FAITHFUL REPRESENTATIONS OF FINITELY GENERATED ABELIAN-BY-POLYCYCLIC GROUPS OVER DIVISION RINGS

1984 ◽  
Vol 35 (3) ◽  
pp. 361-372 ◽  
Author(s):  
B. A. F. WEHRFRITZ
Keyword(s):  
Finitely Generated ◽  
Division Rings ◽  
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

Division Rings Associated with Polycyclic Groups

1984 ◽  
Vol s2-30 (3) ◽  
pp. 465-467
Author(s):  
K. A. Brown ◽  
B. A. F. Wehrfritz
Keyword(s):  
Division Rings ◽  
Polycyclic Groups

scholarly journals Markov semigroups, monoids and groups

2014 ◽  
Vol 24 (05) ◽  
pp. 609-653 ◽  
Author(s):  
Alan J. Cain ◽  
Victor Maltcev
Keyword(s):  
Finite Index ◽  
Regular Language ◽  
Minimal Length ◽  
Direct Products ◽  
Markov Semigroups ◽  
Finitely Generated ◽  
Free Products ◽  
Commutative Semigroups ◽  
Generating Set ◽  
Polycyclic Groups

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.

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.

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.

scholarly journals LINEAR DIVISION RINGS

2009 ◽  
Vol 12 (17) ◽  
pp. 5-11
Author(s):  
Bien Hoang Mai ◽  
Hai Xuan Bui
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Finite Subset ◽  
Multiplicative Group ◽  
Dimensional Vector ◽  
Finitely Generated ◽  
Dimensional Vector Space ◽  
Division Rings ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Let D be a division ring with the center F and suppose that D* is the multiplicative group of D. D is called centrally finite if D is a finite dimensional vector space over F and D is locally centrally finite if every finite subset of D generates over F a division subring which is a finite dimensional vector space over F. We say that D is a linear division ring if every finite subset of D generates over Fa centrally finite division subring. It is obvious that every locally centrally finite division ring is linear. In this report we show that the inverse is not true by giving an example of a linear division ring which is not locally centrally finite. Further, we give some properties of subgroups in linear division rings. In particular, we show that every finitely generated subnormal subgroup in a linear ring is central. An interesting corollary is obtained as the following: If D is a linear division ring and D* is finitely generated, then D is a finite field.

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