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
Therefore
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).)
