Isomorphic rings of monomial representations

Species isomorphisms of fibered Burnside rings

“…On the other hand, the study of isomorphisms between representation rings often reveals common invariants preserved in the rings, in times leading to a positive answer to the isomorphism problem within certain classes of groups, e.g., if G and H are finite groups with isomorphic monomial representation rings, then if G is nilpotent so is H, and if G is either abelian or has square-free order, then G ∼ = H, while the general question remains open (see Müller [5] [6]).…”

Multiplicative induction and units for the ring of monomial representations

“…This ring contains the ordinary Burnside ring Ω(G) as a subring, and is applicable to the representation theory of finite groups. There are some well-known facts about Ω(G, A) (see, e.g., [2,3,12,13,22,23]). Many properties of Burnside rings seem to be extended to monomial Burnside rings; for instance, the prime ideal spectrum of Ω(G, A) was studied in [12] (see also [10]).…”