66.9 Other Operations for Tables of Marks

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.

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.

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> DerivedSubgroupsTomPossible( a5 );
[ ,, 1,,,,,, 9 ]
gap> DerivedSubgroupTom( a5, 2 );
1
gap> DerivedSubgroupsTomPossible( a5 );
[ , 1, 1,,,,,, 9 ]
gap> DerivedSubgroupsTom( a5 );
[ 1, 1, 1, 1, 1, 3, 5, 4, 9 ]

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 ]

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 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

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 ] ]

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 ]

gap> DecomposedFixedPointVector( a5, [ 0, 0, 0, 0, 1, 1 ] );
[ 2/5, -1, -1/2, 0, 1/2, 1 ]

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 secon 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

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 ]

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( SymmetricGroup( [ 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

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.

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 ] ]

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