Burnside Ring - Marks

Much like character theory simplifies working with group representations, marks simplify working with permutation representations and the Burnside ring.

If G acts on X, and H ≤ G (H is a subgroup of G), then the mark of H on X is the number of elements of X that are fixed by every element of H:, where

If H and K are conjugate subgroups, then m_X(H) = m_X(K) for any finite G-set X; indeed, if K = gHg−1 then XK = g · XH.

It is also easy to see that for each H ≤ G, the map Ω(G) → Z : X ↦ m_X(H) is a homomorphism. This means that to know the marks of G, it is sufficient to evaluate them on the generators of Ω(G), viz. the orbits G/H.

For each pair of subgroups H,K ≤ G define

This is m_X(H) for X = G/K. The condition HgK = gK is equivalent to g−1Hg ≤ K, so if H is not conjugate to a subgroup of K then m(K, H) = 0.

To record all possible marks, one forms a table, Burnside's Table of Marks, as follows: Let G₁ (= trivial subgroup), G₂, ..., G_N = G be representatives of the N conjugacy classes of subgroups of G, ordered in such a way that whenever G_i is conjugate to a subgroup of G_j, then i ≤ j. Now define the N × N table (square matrix) whose (i, j)th entry is m(G_i, G_j). This matrix is lower triangular, and the elements on the diagonal are non-zero so it is invertible.

It follows that if X is a G-set, and u its row vector of marks, so u_i = m_X(G_i), then X decomposes as a disjoint union of a_i copies of the orbit of type G_i, where the vector a satisfies,

aM = u,

where M is the matrix of the table of marks. This theorem is due to (Burnside 1897).