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

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.

The Group Ring Of a Class Of Infinite Nilpotent Groups

1955 ◽  
Vol 7 ◽  
pp. 169-187 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Nilpotent Group ◽  
Group Ring ◽  
Characteristic Zero ◽  
Discrete Group ◽  
Nilpotent Groups ◽  
Zero Element ◽  
Free Nilpotent Group ◽  
Image Position ◽  
Torsion Free ◽  
The Ideal

Introduction. In this paper we study the (discrete) group ring Γ of a finitely generated torsion free nilpotent group over a field of characteristic zero. We show that if Δ is the ideal of Γ spanned by all elements of the form G − 1, where G ∈ , thenand the only element belonging to Δw for all w is the zero element (cf. (4.3) below).

An independence theorem for automorphisms of torsion-free groups

1981 ◽  
Vol 90 (3) ◽  
pp. 403-409
Author(s):  
U. H. M. Webb
Keyword(s):  
Automorphism Group ◽  
Nilpotent Group ◽  
Free Group ◽  
Automorphism Groups ◽  
Torsion Free Group ◽  
Free Nilpotent Group ◽  
Free Abelian Groups ◽  
Independence Theorem ◽  
The Relationship ◽  
Torsion Free

This paper considers the relationship between the automorphism group of a torsion-free nilpotent group and the automorphism groups of its subgroups and factor groups. If G2 is the derived group of the group G let Aut (G, G2) be the group of automorphisms of G which induce the identity on G/G2, and if B is a subgroup of Aut G let B¯ be the image of B in Aut G/Aut (G, G2). A p–group or torsion-free group G is said to be special if G2 coincides with Z(G), the centre of G, and G/G2 and G2 are both elementary abelian p–groups or free abelian groups.

scholarly journals On nilpotent and polycyclic groups

1989 ◽  
Vol 40 (1) ◽  
pp. 119-122
Author(s):  
Robert J. Hursey
Keyword(s):  
Nilpotent Group ◽  
Central Series ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Polycyclic Groups ◽  
Torsion Free

A group G is torsion-free, finitely generated, and nilpotent if and only if G is a supersolvable R-group. An ordered polycylic group G is nilpotent if and only if there exists an order on G with respect to which the number of convex subgroups is one more than the length of G. If the factors of the upper central series of a torsion-free nilpotent group G are locally cyclic, then consecutive terms of the series are jumps, and the terms are absolutely convex subgroups.

Generalized partition functions and subgroup growth of free products of nilpotent groups

Analysis ◽  
2005 ◽  
Vol 25 (4) ◽  
Author(s):  
Thomas W. Müller ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Partition Functions ◽  
Free Products ◽  
Subgroup Growth ◽  
Free Nilpotent Group ◽  
Generalized Partition ◽  
Torsion Free

SummaryWe estimate the growth of homomorphism numbers of a torsion-free nilpotent group

SUBSPACES, FREE SUBGROUPS AND A THEOREM OF STALLINGS

2006 ◽  
Vol 16 (03) ◽  
pp. 475-491
Author(s):  
GILBERT BAUMSLAG
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Nilpotent Group ◽  
Homology Group ◽  
Integral Homology ◽  
Free Nilpotent Group ◽  
Theoretic Proof ◽  
Analogous Formula ◽  
Free Subgroups ◽  
Torsion Free

There is a simple group-theoretic formula for the second integral homology group of a group. This is an abelian group and there is an analogous formula for another abelian group, which involves a normal subgroup N of a torsion-free nilpotent group G. Properties of this abelian group translate into properties of G/N. This approach allows one to give a simple purely group-theoretic proof of an old theorem of J. R. Stallings, namely that if Γ is a group, if H1(G,ℤ) is free abelian and if H2(G,ℤ) = 0, then any subset Y of G which is independent modulo the derived group of G, freely generates a free group. The ideas used admit to considerable generalization, yielding in particular, proofs of a number of theorems of U. Stammbach.

scholarly journals Free Group Algebras in Division Rings with Valuation II

2019 ◽  
Vol 72 (6) ◽  
pp. 1463-1504
Author(s):  
Javier Sánchez
Keyword(s):  
Free Group ◽  
Group Algebra ◽  
Neumann Series ◽  
Group Algebras ◽  
Series Ring ◽  
Free Nilpotent Group ◽  
Enveloping Algebra ◽  
Division Rings ◽  
Ore Domain ◽  
Torsion Free

AbstractWe apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings.If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_{L}$ that contains $U(L)$. We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_{L}$ generated by $U(L)$.Let $k$ be a field of characteristic zero, and let $L$ be a nonabelian Lie $k$-algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain, we show that $\mathfrak{D}(L)$ contains (noncommutative) free group algebras. In those same cases, if $L$ is equipped with an involution, we are able to prove that the free group algebra in $\mathfrak{D}(L)$ can be chosen generated by symmetric elements in most cases.Let $G$ be a nonabelian residually torsion-free nilpotent group, and let $k(G)$ be the division subring of the Malcev–Neumann series ring generated by the group algebra $k[G]$. If $G$ is equipped with an involution, we show that $k(G)$ contains a (noncommutative) free group algebra generated by symmetric elements.

Sign in / Sign up

Export Citation Format

Share Document