Isomorphic Burnside rings

Abstract

For any finite group G, let Ω(G) denote the Burnside ring of G. Let A denote the class of finite abelian groups. Let G and H be two groups belonging to the class A . If Ω(G) and Ω(H) are isomorphic as rings, we show that for every prime p, the number of p-subgroups of G is equal to the number of p-subgroups of H. Let A 2 denote the class of finite abelian groups G with the property that for every prime p, the p-primary torsion t p ( G) of G is a direct sum of at most two cyclic groups. For groups G, H in the class A 2 we show that Ω(G) and Ω(H) are isomorphic as rings if and only if G and H are isomorphic as groups. Let Ω + (G) denote the half ring consisting of elements of Ω(G) represented by G-sets. For any prime p, let C ( p) denote the class of finite abelian p-groups. Let G and H be any two groups in C ( p). Then we show that there exists a ring isomorphism α: Ω(G) → Ω(H) further satisfying α( Ω(G) + ) = Ω(H) + if and only if G ∼- H as groups.