منابع مشابه
Elementary classes of groups
For a given family of groups B, we define an elementary class E(B) as the smallest class of groups, that contains B and is closed under the passing to subgroups, quotients, and taking extensions and direct limits. The properties of these classes are considered. The proposed theory has close connections with the theory of elementary amenable groups as well as of Kuroš – Černikov classes.
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In this paper we show that several classes of elementary properties (properties definable by sentences of a first order logic) of groups hold for all nonabelian free groups. These results are obtained by examining special embeddings of these groups into one another which preserve the properties in question. I. It has long been conjectured that the finitely generated nonabelian free groups are e...
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Clearly, any subgroup in which all elements share some fixed point is elementary. We’ve noted how an element of PSL(2,R) takes elements from H to H. However, it follows directly from the closure of the real numbers that PSL(2,R) takes elements from R to R. If we look at the limiting behavior when dealing with ∞, we can say that PSL(2,R) takes elements from R ⋃ {∞} to R ⋃ {∞}. In other words, or...
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There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different: If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the class...
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Let R be a commutative ring, and let l ≥ 2; for l = 2 it is assumed additionally that R has no residue fields of two elements. The subgroups of the general linear group GL(n,R) that contain the elementary symplectic group Ep(2l, R) are described. In the case where R = K is a field, similar results were obtained earlier by Dye, King, and Shang Zhi Li. In the present paper we consider a descripti...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1970
ISSN: 0002-9939
DOI: 10.2307/2036379