Les automorphismes d'un ensemble fortement minimal

1992 ◽  
Vol 57 (1) ◽  
pp. 238-251 ◽  
Author(s):  
Daniel Lascar
Keyword(s):  
Normal Subgroup ◽  
Algebraic Closure ◽  
Finite Set ◽  
Saturated Structure ◽  
Minimal Formula

AbstractLet be a countable saturated structure, and assume that D(v) is a strongly minimal formula (without parameter) such that is the algebraic closure of D(). We will prove the two following theorems:Theorem 1. If G is a subgroup of Aut() of countable index, there exists a finite set A in such that every A-strong automorphism is in G.Theorem 2. Assume that G is a normal subgroup of Aut() containing an element g such that for all n there exists X ⊆ D() such that Dim(g(X)/X) > n. Then every strong automorphism is in G.

Almost strongly minimal theories. I

1972 ◽  
Vol 37 (3) ◽  
pp. 487-493 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Algebraic Closure ◽  
Natural Extension ◽  
Stone Space ◽  
First Order ◽  
Order Language ◽  
Simple Class ◽  
The Universe ◽  
Natural Expansion ◽  
Minimal Formula

In [1] the notions of strongly minimal formula and algebraic closure were applied to the study of ℵ1-categorical theories. In this paper we study a particularly simple class of ℵ1-categorical theories. We characterize this class in terms of the analysis of the Stone space of models of T given by Morley [3].We assume familiarity with [1] and [3], but for convenience we list the principal results and definitions from those papers which are used here. Our notation is the same as in [1] with the following exceptions.We deal with a countable first order language L. We may extend the language L in several ways. If is an L-structure, there is a natural extension of L obtained by adjoining to L a constant a for each (the universe of ). For each sentence A(a1, …, an) ∈ L(A) we say satisfies A(a1, …, an) and write if in Shoenfield's notation If is an L-structure and X is a subset of , then L(X) is the language obtained by adjoining to L a name x for each is the natural expansion of to an L(X)-structure. A structure is an inessential expansion [4, p. 141] of an L-structure if for some .

scholarly journals On the normal structure of a one-point stabilizer in a doubly transitive permutation group

1978 ◽  
Vol 25 (2) ◽  
pp. 145-166
Author(s):  
M. D. Atkinson ◽  
Cheryl E. Praeger
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Closed Subgroup ◽  
Normal Structure ◽  
Subject Classification ◽  
Transitive Permutation Group ◽  
Normal Extension ◽  
Extra Information ◽  
Finite Set ◽  
Point Stabilizer

Let G be a doubly transitive permutation group on a finite set Ω, and let Kα be a normal subgroup of the stabilizer Gα of a point α in Ω. If the action of Gα on the set of orbits of Kα in Ω − {α} is 2-primitive with kernel Kα it is shown that either G is a normal extension of PSL(3, q) or Kα ∩ Gγ is a strongly closed subgroup of Gαγ in Gα, where γ ∈ Ω − {α}. If in addition the action of Gα on the set of orbits of Kα is assumed to be 3-transitive, extra information is obtained using permutation theoretic and centralizer ring methods. In the case where Kα has three orbits in Ω − {α} strong restrictions are obtained on either the structure of G or the degrees of certain irreducible characters of G. Subject classification (Amer. Math. Soc. (MOS) 1970: 20 B 20, 20 B 25.

ON THE AUTOMORPHISM GROUP OF A FREE METABELIAN LIE ALGEBRA

2008 ◽  
Vol 18 (02) ◽  
pp. 209-226 ◽  
Author(s):  
VITALY ROMAN'KOV
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Lie Algebra ◽  
Free Subgroup ◽  
Finite Set ◽  
Free Metabelian Lie Algebra ◽  
Inner Automorphisms ◽  
Infinite Set ◽  
Wild Automorphisms

