns a list of length two, the first entry being a list l of elements of the group stored as UnderlyingGroup (70.7-7) value of tom, the second entry being a list that contains at position i a list of positions in l of generators of a representative of a subgroup in class i.
The GeneratorsSubgroupsTom value is known for all tables of marks that have been computed with TableOfMarks (70.3-1) from a group, and there is a method to compute the value for a table of marks that admits RepresentativeTom (70.10-4).
‣ StraightLineProgramsTom( tom ) | ( attribute ) |
For a table of marks tom with IsTableOfMarksWithGens (70.10-3) value true, StraightLineProgramsTom returns a list that contains at position i either a list of straight line programs or a straight line program (see 37.8), encoding the generators of a representative of the i-th conjugacy class of subgroups of UnderlyingGroup( tom ); in the former case, each straight line program returns a generator, in the latter case, the program returns the list of generators.
There is no default method to compute the StraightLineProgramsTom value of a table of marks if they are not yet stored. The value is known for all tables of marks that belong to the GAP library of tables of marks (see 70.13).
‣ IsTableOfMarksWithGens( tom ) | ( filter ) |
This filter shall express the union of the filters IsTableOfMarks and HasStraightLineProgramsTom and IsTableOfMarks and HasGeneratorsSubgroupsTom. If a table of marks tom has this filter set then tom can be asked to compute information that is in general not uniquely determined by a table of marks, for example the positions of derived subgroups or normalizers of subgroups (see DerivedSubgroupTom (70.9-2), NormalizerTom (70.9-4)).
gap> a5:= TableOfMarks( "A5" );; IsTableOfMarksWithGens( a5 ); true gap> HasGeneratorsSubgroupsTom( a5 ); HasStraightLineProgramsTom( a5 ); false true gap> alt5:= TableOfMarks( AlternatingGroup( 5 ) );; gap> IsTableOfMarksWithGens( alt5 ); true gap> HasGeneratorsSubgroupsTom(alt5); HasStraightLineProgramsTom(alt5); true false gap> progs:= StraightLineProgramsTom( a5 );; gap> OrdersTom( a5 ); [ 1, 2, 3, 4, 5, 6, 10, 12, 60 ] gap> IsCyclicTom( a5, 4 ); false gap> Length( progs[4] ); 2 gap> progs[4][1]; <straight line program> gap> # first generator of an el. ab group of order 4: gap> Display( progs[4][1] ); # input: r:= [ g1, g2 ]; # program: r[3]:= r[2]*r[1]; r[4]:= r[3]*r[2]^-1*r[1]*r[3]*r[2]^-1*r[1]*r[2]; # return value: r[4] gap> x:= [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ];; gap> y:= [ [ Z(2^2), Z(2)^0 ], [ 0*Z(2), Z(2^2)^2 ] ];; gap> res1:= ResultOfStraightLineProgram( progs[4][1], [ x, y ] ); [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2)^2, Z(2)^0 ] ] gap> res2:= ResultOfStraightLineProgram( progs[4][2], [ x, y ] ); [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ] gap> w:= y*x;; gap> res1 = w*y^-1*x*w*y^-1*x*y; true gap> subgrp:= Group( res1, res2 );; Size( subgrp ); IsCyclic( subgrp ); 4 false
‣ RepresentativeTom( tom, sub ) | ( operation ) |
‣ RepresentativeTomByGenerators( tom, sub, gens ) | ( operation ) |
‣ RepresentativeTomByGeneratorsNC( tom, sub, gens ) | ( operation ) |
Let tom be a table of marks with IsTableOfMarksWithGens (70.10-3) value true, and sub a positive integer. RepresentativeTom returns a representative of the sub-th conjugacy class of subgroups of tom.
If the attribute StraightLineProgramsTom (70.10-2) is set in tom then methods for the operations RepresentativeTomByGenerators and RepresentativeTomByGeneratorsNC are available, which return a representative of the sub-th conjugacy class of subgroups of tom, as a subgroup of the group generated by gens. This means that the standard generators of tom are replaced by gens.
RepresentativeTomByGenerators checks whether mapping the standard generators of tom to gens extends to a group isomorphism, and returns fail if not. RepresentativeTomByGeneratorsNC omits all checks. So RepresentativeTomByGenerators is thought mainly for debugging purposes; note that when several representatives are constructed, it is cheaper to construct (and check) the isomorphism once, and to map the groups returned by RepresentativeTom under this isomorphism. The idea behind RepresentativeTomByGeneratorsNC, however, is to avoid the overhead of using isomorphisms when gens are known to be standard generators. In order to proceed like this, the attribute StraightLineProgramsTom (70.10-2) is needed.
gap> RepresentativeTom( a5, 4 ); Group([ (2,3)(4,5), (2,4)(3,5) ])
The following examples require the GAP Character Table Library to be available. If it is not yet loaded then we load it now.
gap> LoadPackage( "ctbllib" ); true
‣ FusionCharTableTom( tbl, tom ) | ( operation ) |
‣ PossibleFusionsCharTableTom( tbl, tom[, options] ) | ( operation ) |
Let tbl be the ordinary character table of the group G, say, and tom the table of marks of G. FusionCharTableTom determines the fusion of the classes of elements from tbl to the classes of cyclic subgroups on tom, that is, a list that contains at position i the position of the class of cyclic subgroups in tom that are generated by elements in the i-th conjugacy class of elements in tbl.
