47 Tables of Marks

The concept of a table of marks was introduced by W.~Burnside in his book Theory of Groups of Finite Order Bur55. Therefore a table of marks is sometimes called a Burnside matrix.

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

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

This chapter describes a function (see TableOfMarks) which restores a The Library of Tables of Marks) or which computes the table of marks for a given group from the subgroup lattice of that group. Moreover this package contains a function to display a table of marks (see DisplayTom), a function to check the consistency of a table of marks (see TestTom), functions which switch between several forms of representation (see Marks, NrSubs, MatTom, and TomMat), functions which derive information about the group from the table of marks (see DecomposedFixedPointVector, NormalizerTom, IntersectionsTom, IsCyclicTom, FusionCharTableTom, PermCharsTom, MoebiusTom, CyclicExtensionsTom, IdempotentsTom, ClassTypesTom, and ClassNamesTom), and some functions for the generic construction of a table of marks (see TomCyclic, TomDihedral, and TomFrobenius).

The functions described in this chapter are implemented in the file LIBNAME/"tom.g".

Subsections

  1. More about Tables of Marks
  2. Table of Marks Records
  3. The Library of Tables of Marks
  4. TableOfMarks
  5. Marks
  6. NrSubs
  7. WeightsTom
  8. MatTom
  9. TomMat
  10. DecomposedFixedPointVector
  11. TestTom
  12. DisplayTom
  13. NormalizerTom
  14. IntersectionsTom
  15. IsCyclicTom
  16. FusionCharTableTom
  17. PermCharsTom
  18. MoebiusTom
  19. CyclicExtensionsTom
  20. IdempotentsTom
  21. ClassTypesTom
  22. ClassNamesTom
  23. TomCyclic
  24. TomDihedral
  25. TomFrobenius
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GAP 3.4.4
April 1997