On the isomorphism problem for the ring of monomial representations of a finite group

Abstract

In this paper we are concerned with the problem of finding properties of a finite group G in the ring D(G) of monomial representations of G. We determine the conductors of the primitiveth root of unity, and prove a structure theorem for the torsion units of D(G). Using these results we show that an abelian group G is uniquely determined by the ring D(G). We state an explicit formula for the primitive idempotents of Z. We get further results for nilpotent and p-nilpotent groups and we obtain properties of Sylow subgroups of G from D(G).

Multiplicative induction and units for the ring of monomial representations

“…Specifically, if G is nilpotent, then the universal result deduces that Ω(G, A) ω ≃ Ω(G) × × H 1 (G, A) (see Example 7.6). This fact is a generalization of [22,Proposition 5.1].…”

“…The unit group Ω(G, A) × of the monomial Burnside ring Ω(G, A) was studied in [2,22]. In Section 7, we show that Ω(G, A) × is a finitely generated abelian group (see Proposition 7.2).…”

“…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]).…”

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]).…”