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) congruent to B(G'), then G congruent to G'.