Solvability of the membership problem in finitely generated solvable matrix groups over number fields

1971 ◽  
Vol 10 (2) ◽  
pp. 108-116 ◽  
Author(s):  
V. M. Kopytov
Keyword(s):  
Number Fields ◽  
Membership Problem ◽  
Finitely Generated ◽  
Matrix Groups

scholarly journals Effective results for unit points on curves over finitely generated domains

2015 ◽  
Vol 158 (2) ◽  
pp. 331-353
Author(s):  
ATTILA BÉRCZES
Keyword(s):  
Algebraic Closure ◽  
Unit Group ◽  
Number Fields ◽  
Finitely Generated ◽  
Quotient Field ◽  
Effective Version ◽  
Formula Group ◽  
Commutative Domain ◽  
Image Position ◽  
Unit Equations

AbstractLet A be a commutative domain of characteristic 0 which is finitely generated over ℤ as a ℤ-algebra. Denote by A* the unit group of A and by K the algebraic closure of the quotient field K of A. We shall prove effective finiteness results for the elements of the set \begin{equation*} \mathcal{C}:=\{ (x,y)\in (A^*)^2 | F(x,y)=0 \} \end{equation*} where F(X, Y) is a non-constant polynomial with coefficients in A which is not divisible over K by any polynomial of the form XmYn - α or Xm - α Yn, with m, n ∈ ℤ⩾0, max(m, n) > 0, α ∈ K*. This result is a common generalisation of effective results of Evertse and Győry [12] on S-unit equations over finitely generated domains, of Bombieri and Gubler [5] on the equation F(x, y) = 0 over S-units of number fields, and it is an effective version of Lang's general but ineffective theorem [20] on this equation over finitely generated domains. The conditions that A is finitely generated and F is not divisible by any polynomial of the above type are essentially necessary.

scholarly journals A FAST ALGORITHM FOR STALLINGS' FOLDING PROCESS

2006 ◽  
Vol 16 (06) ◽  
pp. 1031-1045 ◽  
Author(s):  
NICHOLAS W. M. TOUIKAN
Keyword(s):  
Free Group ◽  
Fast Algorithm ◽  
Free Groups ◽  
Membership Problem ◽  
Finitely Generated ◽  
Folding Process ◽  
Algorithmic Problems ◽  
The Given

Stalling's folding process is a key algorithm for solving algorithmic problems for finitely generated subgroups of free groups. Given a subgroup H = 〈J1,…,Jm〉 of a finitely generated nonabelian free group F = F(x1,…,xn) the folding porcess enables one, for example, to solve the membership problem or compute the index [F : H]. We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stallings' folding process in time O(N log *(N)), where N is the sum of the word lengths of the given generators of H.

scholarly journals Finite presentability of arithmetic groups over global function fields

1987 ◽  
Vol 30 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Helmut Behr
Keyword(s):  
Small Groups ◽  
Function Fields ◽  
Algebraic Groups ◽  
Number Fields ◽  
Arithmetic Groups ◽  
Finitely Generated ◽  
Global Function ◽  
Global Function Fields ◽  
Finite Presentability ◽  
Finitely Presented

Arithmetic subgroups of reductive algebraic groups over number fields are finitely presentable, but over global function fields this is not always true. All known exceptions are “small” groups, which means that either the rank of the algebraic group or the set S of the underlying S-arithmetic ring has to be small. There exists now a complete list of all such groups which are not finitely generated, whereas we onlyhave a conjecture which groups are finitely generated but not finitely presented.

NILPOTENT AND LOCALLY FINITE MAXIMAL SUBGROUPS OF SKEW LINEAR GROUPS

2011 ◽  
Vol 10 (04) ◽  
pp. 615-622 ◽  
Author(s):  
M. RAMEZAN-NASSAB ◽  
D. KIANI
Keyword(s):  
Maximal Subgroup ◽  
Natural Number ◽  
Division Ring ◽  
Subnormal Subgroup ◽  
Maximal Subgroups ◽  
Linear Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Matrix Groups ◽  
Nilpotent Maximal Subgroup

