On properties of (weakly) small groups

Milliet, Cédric

doi:10.2178/jsl/1327068693

Cited by 8 publications

(10 citation statements)

References 21 publications

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“…We go on with the soluble subgroups of a group G having a simple theory. It was shown laboriously in [Mil12,Corollary 5.12] that a soluble subgroup of G whose derived length is ℓ is enveloped by a definable subgroup H that has a finite series H = H 0 H 1 · · · H 2ℓ−1 = {1} of length 2ℓ − 1 consisting of definable subgroups whose factors H i /H i+1 are finite-by-abelian. We provide a better version here.…”

Section: Definable Envelopesmentioning

confidence: 99%

“…Thanks to Tuna Altınel for reminding the author that the problem tackled here had been left opened in [Mil12]. Thanks to Frank Wagner and David Pierce for reading and commenting the paper, to Zoé Chatzidakis, Mark Sapir and Alain Valette for their help towards establishing the Appendix on extraspecial p-groups and to the anonymous referee for his carefull readings and suggestions.…”

Section: Introductionmentioning

confidence: 99%

“…The authors of [EJMR] derive that the soluble radical of G is definable and soluble. If G merely has a simple theory, it is shown in [Mil12] that an abelian subgroup of G is always contained in a definable subgroup H of G that has a normal finite subgroup N such that H/N is abelian (we call such a group H finite-by-abelian).…”

Section: Introductionmentioning

confidence: 99%

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Definable envelopes in groups having a simple theory

Milliet

2017

Journal of Algebra

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Let G be a group having a simple theory. For any nilpotent subgroup N of class n, there is a definable nilpotent subgroup E of G which is virtually 'nilpotent of class at most 2n' and finitely many translates of which cover N . The group E is definable using parameters in N , and normalised by NG(N ). If S is a soluble subgroup of G of derived length ℓ, there is a definable soluble subgroup F which is virtually 'soluble of derived length at most 2ℓ' and contains S. The group F is definable using parameters in S and normalised by NG(S). Analogous results are shown in the more general setting where the ambient group G is defined by the conjunction of infinitely many formulas in a structure having a simple theory. In that case, the envelopes E and F are defined by the conjunction of infinitely many formulas.

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Section: Definable Envelopesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Definable envelopes in groups having a simple theory

Milliet

2017

Journal of Algebra

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“…For convenience of the reader, we state here the main results of [16] that will be needed in the sequel. We refer to the latter paper the reader willing to know more about weakly small groups.…”

Section: (Weakly) Small Toolsmentioning

confidence: 99%

“…In [16], the author has begun the exploration of small and weakly small groups. He noticed that a weakly small group G inherits locally several properties that omega-stable groups share globally.…”

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confidence: 99%

Fields with few types

Milliet

2013

J. symb. log.

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Abstract. According to Belegradek, a first order structure is weakly small if there are countably many 1-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic 2 is a field.In model theory, much attention has been drawn on algebraic structures which lie on the classifiable part of the dividing line provided by Shelah's stability theory, following the criterion: the models of some fixed uncountable cardinality of a theory T should not be classifiable if there are too many of them, such as if T is unstable, or stable non-superstable. Concerning theories with few countable models, less is known, even for N 0 -categorical groups (who have only one countable model up to isomorphism). All the less for the so called small structures who include all possible theories having fewer than continuum many countable models. Known results about small structures concern type-definable equivalence relations [15,8,9] Weakly small structures are introduced by Belegradek in [24] to provide a broad framework for both small and minimal structures: they include omega-stable structures but also Ko-categorical, minimal, and af-minimal ones recently introduced by Poizat in [19].In [24], Wagner shows that infinite small fields are algebraically closed, making the first successful use of the Cantor-Bendixson rank (the very weak analogue of Morley rank) in an algebraic context. He asks whether infinite weakly small fields are algebraically closed [24, Problem 12.5]. Following ideas of [24], an exploration of weakly small groups begins in [13] where it is noticed that a weakly small group G inherits locally several properties that omega-stable groups share globally. For instance G satisfies local descending chain conditions. Every definable subset of

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