scholarly journals Triangles of Baumslag–Solitar Groups

2012 ◽  
Vol 64 (2) ◽  
pp. 241-253 ◽  
Author(s):  
Daniel Allcock
Keyword(s):  
Finite Groups ◽  
Derived Length ◽  
Special Cases ◽  
Finite Quotients ◽  
General Conditions

Abstract Our main result is that many triangles of Baumslag–Solitar groups collapse to finite groups, generalizing a famous example of Hirsch and other examples due to several authors. A triangle of Baumslag–Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of derived length ≤ 3. We obtain a lot of information about finite quotients, even when we cannot determine developability.

scholarly journals GENERALIZED W3-STRINGS FROM FREE FIELDS

1995 ◽  
Vol 10 (06) ◽  
pp. 515-524 ◽  
Author(s):  
J. M. FIGUEROA-O'FARRILL ◽  
C. M. HULL ◽  
L. PALACIOS ◽  
E. RAMOS
Keyword(s):  
Bosonic Strings ◽  
Free Field ◽  
Space Time ◽  
General Construction ◽  
Special Cases ◽  
Free Fields ◽  
Rule Out ◽  
General Conditions ◽  
String Theories

The conventional quantization of w3-strings gives theories which are equivalent to special cases of bosonic strings. We explore whether a more general quantization can lead to new generalized W3-string theories by seeking to construct quantum BRST charges directly without requiring the existence of a quantum W3-algebra. We study W3-like strings with a direct space-time interpretation — that is, with matter given by explicit free field realizations. Special emphasis is placed on the attempt to construct a quantum W-string associated with the magic realizations of the classical w3-algebra. We give the general conditions for the existence of W3-like strings, and comment on how the known results fit into our general construction. Our results are negative: we find no new consistent string theories, and in particular rule out the possibility of critical strings based on the magic realizations.

10. Visitors: entry for temporary purposes

2018 ◽  
Author(s):  
Gina Clayton ◽  
Georgina Firth ◽  
Caroline Sawyer ◽  
Rowena Moffatt ◽  
Helena Wray
Keyword(s):  
Medical Treatment ◽  
Family Members ◽  
Short Term ◽  
Special Cases ◽  
The Law ◽  
The Uk ◽  
General Conditions

Course-focused and comprehensive, the Textbook on series provides an accessible overview of the key areas on the law curriculum. This chapter discusses the law relating to individuals coming to the UK as visitors for short-term or finite purposes such as tourism, business visits, sporting and entertainment engagements, or for private medical treatment. There is a discussion of the withdrawal, reinstatement, and restriction of rights of appeal for those visiting family members in the UK, and the application of Article 8 ECHR to these situations. The revised visitor rules in Appendix V are described in detail. The chapter also discusses the special cases of marriage visitors, carers and transit visitors, and general conditions such as prohibited activities and the need for maintenance and accommodation.

scholarly journals Some challenging group presentations

1999 ◽  
Vol 67 (2) ◽  
pp. 206-213 ◽  
Author(s):  
George Havas ◽  
Derek F. Holt ◽  
P. E. Kenne ◽  
Sarah Rees
Keyword(s):  
Finite Groups ◽  
Infinite Groups ◽  
Derived Length ◽  
Finitely Presented Group ◽  
Group Presentations ◽  
Coset Enumeration ◽  
Finitely Presented

AbstractWe study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations are for finite groups while two are for infinite groups. Thus we answer three explicit questions in the literature and we provide the first published deficiency zero presentation for a group with derived length seven. The tools we use are coset enumeration and Knuth-Bebdix rewriting, which are well-established as methods for proving finiteness or otherwise of a finitely presented group. We briefly comment on their capabilities and compare their performance.

scholarly journals On the laws of certain finite groups

1970 ◽  
Vol 11 (4) ◽  
pp. 441-489 ◽  
Author(s):  
John Cossey ◽  
Sheila Oates MacDonald ◽  
Anne Penfold Street
Keyword(s):  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classification Problems ◽  
Sylow Subgroups ◽  
Local Finiteness ◽  
Special Cases

In recent years a great deal of attention has been devoted to the study of finite simple groups, but one aspect which seems to have been little considered is that of the laws they satisfy. In a recent paper [3], the first two of the present authors gave a basis for laws of PSL(2, 5). The techniques of [3] can be used to show that (modulo certain classification problems) a basis for the laws of PSL(2, pn) can be made up from laws of the following types:(1) an exponent law,(2) laws which determine the Sylow subgroups,(3) laws which determine the normalisers of the Sylow subgroups,(4) in certain special cases, laws which determine subvarieties of smaller exponent, e.g. the subvariety of exponent 12 for those PSL(2, pn) which contain S4,(5) a law implying local finiteness.

