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 Constructing Tables of Marks). For several groups, the table of marks can be restored from the GAP library of tables of marks (see The Library of Tables of Marks).
Given the table of marks of G, one can display it (see Printing Tables of Marks) and derive information about G and its Burnside ring from it (see Attributes of Tables of Marks, Properties of Tables of Marks, Other Operations for Tables of Marks). Moreover, tables of marks in GAP provide an easy access to the classes of subgroups of their underlying groups (see Accessing Subgroups via Tables of Marks).
Let G be a finite group with n conjugacy classes of subgroups C1, C2, …, Cn and representatives Hi ∈ Ci, 1 ≤ i ≤ n. The table of marks of G is defined to be the n ×n matrix M = (mij) where the mark mij is the number of fixed points of the subgroup Hj in the action of G on the right cosets of Hi in G.
Since Hj can only have fixed points if it is contained in a point stablizer the matrix M is lower triangular if the classes Ci are sorted according to the condition that if Hi is contained in a conjugate of Hj then i ≤ j.
Moreover, the diagonal entries mii are nonzero since mii equals the index of Hi 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 mij have further meanings.
If H1 is the trivial subgroup of G then each mark mi1 in the
first column of M is equal to the index of Hi in G
since the trivial subgroup fixes all cosets of Hi.
If Hn = G then each mnj in the last row of M is equal to 1
since there is only one coset of G in G.
In general, mij equals the number of conjugates of Hi containing
Hj, multiplied by the index of Hi in its normalizer in G.
Moreover, the number cij of conjugates of Hj which are contained
in Hi can be derived from the marks mij via the formula
|
Both the marks mij and the numbers of subgroups cij 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 Mutability and Copyability) object tom in the category
IsTableOfMarks (see IsTableOfMarks), with defining attributes
SubsTom (see SubsTom) and MarksTom (see MarksTom).
These two attributes encode the matrix of marks in a compressed form.
The SubsTom 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 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). 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 Other Operations for Tables of Marks), 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) and derived subgroups (see DerivedSubgroupTom) 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 information can be computed from the values of the
attributes NrSubsTom, LengthsTom, OrdersTom, and SubsTom
(see NrSubsTom, LengthsTom, OrdersTom).
NrSubsTom stores a list containing for each entry in the SubsTom
value the corresponding number of conjugates that are contained
in a subgroup, LengthsTom a list containing for each conjugacy class
of subgroups its length, and OrdersTom a list containing for each
class of subgroups their order.
So the MarksTom value of tom may be missing provided that the values
of NrSubsTom, LengthsTom, and OrdersTom 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, DerivedSubgroupTom).
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 Generic Construction of Tables of Marks),
or as table of marks of a factor group from a given table of marks
(see FactorGroupTom).
Access to the group G is provided by the attribute UnderlyingGroup
(see UnderlyingGroup!for tables of marks) 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
(see Accessing Subgroups via Tables of Marks).
Several examples in this chapter require 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 ) A
TableOfMarks( string ) A
TableOfMarks( matrix ) A
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).
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 value true
(see Accessing Subgroups via Tables of Marks).
In the second form, string must be a string, and TableOfMarks gets
the table of marks with name string from the GAP library
(see The Library of Tables of Marks).
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 Table of Marks Objects in GAP)
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
(see Other Operations for Tables of Marks), 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 the Chapter Tables of Marks.
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.
TableOfMarks constructs the
table of marks essentially without the group, instead the knowledge
about the structure of cyclic groups is used.
TableOfMarks computes the table of marks from the lattice
(see TableOfMarksByLattice).
TableOfMarks takes the
lattice of subgroups (see LatticeSubgroups) 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 (see TableOfMarksByLattice).
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.
This is also installed as a method for LatticeSubgroups.
TableOfMarksByLattice( G ) F
TableOfMarksByLattice computes the table of marks of the group G from
the lattice of subgroups of G.
This lattice is computed via LatticeSubgroups (see LatticeSubgroups)
if it is not yet stored in G.
The function TableOfMarksByLattice is installed as a method for
TableOfMarks 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 ) F
LatticeSubgroupsByTom computes the lattice of subgroups of G from the
table of marks of G, using RepresentativeTom
(see RepresentativeTom).
The default ViewObj (see ViewObj) method for tables of marks prints
the string "TableOfMarks", followed by --if known-- the identifier
(see Identifier!for tables of marks) 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.
