Editing Ring (mathematics) (section)

=== Burnside ring of a group ===
To any [[group (mathematics)|group]] is associated its [[Burnside ring]] which uses a ring to describe the various ways the group can [[Group action (mathematics)|act]] on a finite set. The Burnside ring's additive group is the [[free abelian group]] whose basis is the set of transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the [[representation ring]]: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.