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We prove that if two finite groups G and G have isomorphic Burnside rings, then there is a normalized isomorphism between these rings, that is, a ring isomorphism θ : B(G) → B(G ) such that θ(G/1) = G /1. We use this to prove that if two finite groups have isomorphic Burnside rings, then there is a one-to-one correspondence between their families of soluble subgroups which preserves order and conjugacy class of subgroups.  2004 Elsevier Inc. All rights reserved.

where S is the symmetric group on n symbols and k is a field of characteristic n p ) 0. In this paper we answer the question, ''When is the Brauer quotient of a simple F S -module V with respect to a subgroup H of S both simple and 2 n n Ž . projective as an N H rH-module?,'' in some special cases. Remarkably, in each S n Ž . case there is only one such subgroup H up to conjugacy . ᮊ

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 .

Abstract. We prove that most groups of order less than 96 cannot have isomorphic tables of marks unless the groups are isomorphic. Mathematics Subject Classification (2010): 19A22