The default PrintObj (see PrintObj) method for tables of marks
does the same as ViewObj,
except that the group is is Print-ed instead of View-ed.
The default Display (see Display) 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 (see SizeScreen).
The optional second argument arec of Display can be used to change
the default style for displaying a character as shown above.
arec must be a record, its relevant components are the following.
classes
form "subgroups", "supergroups";
in the former case, at position (i,j) of the matrix the number of
conjugates of Hj contained in Hi is printed,
and in the latter case, at position (i,j) the number of conjugates
of Hi which contain Hj 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 ) O
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 Table of Marks Objects in GAP).
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
(see TableOfMarksComponents).
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 ) A
For the table of marks tom of the group G stored as UnderlyingGroup
value of tom (see UnderlyingGroup!for tables of marks),
PermutationTom is a permutation π such that the i-th conjugacy
class of subgroups of G belongs to the iπ-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 (see SortedTom),
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 V
is the info class for computations concerning tables of marks.
IsTableOfMarks( obj ) C
Each table of marks belongs to this category.
TableOfMarksFamily V
Each table of marks belongs to this family.
TableOfMarksComponents V
The list TableOfMarksComponents is used when a table of marks object is
created from a record via ConvertToTableOfMarks
(see ConvertToTableOfMarks).
TableOfMarksComponents contains at position 2i−1 a name of an
attribute and at position 2i the corresponding attribute getter
function.
ConvertToTableOfMarks( record ) F
ConvertToTableOfMarks converts a record with components from
TableOfMarksComponents 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 ) A
SubsTom( tom ) A
The matrix of marks (see More about Tables of Marks) 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) 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 Table of Marks Objects in GAP).
There is no default method for computing the SubsTom value,
and the default MarksTom method needs the values of NrSubsTom and
OrdersTom (see NrSubsTom, OrdersTom).
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 ) A
OrdersTom( tom ) A
Instead of storing the marks (see MarksTom) 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 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 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 ) A
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 ) A
ClassTypesTom distinguishes isomorphism types of the classes of
subgroups of the table of marks tom as far as this is possible
from the SubsTom and MarksTom 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 ) A
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)_tl",
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).
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 ) AM
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 (see Identifier!for tables of marks) 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 The Library of Tables of Marks).
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 ) A
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 ) A
IdempotentsTomInfo( tom ) A
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).
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(
primidems := [ [ 1, -2, -1, 0, 0, 1, 1, 1 ], [ -1, 2, 1, 0, 0, -1, -1, -1,
1 ] ],
fixpointvectors := [ [ 1, 1, 1, 1, 1, 1, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0,
0, 1 ] ] )
Identifier( tom ) A
The identifier of a table of marks tom is a string. It is used for printing the table of marks (see Printing Tables of Marks) and in fusions between tables of marks (see FusionsTom).
If tom is a table of marks from the GAP library of tables of marks
(see The Library of Tables of Marks) then it has an identifier,
and if tom was constructed from a group with Name value (see Name)
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 ) A
MatTom returns the square matrix of marks
(see More about Tables of Marks) of the table of marks tom which is
stored in a compressed form using the attributes MarksTom and SubsTom
(see MarksTom).
This may need substantially more space than the values of MarksTom and
SubsTom.
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 ) A
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 HIO89,
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).
If it is not uniquely determined then the result does not have the
components ex and 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 := [ ] )
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 ) A
WeightsTom extracts the weights from the table of marks tom,
i.e., the diagonal entries of the matrix of marks (see MarksTom),
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] )
IsCyclicTom( tom[, sub] )
IsNilpotentTom( tom[, sub] )
IsPerfectTom( tom[, sub] )
IsSolvableTom( tom[, sub] )
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 ) O
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 ) O
DerivedSubgroupsTom( tom ) F
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 (see DerivedSubgroupsTomPossible).
DerivedSubgroupsTom is just the list of DerivedSubgroupTom values for
all values of sub.
DerivedSubgroupsTomPossible( tom ) AM
DerivedSubgroupsTomUnique( tom ) A
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
(see DerivedSubgroupTom).
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 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.
