Quasi-finitely axiomatizable nilpotent groups

Oger, Francis; Sabbagh, Gabriel

doi:10.1515/jgt.2006.005

Cited by 18 publications

(20 citation statements)

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“…In particular, one obtains an alternative proof that no abelian group is QFA, because, if G is infinite f.g. abelian, then Z(G) = G while ∆(G) is the finite torsion subgroup. The argument in [27] (for this direction) extends the one given above for abelian groups, by introducing ultraproducts.…”

Section: Nilpotent Groups Oger and Sabbaghmentioning

confidence: 87%

“…5.1] using logic, in particular the Mal'cev interpretation of (N, +, ×) in UT 3 (Z) already mentioned when we discussed the proof of Theorem 4.3. In comparison, the characterization in [27] leads further. For instance, it also shows that all non-abelian upper triangular groups over Z, and all free nilpotent non-abelian groups are QFA.…”

Section: Nilpotent Groups Oger and Sabbaghmentioning

confidence: 93%

“…Thus the properties of being QFA and being FA-presentable are incompatible (for f.g. infinite groups, the only relevant case), by the result in [28] that each finitely generated FApresentable group is abelian-by-finite. [27] characterized the property of being QFA for nilpotent groups, in a purely algebraic way. Informally speaking, G is QFA iff the center Z(G) is not too large, in a sense to be specified.…”

Section: Theorem 47 ([22]mentioning

confidence: 99%

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Describing Groups

Nies¹

2007

Bull. symb. log.

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Two ways of describing a group are considered. 1. A group is finite-automaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely axiomatizable if there is a description consisting of a single first-order sentence, together with the information that the group is finitely generated. In the first part of the paper we survey examples of FA-presentable groups, but also discuss theorems restricting this class. In the second part, we give examples of quasi-finitely axiomatizable groups, consider the algebraic content of the notion, and compare it to the notion of a group which is a prime model. We also show that if a structure is bi-interpretable in parameters with the ring of integers, then it is prime and quasi-finitely axiomatizable.

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Section: Nilpotent Groups Oger and Sabbaghmentioning

confidence: 87%

Section: Nilpotent Groups Oger and Sabbaghmentioning

confidence: 93%

Section: Theorem 47 ([22]mentioning

confidence: 99%

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Describing Groups

Nies¹

2007

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“…A. Nies [Nie03b] has proved that the free nilpotent group of class 2 with two generators is QFA prime. F. Oger and G. Sabbagh show that finitely generated nonabelian free nilpotent groups are QFA and prime [OS06]. It is proved in [Nie07] the existence of countinousely many non-isomorphic finitely generated prime groups, which implies that there exists a finitely generated group which is a prime but not QFA.…”

Section: Introductionmentioning

confidence: 99%

Homogeneity and Prime Models in Torsion-Free Hyperbolic Groups

Houcine

2011

Confluentes Math.

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We show that any nonabelian free group F of finite rank is homogeneous; that is for any tuplesā,b ∈ F n , having the same complete n-type, there exists an automorphism of F which sendsā tob.We further study existential types and we show that for any tuplesā,b ∈ F n , ifā andb have the same existential n-type, then eitherā has the same existential type as a power of a primitive element, or there exists an existentially closed subgroup E(ā) (resp. E(b)) of F containingā (resp.b) and an isomorphism σ :We will deal with non-free two-generated torsion-free hyperbolic groups and we show that they are ∃-homogeneous and prime. This gives, in particular, concrete examples of finitely generated groups which are prime and not QFA.

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“…Indeed, F. Oger and G. Sabbagh in [14,Theorem 10], and F. Oger in [13,Theorem 6] proved that a finitely generated nilpotent-by-finite group is QFA if and only if it is prime. 436 C. Lasserre They also gave a complete algebraic characterization of these properties: for each subgroup H of finite index, the center Z.H / is included in the isolator .H / of the derived subgroup H 0 .…”

Section: Introductionmentioning

confidence: 99%

On the direct products of quasi-finitely axiomatizable groups

Lasserre

2014

Journal of Group Theory

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The notions of quasi-finitely axiomatizable (QFA) groups and quasi-axiomatizable (QA) groups were introduced by A. Nies in 2003. We say that a finitely generated group G is QFA if there exists a first-order sentence ' that is satisfied by G such that each finitely generated group that satisfies ' is isomorphic to G. The aim of this paper is to further investigate properties of direct products with respect to these notions, thereby extending F. Oger's result for nilpotent-by-finite groups. More precisely, we study the relations between (1) G and H are QFA, and (2) G H is QFA. We prove that (1) implies (2) under the additional condition that there exist bounds on the commutator width in the central extensions of the center Z.H / by G and Z.G/ by H . We also prove that the converse is true if the groups are Z-replaceable, in particular if they are residually soluble. Thus (1) and (2) are equivalent, for instance, in the classes of finitely generated abelian-by-polycyclic or abelian-by-metanilpotent groups.

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Quasi-finitely axiomatizable nilpotent groups

Cited by 18 publications

References 6 publications

Describing Groups

Describing Groups

Homogeneity and Prime Models in Torsion-Free Hyperbolic Groups

On the direct products of quasi-finitely axiomatizable groups

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