The inverse limit of the Burnside ring for a family of subgroups of a finite group

2018 ◽  
Vol 47 (2) ◽  
pp. 427-444 ◽  
Author(s):  
Yasuhiro HARA ◽  
Masaharu MORIMOTO
Keyword(s):  
Finite Group ◽  
Inverse Limit ◽  
Burnside Ring ◽  
A Finite Group

Cokernels of Homomorphisms from Burnside Rings to Inverse Limits

2017 ◽  
Vol 60 (1) ◽  
pp. 165-172 ◽  
Author(s):  
Masaharu Morimoto
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Inverse Limit ◽  
Minimal Normal Subgroup ◽  
Cartesian Product ◽  
Inverse Limits ◽  
Burnside Ring ◽  
A Finite Group ◽  
Burnside Rings ◽  
Natural Way

AbstractLet G be a finite group and let A(G) denote the Burnside ring of G. Then an inverse limit L(G) of the groups A(H) for proper subgroups H of G and a homomorphism res from A(G) to L(G) are obtained in a natural way. Let Q(G) denote the cokernel of res. For a prime p, let N(p) be the minimal normal subgroup of G such that the order of G/N(p) is a power of p, possibly 1. In this paper we prove that Q(G) is isomorphic to the cartesian product of the groups Q(G/N(p)), where p ranges over the primes dividing the order of G.

scholarly journals Enhanced equivariant Saito duality

2018 ◽  
Vol 17 (10) ◽  
pp. 1850181
Author(s):  
Wolfgang Ebeling ◽  
Sabir M. Gusein-Zade
Keyword(s):  
Fourier Transform ◽  
Finite Group ◽  
Zeta Functions ◽  
Analytic Formula ◽  
Burnside Ring ◽  
Euler Characteristics ◽  
Dual Pairs ◽  
A Finite Group ◽  
Equivariant Version ◽  
Burnside Rings

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring [Formula: see text] of a finite group [Formula: see text] is defined. An element of it is represented by a finite [Formula: see text]-set with a [Formula: see text]-equivariant transformation and with characters of the isotropy subgroups associated to all points. One gives an enhanced version of the equivariant Saito duality. For a complex analytic [Formula: see text]-manifold with a [Formula: see text]-equivariant transformation of it one has an enhanced equivariant Euler characteristic with values in a completion of [Formula: see text]. It is proved that the (reduced) enhanced equivariant Euler characteristics of the Milnor fibers of Berglund–Hübsch dual invertible polynomials are enhanced dual to each other up to sign. As a byproduct, this implies the result about the orbifold zeta functions of Berglund–Hübsch–Henningson dual pairs obtained earlier.

scholarly journals The Hecke Algebra of a Frobenius P-Category

2014 ◽  
Vol 21 (01) ◽  
pp. 1-52 ◽  
Author(s):  
Lluis Puig
Keyword(s):  
Finite Group ◽  
Hecke Algebra ◽  
Canonical Decomposition ◽  
Burnside Ring ◽  
A Finite Group ◽  
The Relationship

We introduce a new avatar of a Frobenius P-category [Formula: see text] under the form of a suitable subring [Formula: see text] of the double Burnside ring of P — called the Hecke algebra of [Formula: see text] — where we are able to formulate: (i) the generalization to a Frobenius P-category of the Alperin Fusion Theorem, (ii) the “canonical decomposition” of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and (iii) the “basic P × P-sets” in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P × P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.

scholarly journals A note on elements of the Burnside ring of a finite group

1981 ◽  
Vol 21 (3) ◽  
pp. 619-623
Author(s):  
Shin Hashimoto ◽  
Shin-ichiro Kakutani
Keyword(s):  
Finite Group ◽  
Burnside Ring ◽  
A Finite Group

Burnside rings of fusion systems and their unit groups

2020 ◽  
Vol 23 (4) ◽  
pp. 709-729
Author(s):  
Jamison Barsotti ◽  
Rob Carman
Keyword(s):  
Finite Group ◽  
Automorphism Groups ◽  
Unit Group ◽  
Fusion System ◽  
Burnside Ring ◽  
Fusion Systems ◽  
A Finite Group ◽  
Burnside Rings

AbstractFor a saturated fusion system {\mathcal{F}} on a p-group S, we study the Burnside ring of the fusion system {B(\mathcal{F})}, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring {B(S)}. We give criteria for an element of {B(S)} to be in {B(\mathcal{F})} determined by the {\mathcal{F}}-automorphism groups of essential subgroups of S. When {\mathcal{F}} is the fusion system induced by a finite group G with S as a Sylow p-group, we show that the restriction of {B(G)} to {B(S)} has image equal to {B(\mathcal{F})}. We also show that, for {p=2}, we can gain information about the fusion system by studying the unit group {B(\mathcal{F})^{\times}}. When S is abelian, we completely determine this unit group.

scholarly journals The slice Burnside ring and the section Burnside ring of a finite group

2012 ◽  
Vol 148 (3) ◽  
pp. 868-906 ◽  
Author(s):  
Serge Bouc
Keyword(s):  
Finite Group ◽  
Prime Spectrum ◽  
Burnside Ring ◽  
Biset Functor ◽  
Primitive Idempotents ◽  
Grothendieck Rings ◽  
A Finite Group ◽  
Burnside Rings

AbstractThis paper introduces two new Burnside rings for a finite group G, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of G-sets and of Galois morphisms of G-sets, respectively. The well-known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural Green biset functor structure. The functorial structure of unit groups of these rings is also discussed.

scholarly journals Brauer relations in finite groups II: Quasi-elementary groups of order paq

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Alex Bartel ◽  
Tim Dokchitser
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Burnside Ring ◽  
Rational Representation ◽  
Generators And Relations ◽  
Representation Ring ◽  
A Finite Group

Abstract.This is the second in a series of papers investigating the space of Brauer relations of a finite group, the kernel of the natural map from its Burnside ring to the rational representation ring. The first paper classified all primitive Brauer relations, that is those that do not come from proper subquotients. In the case of quasi-elementary groups the description is intricate, and it does not specify groups that have primitive relations in terms of generators and relations. In this paper we provide such a classification in terms of generators and relations for quasi-elementary groups of order

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