Burnside ring

Definition
Let $$G$$ be a group. The Burnside ring of $$G$$ is defined in any of the following equivalent ways:


 * It is the ring of all integer-valued functions on conjugacy classes of subgroups in $$G$$
 * It is the ring of $$\mathbb{Z}$$-linear combinations of transitive group actions of $$G$$. The positive linear combinations can be identified with the action on the disjoint union of sets on which it acts transitively.

Note that when $$G$$ is a finite group, or more generally when it has finitely many conjugacy classes of subgroups, the Burnside ring is a free abelian group of rank equal to the number of conjugacy classes of subgroups.