The Burnside ring of a finite group G is constructed from the category of finite G-sets as a Grothendieck group. More precisely, let M(G) be the commutative monoid of isomorphism classes of finite G-sets, with addition the disjoint union of G-sets and identity element the empty set (which is a G-set in a unique way).
The Burnside ring for a finite group G is derived from the category of finite G-sets as a Grothendieck group. Specifically, let M(G) be the commutative monoid of equivalence classes of finite G-sets, with addition defined by disjoint union and identity element being the empty set (which is a G-set in one unique manner).