Burnside Ring - Formal Definition

Formal Definition

Given a finite group G, the elements of its Burnside ring Ω(G) are the formal differences of isomorphism classes of finite G-sets. 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 (disjoint union), where each X_i is a single G-orbit. Choosing any element x_i in X_i creates an isomorphism G/G_i → X_i, where G_i is the stabilizer (isotropy) subgroup of G at x_i. A different choice of representative y_i in X_i gives a conjugate subgroup to G_i as stabilizer. This shows that the generators of Ω(G) as a Z-module are the orbits G/H as H ranges over conjugacy classes of subgroups of G.

In other words, a typical element of Ω(G) is

where a_i in Z and G₁, G₂, ..., G_N are representatives of the conjugacy classes of subgroups of G.