Let K be a field of any characteristic. We prove that a free metabelian Lie algebra M3 of rank 3 over K admits wild automorphisms. Moreover, the subgroup I Aut M3 of all automorphisms identical modulo the derived subalgebra [Formula: see text] cannot be generated by any finite set of IA-automorphisms together with the sets of all inner and all tame IA-automorphisms. In the case if K is finite the group Aut M3 cannot be generated by any finite set of automorphisms together with the sets of all tame, all inner automorphisms and all one-row automorphisms. We present an infinite set of wild IA-automorphisms of M3 which generates a free subgroup F∞ modulo normal subgroup generated by all tame, all inner and all one-row automorphisms of M3.

scholarly journals Solvable normal subgroups of 2-knot groups

2017 ◽  
Vol 26 (11) ◽  
pp. 1750066
Author(s):  
J. A. Hillman
Keyword(s):  
Normal Subgroup ◽  
Normal Subgroups ◽  
Locally Finite ◽  
Torsion Free ◽  
Minimal Formula

If [Formula: see text] is an orientable, strongly minimal [Formula: see text]-complex and [Formula: see text] has one end, then it has no nontrivial locally finite normal subgroup. Hence, if [Formula: see text] is a 2-knot group, then (a) if [Formula: see text] is virtually solvable, then either [Formula: see text] has two ends or [Formula: see text], with presentation [Formula: see text], or [Formula: see text] is torsion-free and polycyclic of Hirsch length 4 (b) either [Formula: see text] has two ends, or [Formula: see text] has one end and the center [Formula: see text] is torsion-free, or [Formula: see text] has infinitely many ends and [Formula: see text] is finite, and (c) the Hirsch–Plotkin radical [Formula: see text] is nilpotent.

A finite set of generators for the homeotopy group of a non-orientable surface

1969 ◽  
Vol 65 (2) ◽  
pp. 409-430 ◽  
Author(s):  
D. R. J. Chillingworth
Keyword(s):  
Normal Subgroup ◽  
Quotient Group ◽  
Closed Surface ◽  
Orientable Surface ◽  
Finite Set

Let X be a closed surface, i.e. a compact connected 2-manifold without boundary. If Gx denotes the group of all homeomorphisms of X to itself, and Nx is the normal subgroup consisting of homeomorphisms which are isotopic to the identity, then the quotient group Gx/Nx is called the homeotopy group of X and is denoted by ∧x.

scholarly journals Almost congruence extension property for subgroups of free groups

2018 ◽  
Vol 21 (1) ◽  
pp. 125-146
Author(s):  
Lev Glebsky ◽  
Nevarez Nieto Saul
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Free Groups ◽  
Extension Property ◽  
Normal Closure ◽  
Sufficient Condition ◽  
Congruence Extension Property ◽  
Finitely Generated ◽  
Finite Set

AbstractLetHbe a subgroup ofFand{\langle\kern-1.422638pt\langle H\rangle\kern-1.422638pt\rangle_{F}}the normal closure ofHinF. We say thatHhas the Almost Congruence Extension Property (ACEP) inFif there is a finite set of nontrivial elements{\digamma\subset H}such that for any normal subgroupNofHone has{H\cap\langle\kern-1.422638pt\langle N\rangle\kern-1.422638pt\rangle_{F}=N}whenever{N\cap\digamma=\emptyset}. In this paper, we provide a sufficient condition for a subgroup of a free group to not possess ACEP. It also shows that any finitely generated subgroup of a free group satisfies some generalization of ACEP.

scholarly journals A FINITENESS PROPERTY FOR PREPERIODIC POINTS OF CHEBYSHEV POLYNOMIALS

2010 ◽  
Vol 06 (05) ◽  
pp. 1011-1025 ◽  
Author(s):  
SU-ION IH ◽  
THOMAS J. TUCKER
Keyword(s):  
Number Field ◽  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Algebraic Closure ◽  
Finiteness Property ◽  
Finite Set

Let K be a number field with algebraic closure [Formula: see text], let S be a finite set of places of K containing the Archimedean places, and let φ be a Chebyshev polynomial. We prove that if [Formula: see text] is not preperiodic, then there are only finitely many preperiodic points [Formula: see text] which are S-integral with respect to α.

