Burnside ring

Summary

The Burnside ring $A(G)$ of a finite group $G$ is a commutative ring built from finite $G$-sets. Its addition is disjoint union and its multiplication is Cartesian product. As an abelian group it is free on the classes of the transitive $G$-sets $G/H$, one for each conjugacy class of subgroups $H$. It records the ways a group can act on finite sets. It maps to the representation ring by sending a $G$-set to its permutation representation.