About Tables of Marks
The concept of the table of marks of a finite group was first introduced by William Burnside in his famous book "Theory of Groups of Finite order." In fact many books refer to a table of marks as 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.
History of the TomLib Library
Tomlib was created by Goetz Pfeiffer and Thomas Merkwitz. It has previously been maintained by Thomas Breuer. We would like to thank him for his contribution and ongoing assistance.
We gratefully acknowledge the support of Science Foundation Ireland under the Research Frontiers program.