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
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.