Let D be a division ring and N be a subnormal subgroup of D*. In this paper we prove that if M is a nilpotent maximal subgroup of N, then M′ is abelian. If, furthermore every element of M is algebraic over Z(D) and M′ ⊈ F* or M/Z(M) or M′ is finitely generated, then M is abelian. The second main result of this paper concerns the subgroups of matrix groups; assume D is a noncommutative division ring, n is a natural number, N is a subnormal subgroup of GLn(D), and M is a maximal subgroup of N. We show that if M is locally finite over Z(D)*, then M is either absolutely irreducible or abelian.

scholarly journals GENERICITY OF FILLING ELEMENTS

2012 ◽  
Vol 22 (02) ◽  
pp. 1250008 ◽  
Author(s):  
BRENT B. SOLIE
Keyword(s):  
Free Group ◽  
Time Complexity ◽  
Linear Time ◽  
Sufficient Condition ◽  
Membership Problem ◽  
Isometric Action ◽  
Finitely Generated ◽  
Generic Subset ◽  
Translation Length

An element of a finitely generated non-Abelian free group F(X) is said to be filling if that element has positive translation length in every very small minimal isometric action of F(X) on an ℝ-tree. We give a proof that the set of filling elements of F(X) is exponentially F(X)-generic in the sense of Arzhantseva and Ol'shanskiı. We also provide an algebraic sufficient condition for an element to be filling and show that there exists an exponentially F(X)-generic subset consisting only of filling elements and whose membership problem has linear time complexity.

scholarly journals An embedding theorem for fields

1976 ◽  
Vol 14 (2) ◽  
pp. 193-198 ◽  
Author(s):  
J.W.S. Cassels
Keyword(s):  
Simple Proof ◽  
Embedding Theorem ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Matrix Groups ◽  
Recurrent Sequences ◽  
Finite Set ◽  
Torsion Free

It is shown that every finitely generated field K of characteristic 0 may be embedded in infinitely many p-adic fields in such a way that the images of any given finite set C of non-zero elements of K are p-adic units. The result is suggested by Lech's proof of his generalization of Mahler's theorem on recurrent sequences. It also provides a simple proof of Selberg's theorem about torsion-free normal subgroups of matrix groups.

scholarly journals Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces

2017 ◽  
Vol 29 (3) ◽  
Author(s):  
Alexander Stasinski ◽  
Christopher Voll
Keyword(s):  
Zeta Functions ◽  
Nilpotent Groups ◽  
Number Fields ◽  
Vector Spaces ◽  
Unipotent Group ◽  
Finitely Generated ◽  
Group Schemes ◽  
Prehomogeneous Vector Spaces ◽  
Rings Of Integers

AbstractWe compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are modelled on certain prehomogeneous vector spaces. Our method is based on evaluating

scholarly journals On the generators of S-unit groups in algebraic number fields

1991 ◽  
Vol 43 (2) ◽  
pp. 325-329 ◽  
Author(s):  
B. Brindza
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Number Fields ◽  
Finitely Generated ◽  
Geometry Of Numbers ◽  
Algebraic Number Fields ◽  
Minimal Set ◽  
Simple Argument ◽  
Small Height ◽  
Multiplicative Subgroup

Given a finitely generated multiplicative subgroup Us in a number field, we employ a simple argument from the geometry of numbers and an inequality on multiplicative dependence in number fields to obtain a minimal set of generators consisting of elements of relatively small height.

scholarly journals On Deciding Finiteness for Matrix Groups over Fields of Positive Characteristic

2001 ◽  
Vol 4 ◽  
pp. 64-72 ◽  
Author(s):  
A. Detinko
Keyword(s):  
Finite Field ◽  
Positive Characteristic ◽  
Deterministic Algorithm ◽  
Transcendence Degree ◽  
Matrix Group ◽  
Finitely Generated ◽  
Matrix Groups ◽  
Field Of Positive Characteristic

AbstractThe author considers the development of algorithms for deciding whether a finitely generated matrix group over a field of positive characteristic is finite. A deterministic algorithm for deciding the finiteness is presented for the case of a field of transcendence degree one over a finite field.

scholarly journals Testing polycyclicity of finitely generated rational matrix groups

2007 ◽  
Vol 76 (259) ◽  
pp. 1669-1683 ◽  
Author(s):  
Björn Assmann ◽  
Bettina Eick
Keyword(s):  
Rational Matrix ◽  
Finitely Generated ◽  
Matrix Groups
Sign in / Sign up

Export Citation Format

Share Document