scholarly journals Constructing uncountably many groups with the same profinite completion

10.53733/89 ◽  
2021 ◽  
Vol 52 ◽  
pp. 765-771
Author(s):  
Nikolay Nikolov ◽  
Dan Segal
Keyword(s):  
Rooted Tree ◽  
The Other ◽  
Profinite Completion ◽  
Soluble Groups ◽  
Derived Length

Two constructions are described: one gives soluble groups of derived length 4, the other uses groups acting on a rooted tree.

On the derived length of subgroups of infinite index in soluble groups

2003 ◽  
Vol 7 (1) ◽  
Author(s):  
Martyn R. Dixon ◽  
Martin J. Evans ◽  
Howard Smith
Keyword(s):  
Infinite Index ◽  
Soluble Groups ◽  
Derived Length

scholarly journals Some bounds for the degree of commutativity of a p-group of maximal class

1995 ◽  
Vol 51 (3) ◽  
pp. 353-367
Author(s):  
Antonio Vera-López ◽  
Gustavo A. Fernández-Alcober
Keyword(s):  
Maximal Subgroup ◽  
Lower Bounds ◽  
Nilpotency Class ◽  
The Other ◽  
Maximal Class ◽  
Derived Length ◽  
Other Hand

In this paper we obtain several lower bounds for the degree of commutativity of a p-group of maximal class of order pm. All the bounds known up to now involve the prime p and are almost useless for small m. We introduce a new invariant b which is related with the commutator structure of the group G and get a bound depending only on b and m, not on p. As a consequence, we bound the derived length of G and the nilpotency class of a certain maximal subgroup in terms of b. On the other hand, we also generalise some results of Blackburn. Examples are given in order to check the sharpness of the bounds.

The ontology of person-centered healthcare: Theoretical essentials to reground medicine within its humanistic framework. Part I - A centered concept of personhood: Some tools to conceptualise subjectivity

2018 ◽  
Vol 6 (1) ◽  
pp. 146
Author(s):  
Thomas Frölich ◽  
F F Bevier ◽  
Alicja Babakhani ◽  
Hannah H Chisholm ◽  
Peter Henningsen ◽  
...  
Keyword(s):  
Graph Theory ◽  
Free Space ◽  
Rooted Tree ◽  
The Other ◽  
Discrete Mathematics ◽  
Soap Bubble ◽  
Soap Bubbles ◽  
In Series ◽  
The One ◽  
Individual Trees

To address subjectivity, as a generally rooted phenomenon, other ways of visualisation must be applied than in conventional objectivistic approaches. Using ‘trees’ as operational metaphors, as employed in Arthur Cayley’s ‘theory of the analytical forms called trees’, one rooted ‘tree’ must be set beneath the other and, if such ‘trees’ are combined, the resulting ‘forest’ is nevertheless made up of individual ‘trees’ and not of a deconstructed mix of ‘roots’, ‘branches’, ‘leaves’ or further categories, each understood as addressable both jointly and individually. The reasons for why we have chosen a graph theory and corresponding discrete mathematics as an approach and application are set out in this first of our three articles. It combines two approaches that, in combination, are quite uncommon and which are therefore not immediately familiar to all readers. But as simple as it is to imagine a tree, or a forest, it is equally simple to imagine a child blowing soap bubbles with the aid of a blow ring. A little more challenging, perhaps, is the additional idea of arranging such blow rings in series, transforming the size of the soap bubble in one ring after the other. To finally combine both pictures, the one of trees and the other of blow rings, goes beyond simple imagination, especially when we prolong the imagined blow ring becoming a tunnel, with a specific inner shape. The inner shape of the blow ring and its expansion as a tunnel are understood as determined by discrete qualities, each forming an internal continuity, depicted as a scale, with the scales combined in the form of a glyph plot. The different positions on these scales determine their length and if the endpoints of the spines are connected with an enveloping line then this corresponds to the free space left open in the tunnel to go through it. Using so many visualisation techniques at once is testing. Nevertheless, this is what we propose here and to facilitate such a visualisation within the imagination, we do it step by step. As the intended result of this ‘juggling of three balls’, as it were, we end up with a concept of how living beings elaborate their principal structure to enable controlled outside-inside communication.

scholarly journals Finite groups of deficiency zero involving the Lucas numbers

1990 ◽  
Vol 33 (1) ◽  
pp. 1-10 ◽  
Author(s):  
C. M. Campbell ◽  
E. F. Robertson ◽  
R. M. Thomas
Keyword(s):  
Finite Groups ◽  
Soluble Group ◽  
Fibonacci Number ◽  
Single Parameter ◽  
The Other ◽  
Finite Soluble Group ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Derived Length ◽  
New Class

In this paper, we investigate a class of 2-generator 2-relator groups G(n) related to the Fibonacci groups F(2,n), each of the groups in this new class also being defined by a single parameter n, though here n can take negative, as well as positive, values. If n is odd, we show that G(n) is a finite soluble group of derived length 2 (if n is coprime to 3) or 3 (otherwise), and order |2n(n + 2)gnf(n, 3)|, where fn is the Fibonacci number defined by f0=0,f1=1,fn+2=fn+fn+1 and gn is the Lucas number defined by g0 = 2, g1 = 1, gn+2 = gn + gn+1 for n≧0. On the other hand, if n is even then, with three exceptions, namely the cases n = 2,4 or –4, G(n) is infinite; the groups G(2), G(4) and G(–4) have orders 16, 240 and 80 respectively.

