The table of marks of a finite group 
records the numbers of group
fixed points for the transitive group actions
of 
on cosets
of its subgroups. Let 
, 
,
..., 
be representatives of the conjugacy classes of
subgroups of 
.
The table of marks is the 
matrix 
with
 |
Thus 
is the number of fixed points of 
in the transitive action of 
on the right cosets of 
(Burnside 1955).
If the subgroup classes are ordered compatibly with containment up to conjugacy, the table is a lower triangular matrix.
Its 
th
diagonal entry is the subgroup index ![[N_G(H_i):H_i]](/images/equations/TableofMarks/Inline14.svg)
of 
in its normalizer, so the
table is a nonsingular matrix. Since every
transitive permutation action of 
is equivalent to a group action
on the cosets of a subgroup, invertibility allows the multiplicities of the transitive
constituents of a finite 
-set
to be recovered from its fixed-point counts. The table also encodes containment information
among conjugacy classes of subgroups (GAP).
For example, using the class order 
, 
,
, 
, the table of marks of the symmetric
group 
is
 |
Tables of marks were introduced by Burnside and are consequently also called Burnside matrices.
