In this section we recall basic facts about finite group actions. The table of marks is defined and its relation to the subgroup structure and to the Burnside ring is described.
Let G be a finite group. Denote by the set of all subgroups of G. Then is a partially ordered set (poset) with incidence relation . The group G acts on by conjugation, i.e. via for and . This G-action respects incidence: if then for all and . We denote the G-orbit of (i.e. the conjugacy class of subgroups of G which contains U) by and usually omit the subscript as long as no confusion can arise. The set of G-orbits also is a poset, with incidence if for and some . We will refer to as the poset structure of G.
A (right) G-set X is a set X together with an action such that and for all and all . Every G-set X decomposes into a disjoint union of orbits , each of which is a G-set itself. A G-set is transitive if it consists of only one orbit. All G-sets in this article are assumed to be finite.
A homomorphism between two G-sets X and Y is a map such that for all and all . Two G-sets X and Y are isomorphic if there exists a bijective homomorphism .
Let be representatives of the conjugacy classes of subgroups of G. Then . For each subgroup the group G acts transitively on the set of right cosets of U in G. Conversely every transitive G-set X is isomorphic to a G-set where U is a point stabilizer of X in G. For every the G-set is isomorphic to . Thus every transitive G-set is isomorphic to for some .
Let and consider the G-set . Then U has fixed points in that action if and only if U is contained in a one point stabilizer, i.e. in at least one conjugate of . Thus the table of marks describes the poset : the incidence matrix of this poset is obtained from M(G) by replacing every nonzero entry by 1.
But M(G) contains far more information about the subgroup structure of G. This is due to the following recalculation of the value of a mark.
The following lemma collects some easy consequences of the above formula. In particular the numbers of incidences between two conjugacy classes of subgroups of G can be derived from M(G).
The table of marks of the alternating group of order 60 in Table 1 serves as an example. has nine conjugacy classes of subgroups. They are distinguished by their orders and have isomorphism types: 1, 2, 3, , 5, , , , and . The rows of the table correspond to the transitive G-sets .
Table 1: The table of marks of .
Denote by the number of conjugates of a subgroup U of G contained in a fixed subgroup V of G. These numbers also are determined by M(G).
On the other hand M(G) is determined by the numbers for all and the additional knowledge of the index for every .
Denote for any G-set X its isomorphism class by [X]. The Burnside ring of G is the free abelian group
generated by the isomorphism classes of transitive G-sets , . Here the sum [X] + [Y] of the isomorphism classes of G-sets X and Y is the isomorphism class of the disjoint union of X and Y. Moreover, their product is the isomorphism class of the Cartesian product of X and Y. This turns into a commutative ring with identity .
Let X and Y be G-sets and let . Then
Thus, if we define for each to be the r-tuple , then the map
is a ring homomorphism from to .
Let . Then can be expressed in terms of the table of marks M(G) as
Moreover, the G-set X is characterized up to isomorphism by .
Let X be a G-set. The permutation character of G on X is defined as for any element . This number, of course, coincides with the mark of the cyclic subgroup generated by g on X. Therefore, the table of marks M(G) contains in the columns corresponding to cyclic subgroups a complete list of transitive permutation characters of G corresponding to the transitive G-sets .
The following proposition (see [Kerber 1991], 3.2.18) provides a way to determine the columns of M(G) which correspond to cyclic subgroups.
Let be the primitive idempotent of corresponding to the conjugacy class of , that is , and write with rational coefficients . (The matrix then is the inverse of M(G).)