scholarly journals The subnormal structure of some classes of coluble groups

1972 ◽  
Vol 13 (3) ◽  
pp. 365-377 ◽  
Author(s):  
D. McDougall
Keyword(s):  
Finite Groups ◽  
Soluble Group ◽  
Subnormal Subgroup ◽  
Negative Integer ◽  
Soluble Groups ◽  
Derived Length ◽  
Subnormal Structure ◽  
A Chain

Finite groups in which normality is transitive have been studied by Best and Taussky, [1], Gaschütz, [3], and Zacher [16]. Infinite soluble groups in which normality is transitive have been studied by Robinson in [9]. A subgroup H of a group G is subnormal in G if H can be connected to G by a chain of r subgroups, in which each is normal in its successor, where r is a non-negative integer. The least such r is called the subnormal index of H in G (or the defect of H in G). Then groups in which normality is transitive are precisely those in which every subnormal subgroup has subnormal index at most one. Thus the structure of soluble groups in which every subnormal subgroup has subnormal index at most n (such a group is said to have bounded subnormal indices) has been dealt with by Robinson in [9] for the case where n is one. However Theorem D of [12] states that a soluble group of derived length n can be embedded in a soluble group in which the subnormal indices are at most n. Therefore we must impose further conditions on the groups if we hope to obtain any worthwhile results for the above problem with n greater than one.

scholarly journals Representations of metabelian groups satisfying the minimal condition for normal subgroups

1976 ◽  
Vol 14 (2) ◽  
pp. 267-278 ◽  
Author(s):  
Howard L. Silcock
Keyword(s):  
Minimal Condition ◽  
Normal Subgroups ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Derived Length ◽  
Indecomposable Representations

A question of John S. Wilson concerning indecomposable representations of metabelian groups satisfying the minimal condition for normal subgroups is answered negatively, by means of an example. It is shown that such representations need not be irreducible, even when the group being represented is an extension of an elementary abelian p–group by a quasicyclic q–group of the type first described by V.S. Čarin, and the characteristic of the field is a prime distinct from both p and q. This implies that certain techniques used in the study of metabelian groups satisfying the minimal condition for normal subgroups are not available for the corresponding class of soluble groups of derived length 3.

scholarly journals The Ultrametric Constraint and its Application to Phylogenetics

2008 ◽  
Vol 32 ◽  
pp. 901-938 ◽  
Author(s):  
N. C. A. Moore ◽  
P. Prosser
Keyword(s):  
Lower Bounds ◽  
Constraint Programming ◽  
Common Ancestor ◽  
Rooted Tree ◽  
The Other ◽  
Recent Common Ancestor ◽  
Evolutionary Relationships ◽  
Construction Problem ◽  
Integer Variables ◽  
Most Recent Common Ancestor

A phylogenetic tree shows the evolutionary relationships among species. Internal nodes of the tree represent speciation events and leaf nodes correspond to species. A goal of phylogenetics is to combine such trees into larger trees, called supertrees, whilst respecting the relationships in the original trees. A rooted tree exhibits an ultrametric property; that is, for any three leaves of the tree it must be that one pair has a deeper most recent common ancestor than the other pairs, or that all three have the same most recent common ancestor. This inspires a constraint programming encoding for rooted trees. We present an efficient constraint that enforces the ultrametric property over a symmetric array of constrained integer variables, with the inevitable property that the lower bounds of any three variables are mutually supportive. We show that this allows an efficient constraint-based solution to the supertree construction problem. We demonstrate that the versatility of constraint programming can be exploited to allow solutions to variants of the supertree construction problem.

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.

THE NUMBER OF PROFINITE GROUPS WITH A SPECIFIED SYLOW SUBGROUP

2015 ◽  
Vol 99 (1) ◽  
pp. 108-127
Author(s):  
COLIN D. REID
Keyword(s):  
Normal Subgroup ◽  
Sylow Subgroup ◽  
The Other ◽  
Profinite Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Other Hand ◽  
Formula Group ◽  
Nontrivial Normal Subgroup

Let $S$ be a finitely generated pro-$p$ group. Let ${\mathcal{E}}_{p^{\prime }}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects nontrivially with every nontrivial normal subgroup of $G$. In this paper, we investigate whether or not there is a bound on $|G:S|$ for $G\in {\mathcal{E}}_{p^{\prime }}(S)$. For instance, we give an example where ${\mathcal{E}}_{p^{\prime }}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $|G:S|$ is bounded in the case where $S$ is just infinite.

scholarly journals Measuring Language Distance of Isolated European Languages

Information ◽  
2020 ◽  
Vol 11 (4) ◽  
pp. 181 ◽  
Author(s):  
Pablo Gamallo ◽  
José Ramom Pichel ◽  
Iñaki Alegria
Keyword(s):  
Natural Language Processing ◽  
Natural Language ◽  
Language Processing ◽  
Historical Linguistics ◽  
Rooted Tree ◽  
The Other ◽  
Historical Evolution ◽  
European Languages ◽  
Shed Light ◽  
European Family

Phylogenetics is a sub-field of historical linguistics whose aim is to classify a group of languages by considering their distances within a rooted tree that stands for their historical evolution. A few European languages do not belong to the Indo-European family or are otherwise isolated in the European rooted tree. Although it is not possible to establish phylogenetic links using basic strategies, it is possible to calculate the distances between these isolated languages and the rest using simple corpus-based techniques and natural language processing methods. The objective of this article is to select some isolated languages and measure the distance between them and from the other European languages, so as to shed light on the linguistic distances and proximities of these controversial languages without considering phylogenetic issues. The experiments were carried out with 40 European languages including six languages that are isolated in their corresponding families: Albanian, Armenian, Basque, Georgian, Greek, and Hungarian.

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