28 found
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  1. Liftez Les Sylows! Une Suite À "Sous-Groupes Périodiques d'Un Groupe Stable".Bruno Poizat & Frank O. Wagner - 2000 - Journal of Symbolic Logic 65 (2):703-704.
    If G is an omega-stable group with a normal definable subgroup H, then the Sylow-2-subgroups of G/H are the images of the Sylow-2-subgroups of G. /// Sei G eine omega-stabile Gruppe und H ein definierbarer Normalteiler von G. Dann sind die Sylow-2-Untergruppen von G/H Bilder der Sylow-2-Untergruppen von G.
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  2. Quasi-Endomorphisms in Small Stable Groups.Frank O. Wagner - 1993 - Journal of Symbolic Logic 58 (3):1044-1051.
    We generalise various properties of quasiendomorphisms from groups with regular generic to small abelian groups. In particular, for a small abelian group such that no infinite definable quotient is connected-by-finite, the ring of quasi-endomorphisms is locally finite. Under some additional assumptions, it decomposes modulo some nil ideal into a sum of finitely many matrix rings.
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  3.  23
    Supersimple Ω-Categorical Groups and Theories.David M. Evans & Frank O. Wagner - 2000 - Journal of Symbolic Logic 65 (2):767-776.
    An ω-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl( $\emptyset$ )-definable subgroup. Every finitely based regular type in a CM-trivial ω-categorical simple theory is non-orthogonal to a type of SU-rank 1. In particular, a supersimple ω-categorical CM-trivial theory has finite SU-rank.
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  4.  10
    Constructing an Almost Hyperdefinable Group.Itay Ben-Yaacov, Ivan Tomašić & Frank O. Wagner - 2004 - Journal of Mathematical Logic 4 (02):181-212.
    This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk.
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  5.  18
    Ample Thoughts.Daniel Palacín & Frank O. Wagner - 2013 - Journal of Symbolic Logic 78 (2):489-510.
    Non-$n$-ampleness as defined by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of $\Sigma$-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.
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  6.  4
    Applications of the Group Configuration Theorem in Simple Theories.Ivan Tomašić & Frank O. Wagner - 2003 - Journal of Mathematical Logic 3 (02):239-255.
    We reconstruct the group action in the group configuration theorem. We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and the geometry associated to a finitely based locally modular regular type is projective geometry over a finite field.
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  7.  23
    The Group Configuration in Simple Theories and its Applications.Itay Ben-Yaacov, Ivan Tomašić & Frank O. Wagner - 2002 - Bulletin of Symbolic Logic 8 (2):283-298.
    In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or in the $\omega$-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity. The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, and (...)
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  8.  10
    Local Supersimplicity and Related Concepts.Enrique Casanovas & Frank O. Wagner - 2002 - Journal of Symbolic Logic 67 (2):744-758.
    We study local strengthenings of the simplicity condition. In particular, we define and study a local Lascar rank, as well as short, low, supershort and superlow theories. An example of a low, non supershort theory is given.
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  9.  23
    Coset-Minimal Groups.Oleg Belegradek, Viktor Verbovskiy & Frank O. Wagner - 2003 - Annals of Pure and Applied Logic 121 (2-3):113-143.
    A totally ordered group G is called coset-minimal if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G{±∞}. Continuing work in Belegradek et al. 1115) and Point and Wagner 261), we study coset-minimality, as well as two weak versions of the notion: eventual and ultimate coset-minimality. These groups are abelian; an eventually coset-minimal group, as a pure ordered group, is an ordered abelian group of finite regular rank. Any (...)
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  10.  10
    Essentially Periodic Ordered Groups.Françoise Point & Frank O. Wagner - 2000 - Annals of Pure and Applied Logic 105 (1-3):261-291.
    A totally ordered group G is essentially periodic if for every definable non-trivial convex subgroup H of G every definable subset of G is equal to a finite union of cosets of subgroups of G on some interval containing an end segment of H; it is coset-minimal if all definable subsets are equal to a finite union of cosets, intersected with intervals. We study definable sets and functions in such groups, and relate them to the quasi-o-minimal groups introduced in Belegradek (...)
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  11.  7
    Small Stable Groups and Generics.Frank O. Wagner - 1991 - Journal of Symbolic Logic 56 (3):1026-1037.
    We define an $\mathfrak{R}$-group to be a stable group with the property that a generic element can only be algebraic over a generic. We then derive some corollaries for $\mathfrak{R}$-groups and fields, and prove a decomposition theorem and a field theorem. As a nonsuperstable example, we prove that small stable groups are $\mathfrak{R}$-groups.
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  12.  11
    Nilpotent Complements and Carter Subgroups in Stable ℜ-Groups.Frank O. Wagner - 1994 - Archive for Mathematical Logic 33 (1):23-34.
    The following theorems are proved about the Frattini-free componentG Φ of a soluble stable ℜ-group: a) If it has a normal subgroupN with nilpotent quotientG Φ/N, then there is a nilpotent subgroupH ofG Φ withG Φ=NH. b) It has Carter subgroups; if the group is small, they are all conjugate. c) Nilpotency modulo a suitable Frattini-subgroup (to be defined) implies nilpotency. The last result makes use of a new structure theorem for the centre of the derivative of the Frattini-free component (...)
