On the Intersection of the Normalizers of the Nilpotent Residuals of All Subgroups of a Finite Group

2013 ◽  
Vol 20 (02) ◽  
pp. 349-360 ◽  
Author(s):  
Lü Gong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Characteristic Subgroup ◽  
A Finite Group ◽  
The Relationship

In this paper, a characteristic subgroup [Formula: see text] of a finite group G is defined, which is the intersection of the normalizers of the nilpotent residuals of all subgroups of G, and the properties of [Formula: see text] and the relationship between [Formula: see text] and the group G are investigated.

Characteristics of GWrSn

1976 ◽  
Vol 79 (3) ◽  
pp. 433-441
Author(s):  
A. G. Williams
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Generating Functions ◽  
Block Structure ◽  
Clifford Theory ◽  
A Finite Group ◽  
The Relationship

The ‘characteristics’ of the wreath product GWrSn, where G is a finite group, are certain polynomials (to be defined in section 2) which are generating functions for the simple characters of GWrSn. Schur (8) first used characteristics of the symmetric group. Specht (9) defined characteristics for GWrSn and found a relation between the characteristics of GWrSn and those of Sn which determined the simple characters of GWrSn. The object of this paper is to describe the p-block structure of GWrSn in the case where p is not a factor of the order of G. We use the relationship between the characteristics of GWrSn and those of Sn, which we deduce from a knowledge of the simple characters of GWrSn (these can be determined, independently of Specht's work, by using Clifford theory).

A Characteristic Subgroup of a p-Stable Group

1968 ◽  
Vol 20 ◽  
pp. 1101-1135 ◽  
Author(s):  
George Glauberman
Keyword(s):  
Finite Group ◽  
Characteristic Subgroup ◽  
Stable Group ◽  
A Finite Group

Let p be a prime, and let S be a Sylow p-subgroup of a finite group G. J. Thompson (13; 14) has introduced a characteristic subgroup JR(S) and has proved the following results:(1.1) Suppose that p is odd. Then G has a normal p-complement if and only if C(Z(S)) and N(JR(S)) have normal p-complements.

Weak Cayley tables and generalized centralizer rings of finite groups

2012 ◽  
Vol 153 (2) ◽  
pp. 281-318 ◽  
Author(s):  
STEPHEN P. HUMPHRIES ◽  
EMMA L. RODE
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Cayley Table ◽  
A Finite Group ◽  
Finite Conjugacy ◽  
Relationship Of ◽  
Cayley Tables ◽  
The Relationship

AbstractFor a finite group G we study certain rings (k)G called k-S-rings, one for each k ≥ 1, where (1)G is the centraliser ring Z(ℂG) of G. These rings have the property that (k+1)G determines (k)G for all k ≥ 1. We study the relationship of (2)G with the weak Cayley table of G. We show that (2)G and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (1)G = Z(ℂG) does not). We also show that (4)G determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.

On class numbers of a finite group and of its subgroups

1993 ◽  
Vol 123 (2) ◽  
pp. 295-301
Author(s):  
A. Vera-López ◽  
J. Sangroniz
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Class Numbers ◽  
Normal Subgroups ◽  
A Finite Group ◽  
The Relationship

SynopsisIn this paper we obtain new results which relate the number of conjugacy classes of л-elements of a finite group and an arbitrary subgroup, which are analogous to some results about normal subgroups. We also prove some new results which show the relationship between class numbers and splitting theorems. Our proofs only involve elementary techniques.

On the normalizers of p-nilpotency-residuals of all subgroups in a finite group

2015 ◽  
Vol 14 (10) ◽  
pp. 1550146
Author(s):  
Xuan Li ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
A Finite Group ◽  
The Relationship

Let G be a finite group. We define the subgroup N𝒩p(G) to be the intersection of the normalizers of the p-nilpotent residuals of all subgroups of G for a prime p. We first investigate the properties of N𝒩p(G), and then study the relationship between ⋂p‖G∣N𝒩p(G) and N𝒩(G), where N𝒩(G) is the intersection of the normalizers of the nilpotent residuals of all subgroups of G.

scholarly journals The Wielandt subgroup of metacyclic p-groups

1990 ◽  
Vol 42 (3) ◽  
pp. 499-510 ◽  
Author(s):  
Elizabeth A. Ormerod
Keyword(s):  
Finite Group ◽  
Nilpotency Class ◽  
Characteristic Subgroup ◽  
Normal Series ◽  
A Finite Group ◽  
Subnormal Subgroups

The Wielandt subgroup is the intersection of the normalisers of all the subnormal subgroups of a group. For a finite group it is a non-trivial characteristic subgroup, and this makes it possible to define an ascending normal series terminating at the group. This series is called the Wielandt series and its length determines the Wielandt length of the group. In this paper the Wielandt subgroup of a metacyclic p–group is identified, and using this information it is shown that if a metacyclic p–group has Wielandt length n, its nilpotency class is n or n + 1.

ON THE σ-NILPOTENT NORM AND THE σ-NILPOTENT LENGTH OF A FINITE GROUP

2020 ◽  
Vol 63 (1) ◽  
pp. 121-132
Author(s):  
BIN HU ◽  
JIANHONG HUANG ◽  
ALEXANDER N. SKIBA
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Normal Subgroups ◽  
Nilpotent Length ◽  
Nilpotent Residual ◽  
A Finite Group ◽  
Primary Groups ◽  
Relationship Of ◽  
The Relationship ◽  
Sigma G

AbstractLet G be a finite group and σ = {σi| i ∈ I} some partition of the set of all primes $\Bbb{P}$ . Then G is said to be: σ-primary if G is a σi-group for some i; σ-nilpotent if G = G1× … × Gt for some σ-primary groups G1, … , Gt; σ-soluble if every chief factor of G is σ-primary. We use $G^{{\mathfrak{N}}_{\sigma}}$ to denote the σ-nilpotent residual of G, that is, the intersection of all normal subgroups N of G with σ-nilpotent quotient G/N. If G is σ-soluble, then the σ-nilpotent length (denoted by lσ (G)) of G is the length of the shortest normal chain of G with σ-nilpotent factors. Let Nσ (G) be the intersection of the normalizers of the σ-nilpotent residuals of all subgroups of G, that is, $${N_\sigma }(G) = \bigcap\limits_{H \le G} {{N_G}} ({H^{{_\sigma }}}).$$ Then the subgroup Nσ (G) is called the σ-nilpotent norm of G. We study the relationship of the σ-nilpotent length with the σ-nilpotent norm of G. In particular, we prove that the σ-nilpotent length of a σ-soluble group G is at most r (r > 1) if and only if lσ (G/ Nσ (G)) ≤ r.

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.

A Note on the p-Deficiency Class of a Finite Group

2012 ◽  
Vol 19 (02) ◽  
pp. 353-358 ◽  
Author(s):  
Tianze Li ◽  
Weigang Xu ◽  
Jiping Zhang
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Finite Groups ◽  
Characteristic P ◽  
P Deficiency ◽  
Class 1 ◽  
A Finite Group ◽  
P Type ◽  
The Relationship

In this note, we explore the relationship between finite groups of characteristic p type and those of p-deficiency class 1. We study the structure of finite groups of characteristic p type. Besides, we show that the p-rank (resp., p-length) of a p-solvable group which is of exact p-deficiency class r(> 0) is bounded by r (resp., a function of r).

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