Burnside ring

In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century, but the algebraic ring structure is a more recent development, due to Solomon (1967).

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 ${\displaystyle X=\cup _{i}X_{i}}$ (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.

...
Wikipedia