Groups with isomorphic Burnside rings

Abstract.

We prove that for some families of finite groups, the isomorphism class of the group is completely determined by its Burnside ring. Namely, we prove the following: if two finite simple groups have isomorphic Burnside rings, then the groups are isomorphic; if G is either Hamiltonian or abelian or a minimal simple group, and G′ is any finite group such that B(G) ≅ B(G′), then G ≅ G′.