scholarly journals Identical relations and decision procedures for groups

1967 ◽  
Vol 7 (1) ◽  
pp. 39-47 ◽  
Author(s):  
Hermann Heineken ◽  
Peter M. Neumann
Keyword(s):  
Structural Property ◽  
Finite Groups ◽  
Decision Procedures ◽  
Nilpotency Class ◽  
Cartesian Products ◽  
Derived Length

Although varieties of groups can in theory be determined as well by the identical relations which the groups all satisfy as by some structural property inherited by subgroups, factor groups and cartesian products which the groups have in common, it seems in practice just as hard to answer questions about properties of a group from knowledge of identical relations as it is from, say, a presentation. Many of the important questions connected with Burnside's problems exemplify this difficulty: we still do not know if there is a bound on the derived length of finite groups of exponent 4, nor whether there is a bound on the nilpotency class of finite groups of exponent p (p ≧ 5, a fixed prime).

scholarly journals The Grothendieck--Teichm\"{u}ller group of~PSL(2,q)

2018 ◽  
Vol 21 (2) ◽  
pp. 241-251
Author(s):  
Pierre Guillot
Keyword(s):  
Finite Groups ◽  
Dihedral Group ◽  
Monodromy Group ◽  
Derived Length

AbstractIn this paper, we show that the Grothendieck–Teichmüller group of{\operatorname{PSL}(2,q)}, or more precisely the group{\mathcal{G\kern-0.569055ptT}_{\kern-1.707165pt1}(\operatorname{PSL}(2,q))}as previously defined by the author, is the product of an elementary abelian 2-group and several copies of the dihedral group of order 8. Moreover, whenqis even, we show that it is trivial. We explain how it follows that the moduli field of any “dessin d’enfant” whose monodromy group is{\operatorname{PSL}(2,q)}has derived length{\leq 3}. This paper can serve as an introduction to the general results on the Grothendieck–Teichmüller group of finite groups obtained by the author.

10. Visitors: entry for temporary purposes

2021 ◽  
pp. 370-384
Author(s):  
Gina Clayton ◽  
Georgina Firth ◽  
Caroline Sawyer ◽  
Rowena Moffatt
Keyword(s):  
Medical Treatment ◽  
Family Members ◽  
Short Term ◽  
Special Cases ◽  
The Law ◽  
The Uk ◽  
General Conditions

This chapter discusses the law relating to individuals coming to the UK as visitors for short-term or finite purposes such as tourism, business visits, sporting and entertainment engagements, or for private medical treatment. There is a discussion of the withdrawal, reinstatement, and restriction of rights of appeal for those visiting family members in the UK, and the application of Article 8 ECHR to these situations. The revised visitor rules in Appendix V are described in detail. The chapter also discusses the special cases of marriage visitors, carers and transit visitors, and general conditions such as prohibited activities and the need for maintenance and accommodation.

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 Right weak Engel conditions on linear groups

2019 ◽  
Vol 114 (1) ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Positive Integer ◽  
Linear Group ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Linear Groups ◽  
Locally Finite ◽  
Class A ◽  
Special Cases ◽  
Prüfer Rank

AbstractIf X is a group-class, a group G is right X-Engel if for all g in G there exists an X-subgroup E of G such that for all x in G there is a positive integer m(x) with [g, nx] ∈ E for all n ≥ m(x). Let G be a linear group. Special cases of our main theorem are the following. If X is the class of all Chernikov groups, or all finite groups, or all locally finite groups, then G is right X-Engel if and only if G has a normal X-subgroup modulo which G is hypercentral. The same conclusion holds if G has positive characteristic and X is one of the following classes; all polycyclic-by-finite groups, all groups of finite Prüfer rank, all minimax groups, all groups with finite Hirsch number, all soluble-by-finite groups with finite abelian total rank. In general the characteristic zero case is more complex.

Hypercentrally embedded subgroups of polycyclic-by-finite groups

2012 ◽  
Vol 15 (6) ◽  
Author(s):  
Rudolf Maier
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finite Quotients

Abstract.A subgroup of a polycyclic-by-finite group is hypercentrally embedded if and only if its projections into the finite quotients have this property.

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