Finite non-cyclic nilpotent group whose number of subgroups is minimal

2021 ◽  
Author(s):  
Wei Meng ◽  
Jiakuan Lu
Keyword(s):  
Nilpotent Group

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

A note on Lie nilpotent group rings

1992 ◽  
Vol 45 (3) ◽  
pp. 503-506 ◽  
Author(s):  
R.K. Sharma ◽  
Vikas Bist
Keyword(s):  
Nilpotent Group ◽  
Group Algebra ◽  
Group Rings ◽  
Group Of Units

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.

Primitivity in representations of polycyclic groups

1980 ◽  
Vol 88 (1) ◽  
pp. 15-31 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Nilpotent Group ◽  
Infinite Dimension ◽  
Finitely Generated ◽  
Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

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.

scholarly journals EQUIVARIANT COMPACTIFICATIONS OF A NILPOTENT GROUP BY G/P

2016 ◽  
Vol 22 (1) ◽  
pp. 163-186
Author(s):  
DAEWOONG CHEONG
Keyword(s):  
Nilpotent Group

A note on FC-nilpotency

2018 ◽  
Vol 17 (09) ◽  
pp. 1850165 ◽  
Author(s):  
Nadja Hempel ◽  
Daniel Palacín
Keyword(s):  
Nilpotent Group

The notion of bounded FC-nilpotent group is introduced and it is shown that any such group is nilpotent-by-finite, generalizing a result of Neumann on bounded FC-groups.

Stability of nilpotent groups of class 2 and prime exponent

1981 ◽  
Vol 46 (4) ◽  
pp. 781-788 ◽  
Author(s):  
Alan H. Mekler
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Stability Class ◽  
Similarity Type ◽  
Nilpotent Class ◽  
Prime Exponent

AbstractLet p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M.Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes.Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class.

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