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. The algebraic ring structure is a more recent development, due to Solomon (1967). Given a finite group G, the generators of its Burnside ring Ω(G) are the formal sums 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. Much as character theory simplifies working with group representations, marks simplify working with permutation representations and the Burnside ring. If G acts on X, and H ≤ G (H is a subgroup of G), then the mark of H on X is the number of elements of X that are fixed by every element of H: , where If H and K are conjugate subgroups, then mX(H) = mX(K) for any finite G-set X; indeed, if K = gHg−1 then XK = g · XH. It is also easy to see that for each H ≤ G, the map Ω(G) → Z : X ↦ mX(H) is a homomorphism. This means that to know the marks of G, it is sufficient to evaluate them on the generators of Ω(G), viz. the orbits G/H. For each pair of subgroups H,K ≤ G define This is mX(H) for X = G/K. The condition HgK = gK is equivalent to g−1Hg ≤ K, so if H is not conjugate to a subgroup of K then m(K, H) = 0.