Lie Nilpotent Group Algebras and Upper Lie Codimension Subgroups

2006 ◽  
Vol 34 (10) ◽  
pp. 3859-3873 ◽  
Author(s):  
Francesco Catino ◽  
Ernesto Spinelli
Keyword(s):  
Nilpotent Group ◽  
Group Algebras

Strongly Lie nilpotent group algebras of index at most 8

2014 ◽  
Vol 13 (07) ◽  
pp. 1450044 ◽  
Author(s):  
Harish Chandra ◽  
Meena Sahai
Keyword(s):  
Nilpotent Group ◽  
Group Algebras ◽  
Characteristic P ◽  
Arbitrary Group

Let K be a field of characteristic p > 0 and let G be an arbitrary group. In this paper, we classify group algebras KG which are strongly Lie nilpotent of index at most 8. We also show that for k ≤ 6, KG is strongly Lie nilpotent of index k if and only if it is Lie nilpotent of index k.

Irreducible Modules for Polycyclic Group Algebras

1981 ◽  
Vol 33 (4) ◽  
pp. 901-914 ◽  
Author(s):  
I. M. Musson
Keyword(s):  
Nilpotent Group ◽  
Irreducible Module ◽  
Polycyclic Group ◽  
Group Algebras ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules

If G is a polycyclic group and k an absolute field then every irreducible kG-module is finite dimensional [10], while if k is nonabsolute every irreducible module is finite dimensional if and only if G is abelian-by-finite [3]. However something more can be said about the infinite dimensional irreducible modules. For example P. Hall showed that if G is a finitely generated nilpotent group and V an irreducible kG-module, then the image of kZ in EndkGV is algebraic over k [3]. Here Z = Z(G) denotes the centre of G. It follows that the restriction Vz of V to Z is generated by finite dimensional kZ-modules. In this paper we prove a generalization of this result to polycyclic group algebras.We introduce some terminology.

A note on Lie nilpotent group algebras

2019 ◽  
Vol 18 (09) ◽  
pp. 1950163
Author(s):  
Meena Sahai ◽  
Bhagwat Sharan
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Group Algebra ◽  
Upper Bound ◽  
Finite Abelian Group ◽  
Index Formula ◽  
Group Algebras ◽  
Arbitrary Group ◽  
Nilpotency Index

Let [Formula: see text] be an arbitrary group and let [Formula: see text] be a field of characteristic [Formula: see text]. In this paper, we give some improvements of the upper bound of the lower Lie nilpotency index [Formula: see text] of the group algebra [Formula: see text]. We also give improved bounds for [Formula: see text], where [Formula: see text] is the number of independent generators of the finite abelian group [Formula: see text]. Furthermore, we give a description of the Lie nilpotent group algebra [Formula: see text] with [Formula: see text] or [Formula: see text]. We also show that for [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] is the upper Lie nilpotency index of [Formula: see text].

scholarly journals Free Symmetric and Unitary Pairs in the Field of Fractions of Torsion-Free Nilpotent Group Algebras

2019 ◽  
Vol 23 (3) ◽  
pp. 605-619
Author(s):  
Vitor O. Ferreira ◽  
Jairo Z. Gonçalves ◽  
Javier Sánchez
Keyword(s):  
Nilpotent Group ◽  
Group Algebras ◽  
Free Nilpotent Group ◽  
Torsion Free

Group algebras of Lie nilpotency index 14

2019 ◽  
Vol 13 (05) ◽  
pp. 2050088
Author(s):  
Suchi Bhatt ◽  
Harish Chandra ◽  
Meena Sahai
Keyword(s):  
Nilpotent Group ◽  
Group Algebras ◽  
Nilpotency Index

Let [Formula: see text] be a group and let [Formula: see text] be a field of characteristic [Formula: see text]. Lie nilpotent group algebras of strong Lie nilpotency index at most 13 have been classified by many authors. In this paper, our aim is to classify the group algebras [Formula: see text] which are strongly Lie nilpotent of index 14.

scholarly journals On the transcendence degree of group algebras of nilpotent groups

1984 ◽  
Vol 25 (2) ◽  
pp. 167-174 ◽  
Author(s):  
Martin Lorenz
Keyword(s):  
Nilpotent Group ◽  
Division Ring ◽  
Nilpotent Groups ◽  
Transcendence Degree ◽  
Group Algebras ◽  
Division Algebras ◽  
Free Nilpotent Group ◽  
Noetherian Domain ◽  
Ring Of Fractions ◽  
Torsion Free

Let G be a finitely generated (f.g.) torsion-free nilpotent group. Then the group algebra k[G] of G over a field k is a Noetherian domain and hence has a classical division ring of fractions, denoted by k(G). Recently, the division algebras k(G) and, somewhat more generally, division algebras generated by f.g. nilpotent groups have been studied in [3] and [5]. These papers are concerned with the question to what extent the division algebra determines the group under consideration. Here we continue the study of the division algebras k(G) and investigate their Gelfand–Kirillov (GK–) transcendence degree.

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?

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