## Burnside ring

For a finite group *G*, the Burnside ring Ω(*G*) of *G* is defined to be the ring generated by formal differences of isomorphism classes of *G*-sets. The ring structure is given by disjoint union and Cartesian product of *G*-sets.

Let \(\mathcal{O}\) be a single orbit of a finite group *G*, and let \(x\in \mathcal{O}\). Then \(\mathcal{O}\) is isomorphic as a *G*-set to *G/H*, where *H*=*G _{x}* (the isotropy subgroup of

*x*). If one chose a different 'starting point' \(y\in\mathcal{O}\), then

*H*would be replaced by a conjugate subgroup. The set of isomorphism classes of

*G*-orbits is in this way in 1-1 correspondence with the set of conjugacy classes of subgroups of

*G*.

The ** Burnside ring** Ω(

*G*) is therefore the

**Z**-module generated by the conjugacy classes of subgroups of

*G*. See the wikipedia page.

Let *S* be a finite *G*-set (a set upon which *G* acts). Then *S* can be decomposed into a finite disjoint union of *G*-orbits, and so corresponds to a sum
$$ S = \sum_i a_i [G/H_i] $$
where the *H _{i}* represent the conjugacy classes of subgroups of

*G*, and the

*a*are (positive) integers.

_{i}### Permutation Representation

If *G* acts on a finite set *S*, then there is associated a permutation representation (over any ring or field, but let us use **Q**) as follows. Let *V* be the vector space whose basis consists of the elements of *S*. Thus a general element of *V* is of the form \(\mathbf{v} = \sum_{s\in S}a_s s\), where the *a _{s}* are elements of the field/ring. Then \(g\in G\) acts by \(g(\mathbf{v}) = \sum_{s\in S} a_s g(s)\), which is a permutation of the coefficients

*a*. The matrix representing

_{s}*g*then has a 1 at the intersection of the column corresponding to

*s*and the row corresponding to

*g*(

*s*), and 0 at other entries of that row and column.

If we fix a base field, this defines a homomorphism from the Burnside ring Ω(*G*) to the ring of representations over that field which we denote $$\beta:\Omega(G)\longrightarrow R(G).$$

#### Properties of β

- β is only surjective for the trivial group and the group of order 2;
- β is seldom injective

#### Table of marks

The element in row *G/K* and column *H* represents the number of points in the orbit (type) *G/K* fixed by *H*. By the definition of the action of *G* on *G*/*K*, a coset *gK* is fixed by *H* whenever *HgK* = *gK*. But this is equivalent to, *g*^{-1}*Hg* < *K* so that
$$m(G/K, H) = \# \left\{ gK \in G/K \mid g^{-1}Hg < K \right\}.$$

- In particular, the diagonal elements are
*m*(*K, K*) = |N_{G}(*K*)/*K*|. - If
*H*is not conjugate to a subgroup of*K*then*m*(*G/K, H*) = 0. - If
*K*is a normal subgroup of*G*and*H < K*then \(g^{-1}Hg \subset K\) for all \(g\in G\), so that \(m(G/K,\,H)=|G/K|\). - If
*G*is Abelian then every subgroup is normal and so, again if*H*<*K*then \(m(G/K,\,H)=|G/K|\), otherwise \(m(G/K,\,H)=0\).