Three cases are handled differently.
The fusion is explicitly stored on tbl. Then nothing has to be done. This happens only if both tbl and tom are tables from the GAP library (see 70.13 and the manual of the GAP Character Table Library).
The UnderlyingGroup (70.7-7) values of tbl and tom are known and equal. Then the group is used to compute the fusion.
There is neither fusion nor group information available. In this case only necessary conditions can be checked, and if they are not sufficient to determine the fusion uniquely then fail is returned by FusionCharTableTom.
PossibleFusionsCharTableTom computes the list of possible fusions from tbl to tom, according to the criteria that have been checked. So if FusionCharTableTom returns a unique fusion then the list returned by PossibleFusionsCharTableTom for the same arguments contains exactly this fusion, and if FusionCharTableTom returns fail then the length of this list is different from 1.
The optional argument options must be a record that may have the following components.
fusionmapa parametrized map which is an approximation of the desired map,
quicka Boolean; if true then as soon as only one possibility remains this possibility is returned immediately; the default value is false.
gap> a5c:= CharacterTable( "A5" );; gap> fus:= FusionCharTableTom( a5c, a5 ); [ 1, 2, 3, 5, 5 ]
‣ PermCharsTom( fus, tom ) | ( operation ) |
‣ PermCharsTom( tbl, tom ) | ( operation ) |
PermCharsTom returns the list of transitive permutation characters from the table of marks tom. In the first form, fus must be the fusion map from the ordinary character table of the group of tom to tom (see FusionCharTableTom (70.11-1)). In the second form, tbl must be the character table of the group of which tom is the table of marks. If the fusion map is not uniquely determined (see FusionCharTableTom (70.11-1)) then fail is returned.
If the fusion map fus is given as first argument then each transitive permutation character is represented by its values list. If the character table tbl is given then the permutation characters are class function objects (see Chapter 72).
gap> PermCharsTom( a5c, a5 ); [ Character( CharacterTable( "A5" ), [ 60, 0, 0, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 30, 2, 0, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 15, 3, 0, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 12, 0, 0, 2, 2 ] ), Character( CharacterTable( "A5" ), [ 10, 2, 1, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 6, 2, 0, 1, 1 ] ), Character( CharacterTable( "A5" ), [ 5, 1, 2, 0, 0 ] ), Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ) ] gap> PermCharsTom( fus, a5 )[1]; [ 60, 0, 0, 0, 0 ]
The following three operations construct a table of marks only from the data given, i.e., without underlying group.
‣ TableOfMarksCyclic( n ) | ( operation ) |
TableOfMarksCyclic returns the table of marks of the cyclic group of order n.
A cyclic group of order n has as its subgroups for each divisor d of n a cyclic subgroup of order d.
gap> Display( TableOfMarksCyclic( 6 ) ); 1: 6 2: 3 3 3: 2 . 2 4: 1 1 1 1
‣ TableOfMarksDihedral( n ) | ( operation ) |
TableOfMarksDihedral returns the table of marks of the dihedral group of order m.
For each divisor d of m, 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 the example below). Otherwise (i.e., if d does not divide n) there is just one class of dihedral subgroups of order d.
gap> Display( TableOfMarksDihedral( 12 ) ); 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
‣ TableOfMarksFrobenius( p, q ) | ( operation ) |
TableOfMarksFrobenius computes the table of marks of a Frobenius group of order p q, where p is a prime and q divides p-1.
gap> Display( TableOfMarksFrobenius( 5, 4 ) ); 1: 20 2: 10 2 3: 5 1 1 4: 4 . . 4 5: 2 2 . 2 2 6: 1 1 1 1 1 1
The GAP package TomLib provides access to several hundred tables of marks of almost simple groups and their maximal subgroups. If this package is installed then the tables from this database can be accessed via TableOfMarks (70.3-1) with argument a string. If also the GAP Character Table Library is installed and contains the ordinary character table of the group for which one wants to fetch the table of marks then one can also call TableOfMarks (70.3-1) with argument the character table.
A list of all names of tables of marks that are provided by the TomLib package can be obtained via AllLibTomNames (TomLib: AllLibTomNames).
gap> names:= AllLibTomNames();; gap> "A5" in names; true
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