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.