scholarly journals On the orbit-stabilizer problem for integral matrix actions of polycyclic groups

2003 ◽  
Vol 72 (243) ◽  
pp. 1511-1530 ◽  
Author(s):  
Bettina Eick ◽  
Gretchen Ostheimer
Keyword(s):  
Integral Matrix ◽  
Polycyclic Groups

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?

On division rings generated by polycyclic groups

1984 ◽  
Vol 47 (2-3) ◽  
pp. 154-164 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Division Rings ◽  
Polycyclic Groups

On some polycyclic groups with small Hirsch length

2017 ◽  
Vol 16 (12) ◽  
pp. 1750237
Author(s):  
Heguo Liu ◽  
Fang Zhou ◽  
Tao Xu
Keyword(s):  
Abelian Subgroup ◽  
Polycyclic Group ◽  
Normal Abelian Subgroup ◽  
Finite Quotient ◽  
Polycyclic Groups

A polycyclic group [Formula: see text] is called an [Formula: see text]-group if every normal abelian subgroup of any finite quotient of [Formula: see text] is generated by [Formula: see text], or fewer, elements and [Formula: see text] is the least integer with this property. In this paper, the structure of [Formula: see text]-groups and [Formula: see text]-groups is determined.

Right-Ordered Polycyclic Groups

1974 ◽  
Vol 17 (2) ◽  
pp. 175-178 ◽  
Author(s):  
Roberta Botto Mura
Keyword(s):  
Additive Group ◽  
Real Numbers ◽  
Ordered Groups ◽  
Polycyclic Groups

One of the features that make right-ordered groups harder to investigate than ordered groups is that their system of convex subgroups may fail to have the following property:(*) if C and C’ are convex subgroups of G and C’ covers C, then C is normal in C’ and C’/C is order-isomorphic to a subgroup of the naturally ordered additive group of real numbers.

scholarly journals The graded Cartan matrix and global dimension of 0-relations algebras

1987 ◽  
Vol 30 (3) ◽  
pp. 351-362 ◽  
Author(s):  
W. D. Burgess
Keyword(s):  
Artinian Ring ◽  
Global Dimension ◽  
Cartan Matrix ◽  
Integral Matrix ◽  
Finite Global Dimension ◽  
Composition Factors

The Cartan matrix C of a left artinian ring A, with indecomposable projectives P1,…,Pn and corresponding simples Si=Pi/JPi, is an n×n integral matrix with entries Cij, the number of copies of the simple sj which appear as composition factors of Pi. A relationship between the invertibility of this matrix (as an integral matrix) and the finiteness of the global dimension has long been known: gl dim A < ∞⇒det C = ± 1 (Eilenberg [3]). More recently Zacharia [9] has shown that gl dim A ≦ 2⇒det C = 1, and in fact no rings of finite global dimension are known with det C = −1. The converse, det C = l⇒gl dim A < ∞, is false, as easy examples show ([[1) or [3]). However if A is left serial, gl dim A < ∞iff det C = l [1]. If A = ⊕n ≧ 0 An is ℤ-graded and the radical J = ⊕n ≧ 0 An, Wilson [8] calls such rings positively graded. Here there is a graded Cartan matrix with entries from ℤ[X] and gl dim A < ∞⇒det = 1 and, hence, det C = l [8, Prop. 2.2].

scholarly journals On a theorem of Rhemtulla on polycyclic groups

2010 ◽  
Vol 214 (10) ◽  
pp. 1898-1900
Author(s):  
B.A.F. Wehrfritz
Keyword(s):  
Polycyclic Groups
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