Burnside ring

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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.