# Burnside ring

From Groupprops

## Definition

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

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

Note that when 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.