On the Isomorphism Problem for Burnside Rings

Brandl, Rolf; Huckle, Thomas

doi:10.2307/2161885

Cited by 2 publications

(2 citation statements)

References 3 publications

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“…Since the Burnside ring B(G) can be embedded in D(G), there is a connection to the similar problem concerning the ring B(G). This problem has been studied in [6,14,16], among others. Considering results for the isomorphism problem for Burnside rings it seems to be useful to work with primitive idempotents of In the second section we give a survey over the construction of D(G), the species and the primitive idempotents of Q(ζ ) ⊗ Z D(G) (ζ ∈ C primitive |G|-th root of unity).…”

Section: Introductionmentioning

confidence: 99%

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

Müller

2011

Journal of Algebra

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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).

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Section: Introductionmentioning

confidence: 99%

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

Müller

2011

Journal of Algebra

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“…In [4, Beispiel 7.1] two non-isomorphic groups of order 5 4 are given which have identical tables of marks. Further counter-examples involving p-groups may be found in [2].…”

Section: Introductionmentioning

confidence: 99%

Irreducible components and isomorphisms of the Burnside ring

Kimmerle¹,

Luca²,

Raggi-Cárdenas³

2008

Journal of Group Theory

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In this paper, we prove that each component of the Burnside ring of a finite group is the soluble component of the Burnside ring of a Weyl subgroup of its corresponding group. We show that groups with isomorphic Burnside rings have the same sublattice of soluble normal subgroups and the same spectrum. This gives for the alternating groups, the sporadic simple groups and many series of simple groups of Lie type a positive answer to Yoshida's question when a finite group is determined by its Burnside ring up to isomorphism.

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On the Isomorphism Problem for Burnside Rings

Abstract: Abstract. Nonisomorphic 3-groups of distinct nilpotency class are constructed with isomorphic Burnside rings.

Cited by 2 publications

References 3 publications

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

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

Irreducible components and isomorphisms of the Burnside ring

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