scholarly journals A finiteness condition on centralizers in locally nilpotent groups

2015 ◽  
Vol 182 (2) ◽  
pp. 289-298 ◽  
Author(s):  
Gustavo A. Fernández-Alcober ◽  
Leire Legarreta ◽  
Antonio Tortora ◽  
Maria Tota
Keyword(s):  
Nilpotent Groups ◽  
Finiteness Condition ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent

Homology and cohomology of locally supersoluble groups

1987 ◽  
Vol 102 (2) ◽  
pp. 233-250 ◽  
Author(s):  
Derek J. S. Robinson
Keyword(s):  
Recent Article ◽  
Nilpotent Groups ◽  
Finiteness Condition ◽  
Vanishing Theorems ◽  
Supersoluble Groups ◽  
Low Dimensions ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Locally Supersoluble Groups

In a recent article [13] a series of vanishing theorems was obtained for the (co)homology of locally nilpotent groups. These results assert that if (co)homology vanishes in low dimensions (0 or 1), then it vanishes in all dimensions, provided that the module satisfies an appropriate finiteness condition.

The complexity of ascendant sequences in locally nilpotent groups

2014 ◽  
Vol 24 (02) ◽  
pp. 189-205 ◽  
Author(s):  
Chris J. Conidis ◽  
Richard A. Shore
Keyword(s):  
Nilpotent Groups ◽  
Low Complexity ◽  
Order Type ◽  
Usual Definition ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Normal Form Theorem ◽  
Algebraic Normal Form ◽  
Definition Of ◽  
Transfinite Recursion

We analyze the complexity of ascendant sequences in locally nilpotent groups, showing that if G is a computable locally nilpotent group and x0, x1, …, xN ∈ G, N ∈ ℕ, then one can always find a uniformly computably enumerable (i.e. uniformly [Formula: see text]) ascendant sequence of order type ω + 1 of subgroups in G beginning with 〈x0, x1, …, xN〉G, the subgroup generated by x0, x1, …, xN in G. This complexity is surprisingly low in light of the fact that the usual definition of ascendant sequence involves arbitrarily large ordinals that index sequences of subgroups defined via a transfinite recursion in which each step is incomputable. We produce this surprisingly low complexity sequence via the effective algebraic commutator collection process of P. Hall, and a related purely algebraic Normal Form Theorem of M. Hall for nilpotent groups.

On locally nilpotent groups

1956 ◽  
Vol 52 (1) ◽  
pp. 5-11 ◽  
Author(s):  
D. H. McLain ◽  
P. Hall
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Product ◽  
Finite Subset ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
A Finite Group

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

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