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.

COMPONENTS AND MINIMAL NORMAL SUBGROUPS OF FINITE AND PSEUDOFINITE GROUPS

2019 ◽  
Vol 84 (1) ◽  
pp. 290-300
Author(s):  
JOHN S. WILSON
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Minimal Normal Subgroup ◽  
Normal Subgroups ◽  
First Order ◽  
Order Language ◽  
Image Position ◽  
A Finite Group

AbstractIt is proved that there is a formula$\pi \left( {h,x} \right)$in the first-order language of group theory such that each component and each non-abelian minimal normal subgroup of a finite groupGis definable by$\pi \left( {h,x} \right)$for a suitable elementhofG; in other words, each such subgroup has the form$\left\{ {x|x\pi \left( {h,x} \right)} \right\}$for someh. A number of consequences for infinite models of the theory of finite groups are described.

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.

Supersoluble Immersion

1959 ◽  
Vol 11 ◽  
pp. 353-369 ◽  
Author(s):  
Reinhold Baer
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Minimal Normal Subgroup ◽  
Independent Interest ◽  
List Type ◽  
Type Number ◽  
Group Of Automorphisms ◽  
A Finite Group ◽  
Subgroup J

Supersoluble immersion of a normal subgroup K of a finite group G shall be defined by the following property:If σ is a homomorphism of G, and if the minimal normal subgroup J of Gσ is part of Kσ then J is cyclic (of order a prime).Our principal aim in the present investigation is the proof of the equivalence of the following three properties of the normal subgroup K of the finite group G:(i)K is supersolubly immersed in G.(ii)K/ϕK is supersolubly immersed in G/ϕK.(iii)If θ is the group of automorphisms induced in the p-subgroup U of K by elements in the normalizer of U in G, then θ' θp-1 is a p-subgroup of θ.Though most of our discussion is concerned with the proof of this theorem, some of our concepts and results are of independent interest.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

scholarly journals On Schur p-Groups of odd order

2017 ◽  
Vol 16 (03) ◽  
pp. 1750045 ◽  
Author(s):  
Grigory Ryabov
Keyword(s):  
Finite Group ◽  
Odd Order ◽  
A Finite Group ◽  
Natural Way

A finite group [Formula: see text] is called a Schur group if any [Formula: see text]-ring over [Formula: see text] is associated in a natural way with a subgroup of [Formula: see text] that contains all right translations. We prove that the groups [Formula: see text], where [Formula: see text], are Schur. Modulo previously obtained results, it follows that every noncyclic Schur [Formula: see text]-group, where [Formula: see text] is an odd prime, is isomorphic to [Formula: see text] or [Formula: see text], [Formula: see text].

scholarly journals A remark on the C-normality of maximal subgroups of finite groups

1997 ◽  
Vol 40 (2) ◽  
pp. 243-246
Author(s):  
Yanming Wang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Primitive Groups ◽  
A Finite Group

A subgroup H is called c-normal in a group G if there exists a normal subgroup N of G such that HN = G and H∩N ≤ HG, where HG =: Core(H) = ∩g∈GHg is the maximal normal subgroup of G which is contained in H. We use a result on primitive groups and the c-normality of maximal subgroups of a finite group G to obtain results about the influence of the set of maximal subgroups on the structure of G.

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