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.

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 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.

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].

Critical maximal subgroups and conjugacy of supplements in finite soluble groups

2018 ◽  
Vol 21 (1) ◽  
pp. 45-63
Author(s):  
Barbara Baumeister ◽  
Gil Kaplan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Maximal Subgroups ◽  
Abelian Normal Subgroup ◽  
Soluble Groups ◽  
A Finite Group

AbstractLetGbe a finite group with an abelian normal subgroupN. When doesNhave a unique conjugacy class of complements inG? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups ofGclosed under conjugation whose intersection equals{\Phi(G)}. In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when{\Phi(G)=1}, these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

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 Formations of Π-soluble groups

1969 ◽  
Vol 10 (1-2) ◽  
pp. 241-250 ◽  
Author(s):  
H. Lausch
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Composition Series ◽  
Soluble Groups ◽  
Hall Subgroups ◽  
A Finite Group

The theory of formations of soluble groups, developed by Gaschütz [4], Carter and Hawkes[1], provides fairly general methods for investigating canonical full conjugate sets of subgroups in finite, soluble groups. Those methods, however, cannot be applied to the class of all finite groups, since strong use was made of the Theorem of Galois on primitive soluble groups. Nevertheless, there is a possiblity to extend the results of the above mentioned papers to the case of Π-soluble groups as defined by Čunihin [2]. A finite group G is called Π-soluble, if, for a given set it of primes, the indices of a composition series of G are either primes belonging to It or they are not divisible by any prime of Π In this paper, we shall frequently use the following result of Čunihin [2]: Ift is a non-empty set of primes, Π′ its complement in the set of all primes, and G is a Π-soluble group, then there always exist Hall Π-subgroups and Hall ′-subgroups, constituting single conjugate sets of subgroups of G respectively, each It-subgroup of G contained in a Hall Π-subgroup of G where each ′-subgroup of G is contained in a Hall Π′-subgroup of G. All groups considered in this paper are assumed to be finite and Π-soluble. A Hall Π-subgroup of a group G will be denoted by G.

scholarly journals A note on subnormal defect in finite soluble groups

1989 ◽  
Vol 39 (2) ◽  
pp. 255-258
Author(s):  
R.A. Bryce
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Derived Length ◽  
A Finite Group ◽  
Subnormal Subgroups

It is shown that for every positive integer n there exists a finite group of derived length n in which all Sylow subgroups are abeian and in which the defect of subnormal subgroups is at most 3.

On Prime Spectrum of Maximal Subgroups in Finite Groups

2018 ◽  
Vol 25 (04) ◽  
pp. 579-584
Author(s):  
Chi Zhang ◽  
Wenbin Guo ◽  
Natalia V. Maslova ◽  
Danila O. Revin
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Prime Spectrum ◽  
Prime Divisors ◽  
A Finite Group ◽  
Theoretical Results

For a positive integer n, we denote by π(n) the set of all prime divisors of n. For a finite group G, the set [Formula: see text] is called the prime spectrum of G. Let [Formula: see text] mean that M is a maximal subgroup of G. We put [Formula: see text] and [Formula: see text]. In this notice, using well-known number-theoretical results, we present a number of examples to show that both K(G) and k(G) are unbounded in general. This implies that the problem “Are k(G) and K(G) bounded by some constant k?”, raised by Monakhov and Skiba in 2016, is solved in the negative.

scholarly journals On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2165
Author(s):  
Abd El-Rahman Heliel ◽  
Mohammed Al-Shomrani ◽  
Adolfo Ballester-Bolinches
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Direct Product ◽  
Prime Numbers ◽  
Chief Factor ◽  
Maximal Subgroups ◽  
Soluble Groups ◽  
Nilpotent Length ◽  
A Finite Group ◽  
Primary Groups

Let σ={σi:i∈I} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by Fσ(G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=Fσ(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by lσ(G). If F is a subgroup-closed saturated formation, we define the σ-F-lengthnσ(G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then nσ(A,F)=nσ(G,F)−i for some i∈{0,1,2}.

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