Normalizing isomorphisms between Burnside rings
Let \(G\) be a finite group and \(B(G)\) its Burnside ring. Given two finite groups \(G\) and \(G'\) one asks when are their corresponding Burnside rings isomorphic. There are infinitely many examples of nonisomorphic groups with isomorphic Burnside rings. However, what can be said about \(G\) and \(G'\) if \(B(G)\cong B(G')\)? The main theorem is the following: Let \(G\) and \(G'\) be finite groups with isomorphic Burnside rings. Then, there exists a normalized isomorphism \(\theta:B(G)\to B(G')\), i.e. \(\theta\) is a ring isomorphism such that \(\theta(G/1)=G'/1\). As an application the authors have the following corollary: Let \(G\) and \(G'\) be finite groups such that their Burnside rings are isomorphic. Then, there is a one-to-one correspondence between the conjugacy classes of soluble subgroups of \(G\) and \(G'\) which preserves order of subgroups and cardinality of the conjugacy class.
scientific article; zbMATH DE number 2105243
scientific article