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  1. Daniel Palacín & Frank O. Wagner (2013). Ample Thoughts. 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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  2. Frank O. Wagner (2013). Quelques Réflexions Inévitables. Archive for Mathematical Logic 52 (1-2):159-171.
    We generalize Frécon’s construction of the inevitable radical to groups in stable and even simple theories.
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  3. Frank O. Wagner (2013). Some Inevitable Reflections. Archive for Mathematical Logic 52 (1-2):159 - 171.
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  4. Itay Ben-Yaacov, Ivan Tomašić & Frank O. Wagner (2004). Constructing an Almost Hyperdefinable Group. Journal of Mathematical Logic 4 (02):181-212.
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  5. Itay Ben-Yaacov & Frank O. Wagner (2004). On Almost Orthogonality in Simple Theories. 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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  6. Frank O. Wagner (2004). Some Remarks on One-Basedness. Journal of Symbolic Logic 69 (1):34-38.
    A type analysable in one-based types in a simple theory is itself one-based.
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  7. Oleg Belegradek, Viktor Verbovskiy & Frank O. Wagner (2003). Coset-Minimal Groups. Annals of Pure and Applied Logic 121 (2-3):113-143.
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  8. Ivan Tomašić & Frank O. Wagner (2003). Applications of the Group Configuration Theorem in Simple Theories. Journal of Mathematical Logic 3 (02):239-255.
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  9. Itay Ben-Yaacov, Ivan Tomasic & Frank O. Wagner (2002). The Group Configuration in Simple Theories and its Applications. 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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  10. Enrique Casanovas & Frank O. Wagner (2002). Local Supersimplicity and Related Concepts. 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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  11. Ziv Shami & Frank O. Wagner (2002). On the Binding Group in Simple Theories. 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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  12. David M. Evans & Frank O. Wagner (2000). Supersimple Ω-Categorical Groups and Theories. 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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  13. Françoise Point & Frank O. Wagner (2000). Essentially Periodic Ordered Groups. Annals of Pure and Applied Logic 105 (1-3):261-291.
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  14. Bruno Poizat & Frank O. Wagner (2000). Liftez Les Sylows! Une Suite À "Sous-Groupes Périodiques d'Un Groupe Stable". 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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  15. Frank O. Wagner (2000). Minimal Fields. Journal of Symbolic Logic 65 (4):1833-1835.
    A minimal field of non-zero characteristic is algebraically closed.
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  16. Frank O. Wagner (1998). CM-Triviality and Stable Groups. 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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  17. Frank O. Wagner (1998). Small Fields. Journal of Symbolic Logic 63 (3):995-1002.
    An infinite field with only countably many pure types is algebraically closed.
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  18. Frank O. Wagner (1997). On the Structure of Stable Groups. Annals of Pure and Applied Logic 89 (1):85-92.
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  19. Frank O. Wagner (1994). A Note on Defining Groups in Stable Structures. 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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  20. Frank O. Wagner (1994). Nilpotent Complements and Carter Subgroups in Stable ℜ-Groups. 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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  21. Frank O. Wagner (1993). Stable Groups, Mostly of Finite Exponent. Notre Dame Journal of Formal Logic 34 (2):183-192.
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  22. Frank O. Wagner (1993). Commutator Conditions and Splitting Automorphisms for Stable Groups. 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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  23. Frank O. Wagner (1993). Quasi-Endomorphisms in Small Stable Groups. 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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  24. Frank O. Wagner (1992). More on ${\Germ R}$. Notre Dame Journal of Formal Logic 33 (2):159-174.
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  25. Frank O. Wagner (1992). A Propos E'equations Generiques. 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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  26. Frank O. Wagner (1992). À Propos d'Équations Génériques. 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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  27. Frank O. Wagner (1991). Small Stable Groups and Generics. 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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