Powerfully nilpotent groups of rank 2 or small order

2020 ◽  
Vol 23 (4) ◽  
pp. 641-658
Author(s):  
Gunnar Traustason ◽  
James Williams
Keyword(s):  
Detailed Analysis ◽  
Special Kind ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Central Series ◽  
Rank 2 ◽  
Full Classification

AbstractIn this paper, we continue the study of powerfully nilpotent groups. These are powerful p-groups possessing a central series of a special kind. To each such group, one can attach a powerful nilpotency class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. In this paper, we will give a full classification of powerfully nilpotent groups of rank 2. The classification will then be used to arrive at a precise formula for the number of powerfully nilpotent groups of rank 2 and order {p^{n}}. We will also give a detailed analysis of the ancestry tree for these groups. The second part of the paper is then devoted to a full classification of powerfully nilpotent groups of order up to {p^{6}}.

Central automorphisms of free nilpotent Lie algebras

2017 ◽  
Vol 16 (11) ◽  
pp. 1750205
Author(s):  
Özge Öztekin ◽  
Naime Ekici
Keyword(s):  
Lie Algebras ◽  
Finite Rank ◽  
Characteristic Zero ◽  
Sufficient Conditions ◽  
Nilpotency Class ◽  
Nilpotent Lie Algebra ◽  
Central Automorphism ◽  
Nilpotent Lie Algebras ◽  
Central Automorphisms

Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.

scholarly journals The class of a nilpotent wreath product

1977 ◽  
Vol 17 (1) ◽  
pp. 53-89 ◽  
Author(s):  
David Shield
Keyword(s):  
Normal Subgroup ◽  
Wreath Product ◽  
Group Ring ◽  
Upper Bound ◽  
Lower Central Series ◽  
Nilpotency Class ◽  
Central Series ◽  
Augmentation Ideal ◽  
Cambridge Philos ◽  
Base Group

Let G be a group with a normal subgroup H whose index is a power of a prime p, and which is nilpotent with exponent a power of p. Gilbert Baumslag (Proc. Cambridge Philos. Soc. 55 (1959), 224–231) has shown that such a group is nilpotent; the main result of this paper is an upper bound on its nilpotency class in terms of parameters of H and G/H. It is shown that this bound is attained whenever G is a wreath product and H its base group.A descending central series, here called the cpp-series, is involved in these calculations more closely than is the lower central series, and the class of the wreath product in terms of this series is also found.Two tools used to obtain the main result, namely a useful basis for a finite p-group and a result about the augmentation ideal of the integer group ring of a finite p-group, may have some independent interest. The main result is applied to the construction of some two-generator groups of large nilpotency class with exponents 8, 9, and 25.

scholarly journals Local nearrings on finite non-abelian $2$-generated $p$-groups

2020 ◽  
Vol 12 (1) ◽  
pp. 199-207
Author(s):  
I.Yu. Raievska ◽  
M.Yu. Raievska
Keyword(s):  
Commutator Subgroup ◽  
Additive Group ◽  
Nilpotency Class ◽  
Invertible Elements ◽  
Cyclic Commutator

It is proved that for ${p>2}$ every finite non-metacyclic $2$-generated p-group of nilpotency class $2$ with cyclic commutator subgroup is the additive group of a local nearring and in particular of a nearring with identity. It is also shown that the subgroup of all non-invertible elements of this nearring is of index $p$ in its additive group.

Left conjugacy closed loops of nilpotency class two

2005 ◽  
Vol 47 (3-4) ◽  
pp. 242-265 ◽  
Author(s):  
Piroska Csörgő ◽  
Aleš Drápal
Keyword(s):  
Nilpotency Class ◽  
Closed Loops

scholarly journals Group Laws Implying Virtual Nilpotence

2003 ◽  
Vol 74 (3) ◽  
pp. 295-312 ◽  
Author(s):  
R. G. Burns ◽  
Yuri Medvedev
Keyword(s):  
Large Class ◽  
Finite Groups ◽  
Additional Condition ◽  
Metabelian Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Locally Graded Groups ◽  
Finite Exponent ◽  
Group Law

AbstractIf ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

scholarly journals The nilpotency class of p-groups in which subgroups have few conjugates

2006 ◽  
Vol 300 (1) ◽  
pp. 160-170 ◽  
Author(s):  
Giovanni Cutolo ◽  
Howard Smith ◽  
James Wiegold
Keyword(s):  
Nilpotency Class

scholarly journals Cohomology of p-groups of nilpotency class smaller than p

2018 ◽  
Vol 21 (2) ◽  
pp. 337-350 ◽  
Author(s):  
Antonio Díaz Ramos ◽  
Oihana Garaialde Ocaña ◽  
Jon González-Sánchez
Keyword(s):  
Prime Number ◽  
Nilpotency Class ◽  
Cohomology Algebra

AbstractLetpbe a prime number, letdbe an integer and letGbe ad-generated finitep-group of nilpotency class smaller thanp. Then the number of possible isomorphism types for the modpcohomology algebra{H^{*}(G;{\mathbb{F}}_{p})}is bounded in terms ofpandd.

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