Burnside ring

Let G be a finite groupMathworldPlanetmath. Recall that by G-set we understand a pair (X,), where X is a set and :G×XX is a group actionMathworldPlanetmath of G on X. For short notation the pair notation will be omitted and G-sets will be simply denoted by capital letters.

Recall that for each subgroupMathworldPlanetmathPlanetmath HG we have canonical G-set G/H={gH;gG} where group action is defined as follows: for any g,kG we have (g,kH)gkH.

Let X and Y be G-sets. Recall that by G-map from X to Y we understand any function F:XY such that for any gG and xX we have F(gx)=gF(x).

It is easy to see that family of all G-sets and G-maps forms a categoryMathworldPlanetmath (with standard comoposition). We shall denote this category by G-𝕊. Moreover, by G-𝕊0 we shall denote full subcategory of G-𝕊 whose objects are all finite G-sets.

From G-sets X and Y one can construct another G-set in two interesting (from our point of view) ways, i.e. by taking disjoint unionMathworldPlanetmathPlanetmath XY with obvious group action and by taking productMathworldPlanetmathPlanetmathPlanetmath X×Y with group action as follows: (g,(x,y))(gx,gy). Moreover it is clear that when X and Y are finite, so are XY and X×Y.

Consider a finite G-set X. Then there exist a natural numberMathworldPlanetmath n, finite family {Hi}i=1n of subgroups of G and an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (in G-𝕊0 category)


Therefore (since G is finite) family of isomorphism classes of G-𝕊0 forms a countable set.

Denote by Ω+(G)={[X];XG-𝕊0} the set of isomorphism classes of category G-𝕊0. Then one can turn Ω+(G) into a semiringMathworldPlanetmath as follows: for any finite G-sets X and Y define


Note that here we treat the empty setMathworldPlanetmath as a G-set (with one and unique group action), therefore Ω+(G) has zero elementMathworldPlanetmath [] (the other way is to formally add the zero to Ω+(G) - this is just technical thing).

Define by Ω(G)=K(Ω+(G)) the Grothendieck group of (Ω+(G),+). If A is an abelian semigroup and f:A×AA is a bilinear map, then it can be uniquely extended to a bilinear map K(f):K(A)×K(A)K(A), therefore Ω(G) can be uniquely turned into a ring from Ω+(G). This ring is called the Burnside ring of G.

Some properties:

(0) each element of Ω(G) can be expressed as a formal diffrence [X]-[Y];

(1) Ω(G) is a commutativePlanetmathPlanetmath, unital ring, where [G/G] is the unity of Ω(G);

(2) Ω can be turned into a contravariant functorMathworldPlanetmath from the category of finite groups to the category of commutative, unital rings;

(3) (Ω+(G),+) is a cancellative semigroup, therefore it embedds into Ω(G);

(4) for the trivial group E there is a ring isomorphism Ω(E);

(5) for any group G there is a ring monomorphism φ:Ω(G)i=1n for some natural number n; this is called the characteristic embedding;

(6) for any two groups G,H we have: if Ω(G) and Ω(H) are isomorphic (as a rings), then |G|=|H|; generally G need not be isomorphic to H.

Title Burnside ring
Canonical name BurnsideRing
Date of creation 2013-03-22 18:08:02
Last modified on 2013-03-22 18:08:02
Owner joking (16130)
Last modified by joking (16130)
Numerical id 10
Author joking (16130)
Entry type Definition
Classification msc 16S99