The DerivedSubgroupsTomUnique value is automatically set when the last
missing unique value is entered in the DerivedSubgroupsTomPossible list
by DerivedSubgroupTom.
gap> a5:= TableOfMarks( "A5" ); TableOfMarks( "A5" ) gap> DerivedSubgroupTom( a5, 2 ); 1 gap> DerivedSubgroupsTom( a5 ); [ 1, 1, 1, 1, 1, 3, 5, 4, 9 ]
NormalizerTom( tom, sub ) O
NormalizersTom( tom ) A
Let tom be the table of marks of a group G, say.
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)
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 A5 (which is a Kleinian four group) is normalized by a subgroup in class 8. This class contains the subgroups of A5 which are isomorphic to A4.
ContainedTom( tom, sub1, sub2 ) O
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.
ContainingTom( tom, sub1, sub2 ) O
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> ContainedTom( a5, 3, 5 ); ContainedTom( a5, 3, 8 ); 0 4 gap> ContainingTom( a5, 3, 5 ); ContainingTom( a5, 3, 8 ); 0 2
CyclicExtensionsTom( tom ) A
CyclicExtensionsTom( tom, p ) O
CyclicExtensionsTom( tom, list ) O
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.
In the second form, CyclicExtensionsTom returns the classes of this
equivalence relation.
In the third 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.
In the first form, the result is the same as in the third form,
with second argument the set of prime divisors of the size of the group
of tom.
(This information is not used by NormalizerTom (see 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 ] ]
DecomposedFixedPointVector( tom, fix ) O
Let tom be the table of marks of the group G, say,
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] ) O
In the first form EulerianFunctionByTom computes the Eulerian
function (see EulerianFunction) 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.
In the second form 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 (see EulerianFunction).
gap> EulerianFunctionByTom( a5, 2 ); 2280 gap> EulerianFunctionByTom( a5, 3 ); 200160 gap> EulerianFunctionByTom( a5, 2, 3 ); 8
IntersectionsTom( tom, sub1, sub2 ) O
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 (A4) of A5 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 ) O
For a table of marks tom of the group G, say,
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 ) A
MaximalSubgroupsTom( tom, sub ) O
In the first form 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 the table of marks tom,
the second entry being the list of class lengths of these groups.
In the second form the same information for the sub-th class of
subgroups is returned.
MinimalSupergroupsTom( tom, sub ) O
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> MaximalSubgroupsTom( s4 ); [ [ 10, 9, 8 ], [ 1, 3, 4 ] ] gap> MaximalSubgroupsTom( s4, 10 ); [ [ 5, 4 ], [ 1, 4 ] ] gap> MinimalSupergroupsTom( s4, 5 ); [ [ 9, 10 ], [ 3, 1 ] ]
An s-tuple of standard generators of a given group G is a vector (g1, g2, …, gs) of elements gi ∈ G satisfying certain conditions (depending on the isomorphism type of G) such that
StandardGeneratorsInfo( G ) A
When called with the group G,
StandardGeneratorsInfo returns a list of records with at least one of
the components script and description.
Each such record defines standard generators of groups isomorphic
to G, the i-th record is referred to as the i-th set of
standard generators for such groups.
The value of script is a dense list of lists, each encoding a command
that has one of the following forms.
StandardGeneratorsFunctions (see StandardGeneratorsFunctions)
that is followed by the list f of strings returns the value v
when it is called with G and
gi1k1 gi2k2 …gimkm.
generators
description script information in human readable form,
in terms of the generators value,
classnames
ATLAS true means that the standard generators coincide with
those defined in Rob Wilson's ATLAS of Group Representations
(see AGR), and false means that this property is not
guaranteed.
There is no default method for an arbitrary isomorphism type, since in general the definition of standard generators is not obvious.
The function StandardGeneratorsOfGroup
(see StandardGeneratorsOfGroup)
can be used to find standard generators of a given group isomorphic
to G.
The generators and description values, if not known, can be computed
by HumanReadableDefinition (see HumanReadableDefinition).
gap> StandardGeneratorsInfo( TableOfMarks( "L3(3)" ) );
[ rec( generators := "a, b",
description := "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4",
script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ],
[ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ],
ATLAS := true ) ]
HumanReadableDefinition( info ) F
ScriptFromString( string ) F
Let info be a record that is valid as value of StandardGeneratorsInfo
(see StandardGeneratorsInfo!for groups).
HumanReadableDefinition returns a string that describes the definition
of standard generators given by the script component of info in
human readable form.
