We identify the Burnside ring 
with its image under the map
in 
.
The only idempotents in 
are 0 and 1. Each idempotent in 
is
of the form 
where 
or 1 for all 
. Let 
be such
that its entry in th i-th position equals 1 and all other entries are 0
and, for any 
, let 
.
The table of marks M(G) provides an efficient means to determine those
subsets 
for which the idempotent 
is an
element of 
.
Let X be a G-set and suppose 
such that H is a normal
subgroup of index p of K for some prime p. Then K acts on the fixed
point set 
with kernel containing H. Since 
is
prime, the orbits of K on 
have either length 1 (corresponding
to 
) or length p. It follows that
in other words, if H lies in the conjugacy class 
of subgroups of
G and if K lies in class 
then the i-th column and the j-th
column of the table of marks M(G) are equal modulo p.
In order to describe the idempotents of 
we define two relations
on the set 
. For any prime p let
such that H is normal of
index p in K, H lies in 
and K lies in 
;
The idempotents of 
are then completely described by the following
proposition (see [Dress 1969], Proposition 1).
In particular, it is possible to decide from the table of marks of G whether G is solvable or not:
Since solvable groups are characterized by the fact that every nontrivial subgroup has a normal subgroup of index p for some prime p we get the following characterization of solvable groups.
