Model theory of finite and pseudofinite groups

Macpherson, Dugald

doi:10.1007/s00153-017-0584-1

Cited by 7 publications

(7 citation statements)

References 71 publications

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“…Recall [8,7] that an infinite structure M is pseudofinite if every sentence true in M has a finite model. Here the theory Th(M) is also called pseudofinite.…”

Section: Preliminariesmentioning

confidence: 99%

“…It is known that the theory T = Th(Z) of the group Z of integers is not pseudofinite [8] and has Szmielew invariants α T p,n = 0, β T p = 0, γ T p = 1, ε T = 1, n ∈ ω, p ∈ P , [13]. In view of Theorem 3.1 the theory T is approximated by theories whose Szmielew invariants converge to γ p = 1, p ∈ P , with torsion-free parameters α p,n = 0 and β p = 0.…”

Section: Illustrationsmentioning

confidence: 99%

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Approximations for Theories of Abelian Groups

Pavlyuk¹,

Sudoplatov²

2020

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Approximations of syntactic and semantic objects play an important role in various fields of mathematics. They can create theories and structures in one given class by means of others, usually simpler. For instance, in certain situations, infinite objects can be approximated by finite or strongly minimal ones. Thus, complicated objects can be collected using simplified ones. Among these objects, Abelian groups, their first order theories, connections and dynamics are of interests. Theories of Abelian groups are characterized by Szmielew invariants leading to the study and descriptions of approximations in terms of these invariants. In the paper we apply a general approach for approximating theories to the class of theories of Abelian groups which characterizes the approximability of a theory of Abelian groups by a given family of theories of Abelian groups in terms of Szmielew invariants and their limits. We describe some forms of approximations for theories of Abelian groups. In particular, approximations of theories of Abelian groups by theories of finite ones are characterized. In addition, we describe approximations by quasi-cyclic and torsion-free Abelian groups and their combinations with respect to given families of prime numbers. Approximations and closures of families of theories with respect to standard Abelian groups for various sets of prime numbers are also described.

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“…Recall [8,7] that an infinite structure M is pseudofinite if every sentence true in M has a finite model. Here the theory Th(M) is also called pseudofinite.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Illustrationsmentioning

confidence: 99%

Approximations for Theories of Abelian Groups

Pavlyuk¹,

Sudoplatov²

2020

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“…For background on pseudofinite groups, see [16]. Some of the results in this section relate profinite groups to pseudofinite groups.…”

Section: Further Observationsmentioning

confidence: 99%

“…Recall that a pseudofinite group is an infinite group that satisfies every first-order sentence that is true of all finite groups; equivalently, it is a group that is elementarily equivalent to an infinite ultraproduct of finite groups. For background on pseudofinite groups; see [15]. Some of the results in this section relate profinite groups to pseudofinite groups.…”

Section: Further Observationsmentioning

confidence: 99%

Profinite groups with NIP theory andp-adic analytic groups

Macpherson

Tent

2016

Bull. London Math. Soc.

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We consider profinite groups as 2-sorted first order structures, with a group sort, and a second sort which acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of all open subgroups, then the first order theory of such a structure is NIP (that is, does not have the independence property) precisely if the group has a normal subgroup of finite index which is a direct product of finitely many compact p-adic analytic groups, for distinct primes p. In fact, the condition NIP can here be weakened to NTP2. We also show that any NIP profinite group, presented as a 2-sorted structure, has an open prosoluble normal subgroup.

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“…Assertions (b), (c) answer negatively questions that have been raised from time to time in the literature (e.g., in Question 3.0.11 in [8]).…”

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confidence: 99%

First-Order Recognizability in Finite and Pseudofinite Groups

Cornulier¹,

Wilson²

2020

J. symb. log.

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It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with ‘solubility’ replaced by ‘nilpotence’ and ‘perfectness’, among others, are false. These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini’s theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

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Model theory of finite and pseudofinite groups

Cited by 7 publications

References 71 publications

Approximations for Theories of Abelian Groups

Approximations for Theories of Abelian Groups

Profinite groups with NIP theory andp-adic analytic groups

First-Order Recognizability in Finite and Pseudofinite Groups

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