Groups with isomorphic fibered Burnside rings

TL;DR

This paper establishes conditions under which fibered Burnside rings of finite groups are isomorphic, revealing that non-isomorphic groups can have isomorphic fibered Burnside rings when the fiber group is non-trivial.

Contribution

It provides a new criterion for fibered Burnside ring isomorphisms and constructs examples of non-isomorphic groups with isomorphic rings, extending known results.

Findings

Identifies conditions for isomorphism of fibered Burnside rings

Constructs examples of non-isomorphic groups with isomorphic rings

Shows that non-trivial fiber groups lead to larger classes of isomorphic rings

Abstract

Let $G$ and $H$ be finite groups. We give a condition on $G$ and $H$ that implies that the $A$-fibered Burnside rings $B^A(G)$ and $B^A(H)$ are isomorphic. As a consequence, we show the existence of non-isomorphic groups $G$ and $H$ such that $B^A(G)$ and $B^A(H)$ are isomorphic rings. Here, the abelian fiber group $A$ can be chosen in a non-trivial way, that is, such that $B^A(G)$ and $B^A(H)$ are strictly bigger than the Burnside rings of $G$ and $H$, for which such counterexamples are already known.