Groups with isomorphic Burnside rings
Let \(G\) be a finite group and \(B(G)\) denote its Burnside ring. One may ask when \(B(G)\) determines the isomorphism class of \(G\). The authors prove that this is the case in the following cases: Let \(G\) and \(G'\) be finite groups with isomorphic Burnside rings then \(G\) is isomorphic to \(G'\) if (Theorem 3.1): \(G\) is Hamiltonian or abelian, or (Theorem 4.1): both \(G\) and \(G'\) are simple, or (Theorem 4.2): \(G\) is minimal simple, where \(G\) is minimal simple if \(G\) is simple non-abelian and its proper subgroups are solvable. These results are based on a previous work by the authors [\textit{A. G. Raggi-Cárdenas} and \textit{L. Valero-Elizondo}, ``Normalizing isomorphisms between Burnside rings'', J. Algebra 277, 643--657 (2004; Zbl 1055.19001)], where they explore more profoundly the consequences on finite groups \(G\) and \(G'\) with isomorphic Burnside rings.
scientific article; zbMATH DE number 2162321