The ideas were introduced by William Burnside at the end of the nineteenth century.
The algebraic ring structure is a more recent development, due to Solomon (1967).
For the ring structure, addition is given by disjoint union of G-sets and multiplication by their Cartesian product.
The Burnside ring is a free Z-module, whose generators are the (isomorphism classes of) orbit types of G. If G acts on a finite set X, then one can write
A different choice of representative yi in Xi gives a conjugate subgroup to Gi as stabilizer.
where ai in Z and G1, G2, ..., GN are representatives of the conjugacy classes of subgroups of G. Much as character theory simplifies working with group representations, marks simplify working with permutation representations and the Burnside ring.
Now define the N × N table (square matrix) whose (i, j)th entry is m(Gi, Gj).
This matrix is lower triangular, and the elements on the diagonal are non-zero so it is invertible.
It follows that if X is a G-set, and u its row vector of marks, so ui = mX(Gi), then X decomposes as a disjoint union of ai copies of the orbit of type Gi, where the vector a satisfies, where M is the matrix of the table of marks.
The ring structure of Ω(G) can be deduced from these tables: the generators of the ring (as a Z-module) are the rows of the table, and the product of two generators has mark given by the product of the marks (so component-wise multiplication of row vectors), which can then be decomposed as a linear combination of all the rows.
The resulting map taking a G-set to the corresponding representation is in general neither injective nor surjective.
The simplest example showing that β is not in general injective is for G = S3 (see table above), and is given by The Burnside ring for compact groups is described in (tom Dieck 1987).