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'.
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