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  13.  9
    On Almost Orthogonality in Simple Theories.Itay Ben-Yaacov & Frank O. Wagner - 2004 - Journal of Symbolic Logic 69 (2):398 - 408.
    1. We show that if p is a real type which is internal in a set $\sigma$ of partial types in a simple theory, then there is a type p' interbounded with p, which is finitely generated over $\sigma$ , and possesses a fundamental system of solutions relative to $\sigma$ . 2. If p is a possibly hyperimaginary Lascar strong type, almost \sigma-internal$ , but almost orthogonal to $\sigma^{\omega}$ , then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts (...)
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  14.  31
    Small Stable Groups and Generics.Frank O. Wagner - 1991 - Journal of Symbolic Logic 56 (3):1026-1037.
    We define an R-group to be a stable group with the property that a generic element (for any definable transitive group action) can only be algebraic over a generic. We then derive some corollaries for R-groups and fields, and prove a decomposition theorem and a field theorem. As a nonsuperstable example, we prove that small stable groups are R-groups.
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  15.  22
    CM-Triviality and Stable Groups.Frank O. Wagner - 1998 - Journal of Symbolic Logic 63 (4):1473-1495.
    We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups.
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  16.  22
    Minimal Fields.Frank O. Wagner - 2000 - Journal of Symbolic Logic 65 (4):1833-1835.
    A minimal field of non-zero characteristic is algebraically closed.
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  17.  8
    More on ${\Germ R}$.Frank O. Wagner - 1992 - Notre Dame Journal of Formal Logic 33 (2):159-174.
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  18.  9
    Stable Groups, Mostly of Finite Exponent.Frank O. Wagner - 1993 - Notre Dame Journal of Formal Logic 34 (2):183-192.
  19.  19
    On the Binding Group in Simple Theories.Ziv Shami & Frank O. Wagner - 2002 - Journal of Symbolic Logic 67 (3):1016-1024.
    We show that if p is a real type which is almost internal in a formula φ in a simple theory, then there is a type p' interalgebraic with a finite tuple of realizations of p, which is generated over φ. Moreover, the group of elementary permutations of p' over all realizations of φ is type-definable.
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  20.  18
    Small Fields.Frank O. Wagner - 1998 - Journal of Symbolic Logic 63 (3):995-1002.
    An infinite field with only countably many pure types is algebraically closed.
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  21.  12
    A Propos E'equations Generiques.Frank O. Wagner - 1992 - Journal of Symbolic Logic 57 (2):548-554.
    We prove that a stable solvable group $G$ which satisfies $x^n = 1$ generically is of finite exponent dividing some power of $n$. Furthermore, $G$ is nilpotent-by-finite. A second result is that in a stable group of finite exponent, involutions either have big centralisers, or invert a subgroup of finite index (which hence has to be abelian).
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  22.  6
    À Propos d'Équations Génériques.Frank O. Wagner - 1992 - Journal of Symbolic Logic 57 (2):548-554.
    We prove that a stable solvable group G which satisfies xn = 1 generically is of finite exponent dividing some power of n. Furthermore, G is nilpotent-by-finite. A second result is that in a stable group of finite exponent, involutions either have big centralisers, or invert a subgroup of finite index (which hence has to be abelian).
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  23.  10
    À la Recherche du Tore Perdu.Thomas Blossier, Amador Martin-Pizarro & Frank O. Wagner - 2016 - Journal of Symbolic Logic 81 (1):1-31.
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  24.  11
    Plus Ultra.Frank O. Wagner - 2015 - Journal of Mathematical Logic 15 (2):1550008.
    We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo [Formula: see text]-invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If [Formula: see text] is an ultraimaginary definable over a tuple [Formula: see text] with [Formula: see text], then [Formula: see text] is eliminable up to rank [Formula: see text]. Finally, we prove some uniform versions of the weak canonical base property.
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  25.  14
    Commutator Conditions and Splitting Automorphisms for Stable Groups.Frank O. Wagner - 1993 - Archive for Mathematical Logic 32 (3):223-228.
    We show that a stable groupG satisfying certain commutator conditions is nilpotent. Furthermore, a soluble stable group with generically splitting automorphism of prime order is nilpotent-by-finite. In particular, a soluble stable group with a generic element of prime order is nilpotent-by-finite.
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  26.  10
    On the Structure of Stable Groups.Frank O. Wagner - 1997 - Annals of Pure and Applied Logic 89 (1):85-92.
    In this paper, we shall survey results about the group-theoretic properties of stable groups. These can be classified into three main categories, according to the strength of the assumptions needed: chain conditions, generic types, and some form of rank. Each category has its typical application: Chain conditions often allow us to deduce global properties from local ones, generic properties are used to get definable groups from undefinable ones, and rank is necessary to interpret fields in certain group actions. While originally (...)
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  27.  11
    A Note on Defining Groups in Stable Structures.Frank O. Wagner - 1994 - Journal of Symbolic Logic 59 (2):575-578.
    If * is a binary partial function which happens to be a group law on some infinite subset of some model of a stable theory, then this subset can be embedded into a definable group such that * becomes the group operation.
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  28. Some Inevitable Reflections.Frank O. Wagner - 2013 - Archive for Mathematical Logic 52 (1-2):159 - 171.
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