Groups with isomorphic Burnside rings

Raggi-Cárdenas, Alberto G.; Valero-Elizondo, Luis

doi:10.1007/s00013-004-1124-x

Cited by 5 publications

(2 citation statements)

References 12 publications

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“…If one imposes further restrictions on the groups, then the answer can become affirmative, an instance of this occurring in [13] where it is shown that the answer to Yoshida's question is positive in the case when both groups are abelian or Hamiltonian, and in [11] in the case when just one of them is hamiltonian or abelian or in the case when both groups are simple.…”

Section: Introductionmentioning

confidence: 91%

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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Section: Introductionmentioning

confidence: 91%

Irreducible components and isomorphisms of the Burnside ring

Kimmerle¹,

Luca²,

Raggi-Cárdenas³

2008

Journal of Group Theory

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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: 98%

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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On reconstructing finite gauge group from fusion rules

Radhakrishnan

2024

J. High Energ. Phys.

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Gauging a finite group 0-form symmetry G of a quantum field theory (QFT) results in a QFT with a Rep(G) symmetry implemented by Wilson lines. The group G determines the fusion of Wilson lines. However, in general, the fusion rules of Wilson lines do not determine G. In this paper, we study the properties of G that can be determined from the fusion rules of Wilson lines and surface operators obtained from higher-gauging Wilson lines. This is in the spirit of Richard Brauer who asked what information in addition to the character table of a finite group needs to be known to determine the group. We show that fusion rules of surface operators obtained from higher-gauging Wilson lines can be used to distinguish infinite pairs of groups which cannot be distinguished using the fusion of Wilson lines. We derive necessary conditions for two non-isomorphic groups to have the same surface operator fusion and find a pair of such groups.

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Groups with isomorphic Burnside rings

Cited by 5 publications

References 12 publications

Irreducible components and isomorphisms of the Burnside ring

Irreducible components and isomorphisms of the Burnside ring

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

On reconstructing finite gauge group from fusion rules

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