scholarly journals Integral division points on curves

2013 ◽  
Vol 149 (12) ◽  
pp. 2011-2035 ◽  
Author(s):  
David Grant ◽  
Su-Ion Ih
Keyword(s):  
Elliptic Curve ◽  
Number Field ◽  
Abelian Varieties ◽  
Algebraic Closure ◽  
Effective Divisor ◽  
Dimensional Case ◽  
Finitely Generated ◽  
General Conjecture ◽  
Finite Set ◽  
Set Of Points

AbstractLet $k$ be a number field with algebraic closure $ \overline{k} $, and let $S$ be a finite set of primes of $k$ containing all the infinite ones. Let $E/ k$ be an elliptic curve, ${\mit{\Gamma} }_{0} $ be a finitely generated subgroup of $E( \overline{k} )$, and $\mit{\Gamma} \subseteq E( \overline{k} )$ the division group attached to ${\mit{\Gamma} }_{0} $. Fix an effective divisor $D$ of $E$ with support containing either: (i) at least two points whose difference is not torsion; or (ii) at least one point not in $\mit{\Gamma} $. We prove that the set of ‘integral division points on $E( \overline{k} )$’, i.e., the set of points of $\mit{\Gamma} $ which are $S$-integral on $E$ relative to $D, $ is finite. We also prove the ${ \mathbb{G} }_{\mathrm{m} } $-analogue of this theorem, thereby establishing the 1-dimensional case of a general conjecture we pose on integral division points on semi-abelian varieties.

Modèles saturés et modèles engendrés par des indiscernables

2001 ◽  
Vol 66 (1) ◽  
pp. 325-348
Author(s):  
Benoît Mariou
Keyword(s):  
Algebraic Closure ◽  
Saturated Model ◽  
Stable Theory ◽  
Independence Property ◽  
Complete Answer ◽  
Unstable Case ◽  
Complete Theories ◽  
Recursively Saturated ◽  
Saturated Structure

AbstractIn the early eighties, answering a question of A. Macintyre, J. H. Schmerl ([13]) proved that every countable recursively saturated structure, equipped with a function β encoding the finite functions, is the β-closure of an infinite indiscernible sequence. This result implies that every countably saturated structure, in a countable but not necessarily recursive language, is an Ehrenfeucht-Mostowski model, by which we mean that the structure expands, in a countable language, to the Skolem hull of an infinite indiscernible sequence (in the new language).More recently, D. Lascar ([5]) showed that the saturated model of cardinality ℵ1 of an ω-stable theory is also an Ehrenfeucht-Mostowski model.These results naturally raise the following problem: which (countable) complete theories have an uncountably saturated Ehrenfeucht-Mostowski model. We study a generalization of this question. Namely, we call ACI-model a structure which can be expanded, in a countable language L′, to the algebraic closure (in L′) of an infinite indiscernible sequence (in L′). And we try to characterize the λ-saturated structures which are ACI-models.The main results are the following. First it is enough to restrict ourselves to ℵ1-saturated structures: if T has an ℵ1-saturated ACI-model then, for every infinite λ, T has a λ-saturated ACI-model. We obtain a complete answer in the case of stable theories: if T is stable then the three following properties are equivalent: (a) T is ω-stable, (b) T has an ℵ1-saturated ACI-model, (c) every saturated model of T is an Ehrenfeucht-Mostowski model. The unstable case is more complicated, however we show that if T has an ℵ1-saturated ACI-model then T doesn't have the independence property.

A note on subgroups of the automorphism group of a saturated model, and regular types

1989 ◽  
Vol 54 (3) ◽  
pp. 858-864 ◽  
Author(s):  
A. Pillay
Keyword(s):  
Automorphism Group ◽  
Algebraic Closure ◽  
Saturated Model ◽  
Superstable Theory ◽  
Alternative Characterization ◽  
Finite Set ◽  
Superstable Theories

AbstractLet M be a saturated model of a superstable theory and let G = Aut(M). We study subgroups H of G which contain G(A), A the algebraic closure of a finite set, generalizing results of Lascar [L] as well as giving an alternative characterization of the simple superstable theories of [P]. We also make some observations about good, locally modular regular types p in the context of p-simple types.

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