66.1 More about Tables of Marks

Let G be a finite group with n conjugacy classes of subgroups C₁, C₂, ¼, C_n and representatives H_i Î 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 stablizer 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 H₁ 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

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.

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