Derived p-Length of a p-Soluble Group with Bounded Indices of Fitting p-Subgroups in Their Normal Closures

2020 ◽  
Vol 72 (3) ◽  
pp. 416-421
Author(s):  
D. V. Gritsuk ◽  
A. A. Trofimuk
Keyword(s):  
Soluble Group

scholarly journals A group variety defined by a semigroup law

1996 ◽  
Vol 60 (2) ◽  
pp. 255-259 ◽  
Author(s):  
A. Yu. Ol'shanskii ◽  
A. Storozhev
Keyword(s):  
Free Group ◽  
Soluble Group ◽  
Group Variety ◽  
Locally Soluble Group

AbstractA group variety defined by one semigroup law in two variables is constructed and it is proved that its free group is not a periodic extension of a locally soluble group.

scholarly journals The stable subgroup graph

2018 ◽  
Vol 36 (3) ◽  
pp. 129-139
Author(s):  
Behnaz Tolue
Keyword(s):  
Abelian Group ◽  
Soluble Group ◽  
Normal Subgroups ◽  
Induced Subgraph ◽  
Vertex Set

In this paper we introduce stable subgroup graph associated to the group $G$. It is a graph with vertex set all subgroups of $G$ and two distinct subgroups $H_1$ and $H_2$ are adjacent if $St_{G}(H_1)\cap H_2\neq 1$ or $St_{G}(H_2)\cap H_1\neq 1$. Its planarity is discussed whenever $G$ is an abelian group, $p$-group, nilpotent, supersoluble or soluble group. Finally, the induced subgraph of stable subgroup graph with vertex set whole non-normal subgroups is considered and its planarity is verified for some certain groups.

scholarly journals Locally soluble groups with min-n.

1974 ◽  
Vol 17 (3) ◽  
pp. 305-318 ◽  
Author(s):  
H. Heineken ◽  
J. S. Wilson
Keyword(s):  
Soluble Group ◽  
Free Groups ◽  
Torsion Group ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Isomorphism Classes ◽  
Torsion Groups ◽  
Torsion Free ◽  
General Method

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.

scholarly journals Lower central depth in finitely generated soluble-by-finite groups

1978 ◽  
Vol 19 (2) ◽  
pp. 153-154 ◽  
Author(s):  
John C. Lennox
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Depth ◽  
Lower Central Series ◽  
Central Series ◽  
Finitely Generated ◽  
Central Depth

We say that a group G has finite lower central depth (or simply, finite depth) if the lower central series of G stabilises after a finite number of steps.In [1], we proved that if G is a finitely generated soluble group in which each two generator subgroup has finite depth then G is a finite-by-nilpotent group. Here, in answer to a question of R. Baer, we prove the following stronger version of this result.

A Class of Three-Generator, Three-Relation, Finite Groups

1970 ◽  
Vol 22 (1) ◽  
pp. 36-40 ◽  
Author(s):  
J. W. Wamsley
Keyword(s):  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Soluble Group ◽  
Finite Nilpotent Group ◽  
Image Position

Mennicke (2) has given a class of three-generator, three-relation finite groups. In this paper we present a further class of three-generator, threerelation groups which we show are finite.The groups presented are defined as:with α|γ| ≠ 1, β|γ| ≠ 1, γ ≠ 0.We prove the following result.THEOREM 1. Each of the groups presented is a finite soluble group.We state the following theorem proved by Macdonald (1).THEOREM 2. G1(α, β, 1) is a finite nilpotent group.1. In this section we make some elementary remarks.

Prefrattini Subgroups and Cover-Avoidance Properties in 𝔘-Groups

1975 ◽  
Vol 27 (4) ◽  
pp. 837-851 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Conjugacy Class ◽  
Soluble Group ◽  
Saturated Formation ◽  
Conjugacy Classes ◽  
Pronormal Subgroup ◽  
Finite Soluble Group ◽  
Factors Associated ◽  
Chief Factors

W. Gaschutz [5] introduced a conjugacy class of subgroups of a finite soluble group called the prefrattini subgroups. These subgroups have the property that they avoid the complemented chief factors of G and cover the rest. Subsequently, these results were generalized by Hawkes [12], Makan [14; 15] and Chambers [2]. Hawkes [12] and Makan [14] obtained conjugacy classes of subgroups which avoid certain complemented chief factors associated with a saturated formation or a Fischer class. Makan [15] and Chambers [2] showed that if W, D and V are the prefrattini subgroup, 𝔍-normalizer and a strongly pronormal subgroup associated with a Sylow basis S, then any two of W, D and V permute and the products and intersections of these subgroups have an explicit cover-avoidance property.

scholarly journals Enumerating subgroups

1987 ◽  
Vol 43 (2) ◽  
pp. 220-223 ◽  
Author(s):  
R. J. Cook ◽  
James Wiecold ◽  
A. G. Wellamson
Keyword(s):  
Prime Divisor ◽  
Soluble Group ◽  
Maximal Subgroups ◽  
Finite Soluble Group

AbstractIt is proved that a finite soluble group of order n has at most (n − 1)/(q − 1) maximal subgroups, where q is the smallest prime divisor of n.

scholarly journals The Engel elements of a soluble group

1959 ◽  
Vol 3 (2) ◽  
pp. 151-168 ◽  
Author(s):  
K. W. Gruenberg
Keyword(s):  
Soluble Group ◽  
Engel Elements

scholarly journals A conjecture of Lennox and Wiegold concerning supersoluble groups

1983 ◽  
Vol 35 (2) ◽  
pp. 218-220 ◽  
Author(s):  
J. R. J. Groves
Keyword(s):  
Soluble Group ◽  
Infinite Subset ◽  
Supersoluble Group ◽  
Finitely Generated ◽  
Supersoluble Groups

AbstractWe prove a conjecture of Lennox and Wiegold that a finitely generated soluble group, in which every infinite subset contains two elements generating a supersoluble group, is finite-by-supersoluble.

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