scholarly journals ON THE RANK OF A VERBAL SUBGROUP OF A FINITE GROUP

2021 ◽  
pp. 1-15
Author(s):  
ELOISA DETOMI ◽  
MARTA MORIGI ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Nilpotent Subgroup ◽  
Finite Soluble Group ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
A Finite Group ◽  
Commutator Word

Abstract We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

scholarly journals ON THE EXPONENT OF A VERBAL SUBGROUP IN A FINITE GROUP

2012 ◽  
Vol 93 (3) ◽  
pp. 325-332 ◽  
Author(s):  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Nilpotent Subgroup ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
A Finite Group ◽  
Commutator Word

AbstractLet $w$ be a multilinear commutator word. We prove that if $e$ is a positive integer and $G$ is a finite group in which any nilpotent subgroup generated by $w$-values has exponent dividing $e$, then the exponent of the corresponding verbal subgroup $w(G)$ is bounded in terms of $e$ and $w$only.

scholarly journals On finite groups in which commutators are covered by Engel subgroups

2019 ◽  
Vol 22 (6) ◽  
pp. 1049-1057
Author(s):  
Pavel Shumyatsky ◽  
Danilo Silveira
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Bounded Number ◽  
A Finite Group ◽  
Commutator Word ◽  
Positive Integers

Abstract Let {m,n} be positive integers and w a multilinear commutator word. Assume that G is a finite group having subgroups {G_{1},\ldots,G_{m}} whose union contains all w-values in G. Assume further that all elements of the subgroups {G_{1},\ldots,G_{m}} are n-Engel in G. It is shown that the verbal subgroup {w(G)} is s-Engel for some {\{m,n,w\}} -bounded number s.

scholarly journals The formation generated by a finite group

1970 ◽  
Vol 2 (3) ◽  
pp. 347-357 ◽  
Author(s):  
R. M. Bryant ◽  
R. A. Bryce ◽  
B. Hartley
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Saturated Formation ◽  
Finite Soluble Group ◽  
Arbitrary Finite Group ◽  
A Finite Group

We prove here that the (saturated) formation generated by a finite soluble group has only finitely many (saturated) subformations. This answers a question asked by Professor W. Gaschütz. Some partial results are also given in the case of a formation generated by an arbitrary finite group.

On a theorem of Gagola and Lewis

2016 ◽  
Vol 16 (08) ◽  
pp. 1750158 ◽  
Author(s):  
Jiakuan Lu
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Finite Soluble Group ◽  
Irreducible Characters ◽  
A Finite Group

Gagola and Lewis proved that a finite group [Formula: see text] is nilpotent if and only if [Formula: see text] divides [Formula: see text] [Formula: see text] [Formula: see text] for all irreducible characters [Formula: see text] of [Formula: see text]. In this paper, we prove that a finite soluble group [Formula: see text] is nilpotent if and only if [Formula: see text] divides [Formula: see text] [Formula: see text] [Formula: see text] for all irreducible monomial characters [Formula: see text] of [Formula: see text].

scholarly journals ON GROUPS ADMITTING A WORD WHOSE VALUES ARE ENGEL

2013 ◽  
Vol 23 (01) ◽  
pp. 81-89 ◽  
Author(s):  
RAIMUNDO BASTOS ◽  
PAVEL SHUMYATSKY ◽  
ANTONIO TORTORA ◽  
MARIA TOTA
Keyword(s):  
Finite Group ◽  
Multilinear Commutator ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Verbal Subgroup ◽  
Locally Nilpotent ◽  
Locally Graded Group ◽  
Commutator Word ◽  
Positive Integers

Let m, n be positive integers, v a multilinear commutator word and w = vm. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent.

The Structure of G/ Φ (G) in Terms of σ (G)

2006 ◽  
Vol 13 (01) ◽  
pp. 1-8
Author(s):  
Alireza Jamali ◽  
Hamid Mousavi
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Soluble Group ◽  
Maximal Subgroups ◽  
Finite Soluble Group ◽  
A Finite Group

Let G be a finite group. We let [Formula: see text] and σ (G) denote the number of maximal subgroups of G and the least positive integer n such that G is written as the union of n proper subgroups, respectively. In this paper, we determine the structure of G/ Φ (G) when G is a finite soluble group with [Formula: see text].

scholarly journals Finite one-relator products of two cyclic groups with the relator of arbitrary length

1992 ◽  
Vol 53 (3) ◽  
pp. 352-368 ◽  
Author(s):  
C. M. Campbell ◽  
P. M. Heggie ◽  
E. F. Robertson ◽  
R. M. Thomas
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Finite Soluble Group ◽  
Arbitrary Length ◽  
Cyclic Groups ◽  
Image Position ◽  
A Finite Group

AbstractIn this paper we consider the groups G = G(α, n) defined by the presentations . We derive a formula for [G′: ″] and determine the order of G whenever n ≦ 7. We show that G is a finite soluble group if n is odd, but that G can be infinite when n is even, n ≧ 8. We also show that G(6, 10) is a finite insoluble group involving PSU(3, 4), and that the group H with presentation is a finite group of deficiency zero of order at least 114,967,210,176,000.

A Note on the σ-Covers of Finite Soluble Groups

2005 ◽  
Vol 12 (04) ◽  
pp. 691-697 ◽  
Author(s):  
Alireza Jamali ◽  
Hamid Mousavi
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Soluble Group ◽  
Maximal Subgroups ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
A Finite Group

A cover for a finite group is a set of proper subgroups whose union is the whole group. For a finite group G, we denote by σ (G) the least positive integer n such that G has a cover of size n. This paper deals with the covers of a finite soluble group G having σ (G) elements of maximal subgroups.

scholarly journals Finite soluble groups have large centralisers

1987 ◽  
Vol 35 (2) ◽  
pp. 291-298 ◽  
Author(s):  
John Cossey
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Central Element ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
A Finite Group

We say that a finite group G has a large centraliser if G contains a non-central element x with |CG (x)| > |G|½. We prove that every finite soluble group has a large centraliser, confirming a conjecture of Bertram and Herzog.

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.

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