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.

scholarly journals On the structure of some locally nilpotent groups without contranormal subgroups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Leonid A. Kurdachenko ◽  
Patrizia Longobardi ◽  
Mercede Maj
Keyword(s):  
Finite Group ◽  
Nilpotent Groups ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
A Finite Group

Abstract Following J. S. Rose, a subgroup 𝐻 of a group 𝐺 is said to be contranormal in 𝐺 if G = H G G=H^{G} . It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. We study nilpotent-by-Černikov groups with no proper contranormal subgroups. Furthermore, we study the structure of groups with a finite proper contranormal subgroup.

scholarly journals Residual dimension of nilpotent groups

2020 ◽  
Vol 23 (5) ◽  
pp. 801-829
Author(s):  
Mark Pengitore
Keyword(s):  
New York ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Maximum Order ◽  
Asymptotic Bounds ◽  
Group Morphism ◽  
A Finite Group

AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

The normalizer property for integral group rings of holomorphs of finite nilpotent groups and the symmetric groups

2017 ◽  
Vol 16 (02) ◽  
pp. 1750025 ◽  
Author(s):  
Jinke Hai ◽  
Shengbo Ge ◽  
Weiping He
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Nilpotent Group ◽  
Symmetric Groups ◽  
Nilpotent Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
A Finite Group ◽  
The Normalizer Property

Let [Formula: see text] be a finite group and let [Formula: see text] be the holomorph of [Formula: see text]. If [Formula: see text] is a finite nilpotent group or a symmetric group [Formula: see text] of degree [Formula: see text], then the normalizer property holds for [Formula: see text].

A Note on Special Local 2-Nilpotent Groups and the Solvability of Finite Groups

2018 ◽  
Vol 25 (04) ◽  
pp. 541-546
Author(s):  
Jiangtao Shi ◽  
Klavdija Kutnar ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Quaternion Group ◽  
A Finite Group

A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, [Formula: see text] and NG(P) is 2-nilpotent. In this paper, it is shown that SL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if [Formula: see text], and that GL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555–1560].

Planarity and fixing number of inclusion graph of a nilpotent group

2019 ◽  
Vol 19 (12) ◽  
pp. 2150001
Author(s):  
Shikun Ou ◽  
Dein Wong ◽  
Hailin Liu ◽  
Fenglei Tian
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Undirected Graph ◽  
Nilpotent Groups ◽  
Cyclic Groups ◽  
Vertex Set ◽  
A Finite Group

The inclusion graph of a finite group [Formula: see text], written as [Formula: see text], is defined to be an undirected graph whose vertices are all nontrivial subgroups of [Formula: see text], and two distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. For a graph [Formula: see text] with vertex set [Formula: see text], a set of vertices [Formula: see text] is called a fixing set of [Formula: see text] if the only automorphism of [Formula: see text] that fixes every element in [Formula: see text] is the identity. The fixing number of [Formula: see text] is the smallest size of a fixing set of [Formula: see text]. In this paper, we determine the finite nilpotent groups whose inclusion graphs are planar. Moreover, using the technique of characteristic matrices, we characterize the fixing sets and give the exact value on the fixing number of the inclusion graphs for finite cyclic groups.

A Finite Index Property of Certain Solvable Groups

1984 ◽  
Vol 27 (4) ◽  
pp. 485-489
Author(s):  
A. H. Rhemtulla ◽  
H. Smith
Keyword(s):  
Solvable Group ◽  
Finite Index ◽  
Finite Rank ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Torsion Free

AbstractA group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.

Locally Finite Products of Totally Permutable Nilpotent Groups

2009 ◽  
Vol 16 (03) ◽  
pp. 535-540 ◽  
Author(s):  
Maria De Falco ◽  
Francesco de Giovanni ◽  
Carmela Musella
Keyword(s):  
Finite Group ◽  
Nilpotent Groups ◽  
Infinite Groups ◽  
Locally Finite ◽  
Finite Products ◽  
A Finite Group

A group G=AB is said to be totally factorized by its subgroups A and B if XY=YX for all subgroups X of A and Y of B. It is known that any finite group totally factorized by supersoluble subgroups is supersoluble, and that a finite group totally factorized by nilpotent subgroups is abelian-by-nilpotent. This latter result is extended here to certain classes of infinite groups.

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