On semi-σ-nilpotent finite groups

2019 ◽  
Vol 18 (10) ◽  
pp. 1950200
Author(s):  
Chi Zhang ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Product Formula ◽  
Nilpotent Subgroup ◽  
Chief Factor ◽  
Group A ◽  
A Finite Group

Let [Formula: see text] be a partition of the set [Formula: see text] of all primes and [Formula: see text] a finite group. A chief factor [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-central if the semidirect product [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. [Formula: see text] is called [Formula: see text]-nilpotent if every chief factor of [Formula: see text] is [Formula: see text]-central. We say that [Formula: see text] is semi-[Formula: see text]-nilpotent (respectively, weakly semi-[Formula: see text]-nilpotent) if the normalizer [Formula: see text] of every non-normal (respectively, every non-subnormal) [Formula: see text]-nilpotent subgroup [Formula: see text] of [Formula: see text] is [Formula: see text]-nilpotent. In this paper we determine the structure of finite semi-[Formula: see text]-nilpotent and weakly semi-[Formula: see text]-nilpotent groups.

scholarly journals Finite groups with given systems of generalised σ-permutable subgroups

2021 ◽  
pp. 25-33
Author(s):  
Viktoria S. Zakrevskaya
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Chief Factor ◽  
Permutable Subgroup ◽  
Permutable Subgroups ◽  
Group A ◽  
A Finite Group

Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. A set ℋ  of subgroups of G is said to be a complete Hall σ-set of G if every member ≠1 of ℋ  is a Hall σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ⌒ π(G)  ≠ ∅.  A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ  such that AH x = H  xA for all H ∈ ℋ  and all x ∈ G; σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ … ≤ At = G such that either Ai − 1 ⊴ Ai or Ai /(Ai − 1)Ai is σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = < A, B >; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised.

Finite groups with only 𝔉 {\mathfrak{F}} -normal and 𝔉 {\mathfrak{F}} -abnormal subgroups

2019 ◽  
Vol 22 (5) ◽  
pp. 915-926 ◽  
Author(s):  
Bin Hu ◽  
Jianhong Huang ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Finite Groups ◽  
Chief Factor ◽  
A Finite Group

Abstract Let G be a finite group, and let {\mathfrak{F}} be a class of groups. A chief factor {H/K} of G is said to be {\mathfrak{F}} -central (in G) if the semidirect product {(H/K)\rtimes(G/C_{G}(H/K))\in\mathfrak{F}} . We say that a subgroup A of G is {\mathfrak{F}} -normal in G if every chief factor {H/K} of G between {A_{G}} and {A^{G}} is {\mathfrak{F}} -central in G and {\mathfrak{F}} -abnormal in G if V is not {\mathfrak{F}} -normal in W for every two subgroups {V<W} of G such that {A\leq V} . We give a description of finite groups in which every subgroup is either {\mathfrak{F}} -normal or {\mathfrak{F}} -abnormal.

ON A LATTICE CHARACTERISATION OF FINITE SOLUBLE PST-GROUPS

2019 ◽  
Vol 101 (2) ◽  
pp. 247-254 ◽  
Author(s):  
ZHANG CHI ◽  
ALEXANDER N. SKIBA
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Nilpotent Groups ◽  
Chief Factor ◽  
Finite Soluble Group ◽  
A Finite Group

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a PST-group if and only if $A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.

scholarly journals Finite Groups with Certain Permutability Criteria

2019 ◽  
Vol 12 (2) ◽  
pp. 571-576 ◽  
Author(s):  
Rola A. Hijazi ◽  
Fatme M. Charaf
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is said to be S-permutable in G if itpermutes with all Sylow subgroups of G. In this note we prove that if P, the Sylowp-subgroup of G (p > 2), has a subgroup D such that 1 <|D|<|P| and all subgroups H of P with |H| = |D| are S-permutable in G, then G′ is p-nilpotent.

On SS-supplemented subgroups of some sylow subgroups of finite groups

2021 ◽  
Author(s):  
Xuanli He ◽  
Qinghong Guo ◽  
Muhong Huang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Necessary Conditions ◽  
Saturated Formation ◽  
Sylow Subgroups ◽  
Group A ◽  
Sufficient And Necessary Conditions ◽  
A Finite Group

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] permutes with all Sylow subgroups of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-supplemented in [Formula: see text] if there exists a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate [Formula: see text]-nilpotency of a finite group. As applications, we give some sufficient and necessary conditions for a finite group belongs to a saturated formation.

scholarly journals Impartial achievement games for generating nilpotent groups

2019 ◽  
Vol 22 (3) ◽  
pp. 515-527
Author(s):  
Bret J. Benesh ◽  
Dana C. Ernst ◽  
Nándor Sieben
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Generating Set ◽  
Odd Order ◽  
Impartial Game ◽  
A Finite Group

AbstractWe study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form{T\times H}, whereTis a 2-group andHis a group of odd order. This includes all nilpotent and hence abelian groups.

A Note on Special Local 2-Nilpotent Groups and the Solvability of Finite Groups

2018 ◽  
Vol 25 (04) ◽  
pp. 541-546
Author(s):  
Jiangtao Shi ◽  
Klavdija Kutnar ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Quaternion Group ◽  
A Finite Group

A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, [Formula: see text] and NG(P) is 2-nilpotent. In this paper, it is shown that SL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if [Formula: see text], and that GL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555–1560].

On the 𝔉-hypercenter and the intersection of 𝔉-maximal subgroups of a finite group

2018 ◽  
Vol 21 (3) ◽  
pp. 463-473
Author(s):  
Viachaslau I. Murashka
Keyword(s):  
Finite Group ◽  
Nilpotent Groups ◽  
Chief Factor ◽  
Maximal Subgroups ◽  
Inner Automorphism ◽  
Formula Group ◽  
Saturated Formations ◽  
A Finite Group

Abstract Let {\mathfrak{X}} be a class of groups. A subgroup U of a group G is called {\mathfrak{X}} -maximal in G provided that (a) {U\in\mathfrak{X}} , and (b) if {U\leq V\leq G} and {V\in\mathfrak{X}} , then {U=V} . A chief factor {H/K} of G is called {\mathfrak{X}} -eccentric in G provided {(H/K)\rtimes G/C_{G}(H/K)\not\in\mathfrak{X}} . A group G is called a quasi- {\mathfrak{X}} -group if for every {\mathfrak{X}} -eccentric chief factor {H/K} and every {x\in G} , x induces an inner automorphism on {H/K} . We use {\mathfrak{X}^{*}} to denote the class of all quasi- {\mathfrak{X}} -groups. In this paper we describe all hereditary saturated formations {\mathfrak{F}} containing all nilpotent groups such that the {\mathfrak{F}^{*}} -hypercenter of G coincides with the intersection of all {\mathfrak{F}^{*}} -maximal subgroups of G for every group G.

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”.)

The Influence of Certain Abelian Subgroups on the Structure of Finite Groups

2008 ◽  
Vol 15 (03) ◽  
pp. 479-484 ◽  
Author(s):  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
Abelian Subgroups ◽  
A Finite Group

Let G be a finite group. A subgroup K of a group G is called an [Formula: see text]-subgroup of G if NG(K) ∩ Kx ≤ K for all x ∈ G. The set of all [Formula: see text]-subgroups of G is denoted by [Formula: see text]. In this paper, we investigate the structure of a group G under the assumption that certain abelian subgroups of prime power order belong to [Formula: see text].

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