The names of the generators are taken from the generators component
(default names "a", "b" etc. are computed if necessary),
and the result is stored in the description component.
ScriptFromString does the converse of HumanReadableDefinition, i.e.,
it takes a string string as returned by HumanReadableDefinition,
and returns a corresponding script list.
If ``condition'' lines occur in the script
(see StandardGeneratorsInfo!for groups)
then the functions that occur must be contained in
StandardGeneratorsFunctions (see StandardGeneratorsFunctions).
gap> scr:= ScriptFromString( "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4" );
[ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], [ 1, 1, 2, 1, 13 ],
[ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ]
gap> info:= rec( script:= scr );
rec( script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ],
[ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ] )
gap> HumanReadableDefinition( info );
"|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4"
gap> info;
rec( script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ],
[ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ],
generators := "a, b",
description := "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4" )
StandardGeneratorsFunctions V
StandardGeneratorsFunctions is a list of even length.
At position 2i−1, a function of two arguments is stored,
which are expected to be a group and a group element.
At position 2i a list of strings is stored such that first inserting a
generator name in all holes and then forming the concatenation yields
a string that describes the function at the previous position;
this string must contain the generator enclosed in round brackets (
and ).
This list is used by the functions StandardGeneratorsInfo
(see StandardGeneratorsInfo!for groups), HumanReadableDefinition, and
ScriptFromString (see HumanReadableDefinition).
Note that the lists at even positions must be pairwise different.
gap> StandardGeneratorsFunctions{ [ 1, 2 ] };
[ function( G, g ) ... end, [ "|C(",, ")|" ] ]
IsStandardGeneratorsOfGroup( info, G, gens ) F
Let info be a record that is valid as value of StandardGeneratorsInfo
(see StandardGeneratorsInfo!for groups), G a group, and gens a list
of generators for G.
In this case, IsStandardGeneratorsOfGroup returns true if gens
satisfies the conditions of the script component of info,
and false otherwise.
Note that the result true means that gens is a list of standard
generators for G only if G has the isomorphism type for which info
describes standard generators.
StandardGeneratorsOfGroup( info, G[, randfunc] ) F
Let info be a record that is valid as value of StandardGeneratorsInfo
(see StandardGeneratorsInfo!for groups),
and G a group of the isomorphism type for which info describes
standard generators.
In this case, StandardGeneratorsOfGroup returns a list of standard
generators (see Section Standard Generators of Groups) of G.
The optional argument randfunc must be a function that returns an
element of G when called with G; the default is PseudoRandom.
In each call to StandardGeneratorsOfGroup,
the script component of info is scanned line by line.
randfunc is used to find an element of the prescribed order
whenever a definition line is met,
and for the relation and condition lines in the script list,
the current generator candidates are checked;
if a condition is not fulfilled, all candidates are thrown away,
and the procedure starts again with the first line.
When the conditions are fulfilled after processing the last line
of the script list, the standard generators are returned.
Note that if G has the wrong isomorphism type then
StandardGeneratorsOfGroup returns a list of elements in G
that satisfy the conditions of the script component of info
if such elements exist, and does not terminate otherwise.
In the former case, obviously the returned elements need not be standard
generators of G.
gap> a5:= AlternatingGroup( 5 ); Alt( [ 1 .. 5 ] ) gap> info:= StandardGeneratorsInfo( TableOfMarks( "A5" ) )[1]; rec( generators := "a, b", description := "|a|=2, |b|=3, |ab|=5", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] ], ATLAS := true ) gap> IsStandardGeneratorsOfGroup( info, a5, [ (1,3)(2,4), (3,4,5) ] ); true gap> IsStandardGeneratorsOfGroup( info, a5, [ (1,3)(2,4), (1,2,3) ] ); false gap> s5:= SymmetricGroup( 5 );; gap> RepresentativeAction( s5, [ (1,3)(2,4), (3,4,5) ], > StandardGeneratorsOfGroup( info, a5 ), OnPairs ) <> fail; true
Let tom be the table of marks of the group G,
and assume that tom has access to G via the UnderlyingGroup value
(see UnderlyingGroup!for tables of marks).
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 value of
tom is true (see IsTableOfMarksWithGens).
In this case, RepresentativeTom (see RepresentativeTom) 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
(see GeneratorsSubgroupsTom), which uses explicit generators.
