Third-Engel 2-Groups are Soluble

1972 ◽  
Vol 15 (4) ◽  
pp. 523-524 ◽  
Author(s):  
Narain Gupta
Keyword(s):  
Soluble Group ◽  
Engel Group

AbstractIt is shown that a 3rd-Engel group is an extension of a soluble group by a group of exponent 5.

RINGS SATISFYING GENERALIZED ENGEL CONDITIONS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250121 ◽  
Author(s):  
M. RAMEZAN-NASSAB ◽  
D. KIANI
Keyword(s):  
Soluble Group ◽  
Associative Ring ◽  
Commutator Ideal ◽  
Engel Group ◽  
Algebra Over A Field

Let R be an associative ring and let x, y ∈ R. Define the generalized commutators as follows: [x, 0y] = x and [x, ky] = [x, k-1y]y - y[x, k-1y](k = 1, 2, …). In this paper we study some generalized Engel rings, i.e. [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)y] = 0), [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)yn(x, y)] = 0) and [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)yn(x, y)]r(x, y) = 0). Among other results, it is proved that every Artinian [Formula: see text]-ring is strictly Lie-nilpotent. Also, we show that in each of the following cases R has nil commutator ideal: (1) if R is a [Formula: see text]-ring with unity and k, n independent of y; (2) if R is a locally bounded [Formula: see text]-ring (defined below); (3) if R is an algebraic algebra over a field in which R* is a bounded Engel group or a soluble group.

On 4-Engel Groups

2007 ◽  
Vol 10 ◽  
pp. 341-353 ◽  
Author(s):  
Michael Vaughan-Lee
Keyword(s):  
Normal Closure ◽  
Engel Group

In this note, the author proves that a group G is a 4-Engel group if and only if the normal closure of every element g ∈ G is a 3-Engel 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.

Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group

2011 ◽  
Vol 202 (11) ◽  
pp. 1593-1615 ◽  
Author(s):  
Andrei A Ardentov ◽  
Yurii L Sachkov
Keyword(s):  
Engel Group ◽  
Extremal Trajectories

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.

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