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 Xi is a single G-orbit. Choosing any element xi in Xi creates an isomorphism G/Gi → Xi, where Gi is the stabilizer (isotropy) subgroup of G at xi. A different choice of representative yi in Xi gives a conjugate subgroup to Gi 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 ai in Z and G1, G2, ..., GN are representatives of the conjugacy classes of subgroups of G.
Read more about this topic: Burnside Ring
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