The second, more general, possibility is implemented by the attributes
StraightLineProgramsTom (see StraightLineProgramsTom) and
StandardGeneratorsInfo (see StandardGeneratorsInfo!for tables of marks).
The StraightLineProgramsTom value encodes the generators as
straight line programs (see Straight Line Programs) 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.
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 RepresentativeTomByGenerators for details and an example).
GeneratorsSubgroupsTom( tom ) A
Let tom be a table of marks with IsTableOfMarksWithGens 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 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 (see TableOfMarks) from a group,
and there is a method to compute the value for a table of marks that
admits RepresentativeTom (see RepresentativeTom).
StraightLineProgramsTom( tom ) A
For a table of marks tom with IsTableOfMarksWithGens value true,
StraightLineProgramsTom returns a list that contains at position i
either a list of straight line programs or a straight line program
(see Straight Line Programs), 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 The Library of Tables of Marks).
IsTableOfMarksWithGens( tom ) F
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, NormalizerTom).
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> Display( progs[4][1] ); # first generator of an el. ab group of order 4 # 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 ) O
RepresentativeTomByGenerators( tom, sub, gens ) O
RepresentativeTomByGeneratorsNC( tom, sub, gens ) O
Let tom be a table of marks with IsTableOfMarksWithGens value true
(see IsTableOfMarksWithGens), and sub a positive integer.
RepresentativeTom returns a representative of the sub-th conjugacy
class of subgroups of tom.
RepresentativeTomByGenerators and RepresentativeTomByGeneratorsNC
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.
gap> RepresentativeTom( a5, 4 ); Group([ (2,3)(4,5), (2,4)(3,5) ])
StandardGeneratorsInfo( tom ) A
For a table of marks tom, a stored value of StandardGeneratorsInfo
equals the value of this attribute for the underlying group
(see UnderlyingGroup!for tables of marks) of tom,
cf. Section Standard Generators of Groups.
In this case, the GeneratorsOfGroup value of the underlying group G
of tom is assumed to be in fact a list of standard generators for G;
So one should be careful when setting the StandardGeneratorsInfo value
by hand.
There is no default method to compute the StandardGeneratorsInfo value
of a table of marks if it is not yet stored.
gap> std:= StandardGeneratorsInfo( a5 );
[ rec( generators := "a, b", description := "|a|=2, |b|=3, |ab|=5",
script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] ], ATLAS := true ) ]
gap> # Now find standard generators of an isomorphic group.
gap> g:= SL(2,4);;
gap> repeat
> x:= PseudoRandom( g );
> until Order( x ) = 2;
gap> repeat
> y:= PseudoRandom( g );
> until Order( y ) = 3 and Order( x*y ) = 5;
gap> # Compute a representative w.r.t. these generators.
gap> RepresentativeTomByGenerators( a5, 4, [ x, y ] );
Group([ [ [ Z(2)^0, Z(2^2)^2 ], [ 0*Z(2), Z(2)^0 ] ],
[ [ Z(2)^0, Z(2^2) ], [ 0*Z(2), Z(2)^0 ] ] ])
gap> # Show that the new generators are really good.
gap> grp:= UnderlyingGroup( a5 );;
gap> iso:= GroupGeneralMappingByImages( grp, g,
> GeneratorsOfGroup( grp ), [ x, y ] );;
gap> IsGroupHomomorphism( iso );
true
gap> IsBijective( iso );
true
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 ) O
PossibleFusionsCharTableTom( tbl, tom[, options] ) O
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.
UnderlyingGroup values of tbl and tom are known and
equal.
Then the group is used to compute the fusion.
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.
fusionmap
quick 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 ) O
PermCharsTom( tbl, tom ) O
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).
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)
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 Class Functions).
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 ) O
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.
TableOfMarksDihedral( n ) O
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.
TableOfMarksFrobenius( p, q ) O
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( TableOfMarksCyclic( 6 ) ); 1: 6 2: 3 3 3: 2 . 2 4: 1 1 1 1 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 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 with argument a string (see TableOfMarks).
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 with argument the character
table.
A list of all names of tables of marks in the database can be obtained via
AllLibTomNames.
gap> names:= AllLibTomNames();; gap> "A5" in names; true
[Top] [Up] [Previous] [Next] [Index]
GAP 4 manual
March 2006