Model theoretic connected components of groups
نویسندگان
چکیده
منابع مشابه
Model theoretic connected components of groups
We give a general exposition of model theoretic connected components of groups. We show that if a group G has NIP, then there exists the smallest invariant (over some small set) subgroup of G with bounded index (Theorem 5.4). This result extends theorem of Shelah from [14]. We consider also in this context the multiplicative and the additive groups of an infinite field.
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We prove that for a finitely generated infinite nilpotent group G with a first order structure (G, ·, . . .), the connected component G of a sufficiently saturated extension G of G exists and equals ⋂ n∈N {g : g ∈ G}. We construct a first order expansion of Z by a predicate (Z,+, P ) such that the type-connected component Z ∅ is strictly smaller than Z. We generalize this to finitely generated ...
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We give examples of definable groups G (in a saturated model, sometimes o-minimal) such that G00 6= G000, yielding also new examples of “non G-compact” theories. We also prove that for G definable in a (saturated) o-minimal structure, G has a “bounded orbit” (i.e. there is a type of G whose stabilizer has bounded index) if and only if G is definably amenable, giving a positive answer to a conje...
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In this sequel to [3] we try to give a comprehensive account of the “connected components” G00 and G000 as well as the various quotients G/G00, G/G000, G00/G000, for G a group definable in a (saturated) ominimal expansion of a real closed field. Key themes are the structure of G00/G000 and the problem of “exactness” of the G 7→ G00 functor. We prove that the examples produced in [3] are typical...
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the divisibility graph $mathscr{d}(g)$ for a finite group $g$ is a graph with vertex set ${rm cs}(g)setminus{1}$ where ${rm cs}(g)$ is the set of conjugacy class sizes of $g$. two vertices $a$ and $b$ are adjacent whenever $a$ divides $b$ or $b$ divides $a$. in this paper we will find the number of connected components of $mathscr{d}(g)$ where $g$ is a simple zassenhaus group or an sp...
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ژورنال
عنوان ژورنال: Israel Journal of Mathematics
سال: 2011
ISSN: 0021-2172,1565-8511
DOI: 10.1007/s11856-011-0067-8