scholarly journals Most finitely generated subgroups of infinite unitriangular matrices are free

2002 ◽  
Vol 66 (3) ◽  
pp. 419-423 ◽  
Author(s):  
Waldemar Hołubowski
Keyword(s):  
Finite Field ◽  
Free Groups ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Suitable Sense ◽  
Countable Subgroup ◽  
Almost All ◽  
Free Subgroups

In this note we prove that the group G of infinite dimensional upper unitriangular matrices over a finite field contains an explicit countable subgroup ‘full’ of free subgroups. We deduce from this fact that, in a suitable sense, almost all k–generator subgroups of G are free groups of rank k.

ON THE INTERSECTION OF FINITELY GENERATED SUBGROUPS IN FREE PRODUCTS OF GROUPS

1999 ◽  
Vol 09 (05) ◽  
pp. 521-528 ◽  
Author(s):  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Free Groups ◽  
Product Formula ◽  
Finitely Generated ◽  
Free Products ◽  
Products Of Groups ◽  
Free Subgroups

A subgroup H of a free product [Formula: see text] of groups Gα, α∈ I, is called factor free if for every [Formula: see text] and β ∈ I one has S H S-1∩ Gβ = {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free). If K is a finitely generated free group, denote [Formula: see text], where r(K) is the rank of K. It is proven that if H, K are finitely generated factor free subgroups of a free product [Formula: see text] then [Formula: see text]. It is also shown that the inequality [Formula: see text] of Hanna Neumann conjecture on subgroups of free groups does not hold for factor free subgroups of free products.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

scholarly journals On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

scholarly journals Isometric Group Actions on Hilbert Spaces: Structure of Orbits

2008 ◽  
Vol 60 (5) ◽  
pp. 1001-1009 ◽  
Author(s):  
Yves de Cornulier ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Nilpotent Group ◽  
Hilbert Spaces ◽  
Metabelian Group ◽  
Isometric Action ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Isometric Group Actions ◽  
Dense Orbits

AbstractOur main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

2010 ◽  
Vol 06 (03) ◽  
pp. 579-586 ◽  
Author(s):  
ARNO FEHM ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Galois Extension ◽  
Characteristic Zero ◽  
Rational Point ◽  
Abelian Varieties ◽  
Finitely Generated ◽  
Infinite Rank ◽  
Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

scholarly journals Generating Adjoint Groups

2019 ◽  
Vol 62 (3) ◽  
pp. 733-738 ◽  
Author(s):  
Be'eri Greenfeld
Keyword(s):  
Open Problem ◽  
Jacobson Radical ◽  
The Other ◽  
Adjoint Group ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Other Hand ◽  
Nil Ring

AbstractWe prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated. We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated and, on the other hand, construct a finitely generated, infinite-dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.

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 Finitely generated cyclic extensions of free groups are residually finite

1971 ◽  
Vol 5 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Free Group ◽  
Principal Ideal ◽  
Free Groups ◽  
Cyclic Extension ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Residually Finite ◽  
Ideal Domain ◽  
Direct Sum

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

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