Groups whose irreducible representations have finite degree

1981 ◽  
Vol 90 (3) ◽  
pp. 411-421 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Positive Solution ◽  
Finite Dimension ◽  
Systematic Investigation ◽  
Polycyclic Group ◽  
Irreducible Representations ◽  
Finitely Generated ◽  
Finite Degree ◽  
Commutative Field ◽  
Locally Finite

Throughout this paper F denotes a (commutative) field. Let XF denote the class of all groups G such that every irreducible FC-module has finite dimension over F. In (1) P. Hall showed that if F is not locally finite and if G is polycyclic, then G∈XF if and only if G is abelian-by-finite. Also in (1), if F is locally finite he proved that every finitely generated nilpotent group is in XF and he conjectured that XF should contain every polycyclic group. This turned out to be very difficult, but a positive solution was eventually found by Roseblade, see (8). Meanwhile Levič in (4) had started a systematic investigation of the classes XF. Although his paper contains a number of errors, obscurities and omissions, it remains an interesting work, and it, or more accurately its recent translation, stimulated this present paper.

scholarly journals Groups whose irreducible representations have finite degree II

1982 ◽  
Vol 25 (3) ◽  
pp. 237-243 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Characteristic Zero ◽  
Finite Dimension ◽  
Irreducible Representations ◽  
Finitely Generated ◽  
Finite Degree ◽  
Commutative Field ◽  
Soluble Groups

If F is a (commutative) field let denote the class of all groups G such that every irreducible FG-module has finite dimension over F. The introduction to [7] contains motivation for considering these classes and surveys some of the results to date concerning them. In [7] for every field F we determined the finitely generated soluble groups in . Here, for fields F of characteristic zero, we determine, at least in principle, the soluble groups in . Our main result is the following.

Groups whose irreducible representations have finite degree: III

1982 ◽  
Vol 91 (3) ◽  
pp. 397-406 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Characteristic Zero ◽  
Positive Characteristic ◽  
Finite Dimension ◽  
Irreducible Representations ◽  
Finitely Generated ◽  
Finite Degree ◽  
Commutative Field

If F is a (commutative) field let XF denote the class of all groups G such that every irreducible FG module has finite dimension over F. In the first paper (9) of this series we classified finitely generated soluble XF-groups for each field F and in the second (10) we characterized soluble XF-groups for each field F of characteristic zero. Here we consider soluble XF-groups over fields F of positive characteristic.

scholarly journals Finitely generated nilpotent group C*-algebras have finite nuclear dimension

2018 ◽  
Vol 2018 (738) ◽  
pp. 281-298 ◽  
Author(s):  
Caleb Eckhardt ◽  
Paul McKenney
Keyword(s):  
Irreducible Representation ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Irreducible Representations ◽  
Finitely Generated ◽  
C Algebra ◽  
C Algebras ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
Torsion Free

Abstract We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its ordered K-theory and is approximately subhomogeneous. We observe that all C*-algebras generated by faithful irreducible representations of finitely generated, torsion free nilpotent groups satisfy the universal coefficient theorem.

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.

Primitive irreducible representations of nilpotent groups

1977 ◽  
Vol 82 (2) ◽  
pp. 241-247 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Irreducible Representations ◽  
Finitely Generated

Here we answer a question raised by Zaleskii (5) and reiterated by Segal (3) of whether a finitely generated nilpotent group can have primitive irreducible representations.

BREADTH IN POLYCYCLIC GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1073-1083 ◽  
Author(s):  
AVINOAM MANN ◽  
DAN SEGAL
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Nilpotent Group ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Finite Group Theory ◽  
Polycyclic Groups

The breadth of a polycyclic group is the maximum of h(G) - h(CG(x)) for x ∈ G, where h(G) is the Hirsch length. We prove a number of results that bound the class of a finitely generated nilpotent group, and the Hirsch length of the derived group in a polycyclic group, in terms of the breadth. These results are analogues of well-known results in finite group theory.

Locally finite groups and configurations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mahdi Meisami ◽  
Ali Rejali ◽  
Meisam Soleimani Malekan ◽  
Akram Yousofzadeh
Keyword(s):  
Finite Group ◽  
Positive Solution ◽  
Finite Groups ◽  
Discrete Group ◽  
Locally Finite Group ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Normalized Solution

Abstract Let 𝐺 be a discrete group. In 2001, Rosenblatt and Willis proved that 𝐺 is amenable if and only if every possible system of configuration equations admits a normalized solution. In this paper, we show independently that 𝐺 is locally finite if and only if every possible system of configuration equations admits a strictly positive solution. Also, we give a procedure to get equidecomposable subsets 𝐴 and 𝐵 of an infinite finitely generated or a locally finite group 𝐺 such that A ⊊ B A\subsetneq B , directly from a system of configuration equations not having a strictly positive solution.

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.

Characterization of finitely generated groups by types

2018 ◽  
Vol 28 (08) ◽  
pp. 1613-1632 ◽  
Author(s):  
A. G. Myasnikov ◽  
N. S. Romanovskii
Keyword(s):  
Free Surface ◽  
Nilpotent Group ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Burnside Groups ◽  
Text Types ◽  
Rigid Group

In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups [Formula: see text] are fully characterized in the class of all groups by the set [Formula: see text] of types realized in [Formula: see text]. Furthermore, it turns out that these groups [Formula: see text] are fully characterized already by some particular rather restricted fragments of the types from [Formula: see text]. In particular, every finitely generated nilpotent group is completely defined by its [Formula: see text]-types, while a finitely generated rigid group is completely defined by its [Formula: see text]-types, and a finitely generated metabelian or polycyclic group is completely defined by its [Formula: see text]-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.

Irreducible representations of finitely generated nilpotent groups

1977 ◽  
Vol 81 (2) ◽  
pp. 201-208 ◽  
Author(s):  
Daniel Segal
Keyword(s):  
Irreducible Representation ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Irreducible Representations ◽  
Closed Field ◽  
Finitely Generated ◽  
Dimensional Representation ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation

1. Introduction. It is well known that every finite-dimensional irreducible representation of a nilpotent group over an algebraically closed field is monomial, that is induced from a 1-dimensional representation of some subgroup. However, even a finitely generated nilpotent group in general has infinite-dimensional irreducible representations, and as a first step towards an understanding of these one wants to discover whether they too are necessarily monomial. The main point of this note is to show how far they can fail to be so.

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