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).

scholarly journals Modular Representations of Finite Groups with Unsaturated Split (B,N)-Pairs

1980 ◽  
Vol 32 (3) ◽  
pp. 714-733 ◽  
Author(s):  
N. B. Tinberg
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Prime Number ◽  
Semidirect Product ◽  
Finite Groups ◽  
Modular Representations ◽  
Characteristic P ◽  
Representations Of Finite Groups ◽  
A Finite Group

1. Introduction.Let p be a prime number. A finite group G = (G, B, N, R, U) is called a split(B, N)-pair of characteristic p and rank n if(i) G has a (B, N)-pair (see [3, Definition 2.1, p. B-8]) where H= B ⋂ N and the Weyl group W= N/H is generated by the set R= ﹛ω 1,… , ω n) of “special generators.”(ii) H= ⋂n∈N n-1Bn(iii) There exists a p-subgroup U of G such that B = UH is a semidirect product, and H is abelian with order prime to p.A (B, N)-pair satisfying (ii) is called a saturated (B, N)-pair. We call a finite group G which satisfies (i) and (iii) an unsaturated split (B, N)- pair. (Unsaturated means “not necessarily saturated”.)

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.

Finite groups with some s-conditionally permutable subgroups

2017 ◽  
Vol 16 (12) ◽  
pp. 1750224
Author(s):  
S. E. Mirdamadi ◽  
G. R. Rezaeezadeh
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Permutable Subgroups ◽  
A Finite Group

A subgroup [Formula: see text] of a finite group [Formula: see text] is said to be [Formula: see text]-conditionally permutable in [Formula: see text] if for every Sylow subgroup [Formula: see text] of [Formula: see text], there exists an element [Formula: see text] such that [Formula: see text]. In this paper, the structure of solvable group [Formula: see text] in which every [Formula: see text]-subgroup of [Formula: see text] or every subnormal subgroup of [Formula: see text] is [Formula: see text]-conditionally permutable in [Formula: see text] is described. Let [Formula: see text] be a solvable group and [Formula: see text] the largest prime dividing [Formula: see text]. Suppose further that [Formula: see text] is the Sylow [Formula: see text]-subgroup of [Formula: see text] and [Formula: see text]. We are going to show that [Formula: see text] is a PST-group if and only if every subnormal subgroup of [Formula: see text] is [Formula: see text]-conditionally permutable in [Formula: see text].

scholarly journals ON THE REGULARITY CONJECTURE FOR THE COHOMOLOGY OF FINITE GROUPS

2008 ◽  
Vol 51 (2) ◽  
pp. 273-284 ◽  
Author(s):  
David J. Benson
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Finite Groups ◽  
Cohomology Ring ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Characteristic P ◽  
Residue Classes ◽  
The Difference ◽  
A Finite Group

AbstractLet $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo–Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

On strongly associative group algebras of p-solvable groups

2015 ◽  
Vol 14 (06) ◽  
pp. 1550085 ◽  
Author(s):  
Jonas Gonçalves Lopes
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Group Algebra ◽  
Modular Group ◽  
Solvable Groups ◽  
Crossed Product ◽  
Group Algebras ◽  
Characteristic P ◽  
Partial Action ◽  
A Finite Group

Given a partial action α of a group G on the group algebra FH, where H is a finite group and F is a field whose characteristic p divides the order of H, we investigate the associativity question of the partial crossed product FH *α G. If FH *α G is associative for any G and any α, then FH is called strongly associative. We characterize the strongly associative modular group algebras FH with H being a p-solvable group.

scholarly journals Fusion systems and group actions with abelian isotropy subgroups

2013 ◽  
Vol 56 (3) ◽  
pp. 873-886 ◽  
Author(s):  
Özgün Ünlü ◽  
Ergün Yalçin
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Finite Solvable Group ◽  
Recursive Method ◽  
Fusion Systems ◽  
Product Of Spheres ◽  
A Finite Group ◽  
Positive Integers

AbstractWe prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × $\mathbb{S}^{n_1}\$ × … × $\mathbb{S}^{n_k}$ for some positive integers n1, …, nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology.

scholarly journals PRIMES OF HEIGHT ONE AND A CLASS OF NOETHERIAN FINITELY PRESENTED ALGEBRAS

2007 ◽  
Vol 17 (07) ◽  
pp. 1465-1491 ◽  
Author(s):  
ISABEL GOFFA ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Ideals ◽  
Semigroup Algebras ◽  
Prime Spectrum ◽  
Maximal Orders ◽  
Identity Based ◽  
A Finite Group ◽  
The Relationship ◽  
Finitely Presented

Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra. So, it is natural to consider such algebras as semigroup algebras K[S] and to investigate the structure of the monoid S. The relationship between the prime ideals of the algebra and those of the monoid S is one of the main tools. Results analogous to fundamental facts known for the prime spectrum of algebras graded by a finite group are obtained. This is then applied to characterize a large class of prime Noetherian maximal orders that satisfy a polynomial identity, based on a special class of submonoids of polycyclic-by-finite groups. The main results are illustrated with new constructions of concrete classes of finitely presented algebras of this type.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

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〉.

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