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ON THE ISOLATED POINTS IN THE SPACE OF GROUPS
, 2009
"... Abstract. We investigate the isolated points in the space of finitely generated groups. We give a workable characterization of isolated groups and study their hereditary properties. Various examples of groups are shown to yield isolated groups. We also discuss a connection between isolated groups an ..."
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Abstract. We investigate the isolated points in the space of finitely generated groups. We give a workable characterization of isolated groups and study their hereditary properties. Various examples of groups are shown to yield isolated groups. We also discuss a connection between isolated groups and solvability of the word problem.
Definability of Group Theoretic Notions
, 2000
"... We consider logical definability of the grouptheoretic notions of simplicity, nilpotency and solvability on the class of finite groups. On one hand, we show that these notions are definable by sentences of deterministic transitive closure logic DTC. These results are based on known grouptheoretic ..."
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We consider logical definability of the grouptheoretic notions of simplicity, nilpotency and solvability on the class of finite groups. On one hand, we show that these notions are definable by sentences of deterministic transitive closure logic DTC. These results are based on known grouptheoretic results. On the other hand, we prove that simplicity, nilpotency and the normal closure of a subset of a group are not definable by single sentences of first order logic FO. In addition, we show that an isomorphism between two arbitrary finite abelian groups can be expressed by a sentence of DTC(I), where DTC(I) is DTC enhanced with the equicardinality quantifier I, and that it is not expressible by a sentence of L ! 1! . 1 Introduction Descriptive complexity theory is a branch of Finite model theory in which expressive power of various logics is studied on the class of finite models. Descriptive complexity theory and computational complexity theory have a close connection. For example o...
Existentially closed groups in specific classes. In Finite and locally finite groups
, 1994
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A REDUCTION OF THE OPEN SENTENCE PROBLEM FOR FINITE GROUPS Dedicated to the memory of Reinhold Baer
"... If K is a class of models for the first order language if, we denote the set of all sentences of if that are valid in K by &~K and call it the theory of K. The open theory VK consists of all quantifier free formulas of $ £ whose universal closures are theorems of K. We let 3K stand for the exist ..."
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If K is a class of models for the first order language if, we denote the set of all sentences of if that are valid in K by &~K and call it the theory of K. The open theory VK consists of all quantifier free formulas of $ £ whose universal closures are theorems of K. We let 3K stand for the existential theory ofK, that is, the set of
THE WORD PROBLEM IN QUOTIENTS OF A GROUP
, 2002
"... Every finitely presented group G has a quotient group with solvable word problem – namely the trivial group. We will construct a finitely presented group G such that every nontrivial quotient of G has unsolvable word problem (of degree 0 ′ in case the quotient is recursively presented) and such tha ..."
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Every finitely presented group G has a quotient group with solvable word problem – namely the trivial group. We will construct a finitely presented group G such that every nontrivial quotient of G has unsolvable word problem (of degree 0 ′ in case the quotient is recursively presented) and such that every countable group is embedded in some quotient of G. A presentation of a group is an ordered pair 〈S; D 〉 where S is a set and D a collection of words on the elements of S and their inverses. The group G presented by 〈S; D 〉 is the quotient of the free group on S by the normal closure of the words in D, written G = 〈S; D〉. We usually will not distinguish between a group (as an abstract algebraic object) and its presentation. G = 〈S; D 〉 is said to be finitely generated if S is finite and finitely presented when S and D are both finite. In case S is finite and D is a recursively enumerable (r.e.) set of words, G is said to be recursively
A FEW REMARKS ON nINFINITE FORCING COMPANIONS
"... Abstract. We show that the basic properties of Robinson’s infinite forcing companions are naturally transmitted to the so called ninfinite forcing companions and start with the examination of mutual relations of ninfinite forcing companions of Peano arithmetic. 1. Preliminaries Throughout the art ..."
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Abstract. We show that the basic properties of Robinson’s infinite forcing companions are naturally transmitted to the so called ninfinite forcing companions and start with the examination of mutual relations of ninfinite forcing companions of Peano arithmetic. 1. Preliminaries Throughout the article L is a first order language. In general discussions mostly it is irrelevant whether it is with equality or not; however, in some cases, for instance when it comes to finite models, the supposition of the existence of the equality relation could be of significance – see 2.6. For a theory T of the language L, µ(T) will be the slass of all its models (as usual, by a theory we assume a consistent deductively closed set of sentences – thus, T ϕ means ϕ ∈ T). By Σnformula we mean any formula equivalent to a formula in prenex normal form whose prenex consists of n blocks of quantifiers, the first one is the block of existential quantifiers (Πnformulas are defined analoguosly). The models (of the language L) will be denote by A,B..., while their domains will be A,B,.... For a model A, Diagn(A) is the set of all Σn, Πnsenteneces of the language L(A) (the simple expansion of the language L obtained by adding a new set of constants which is in one to one correspendence with domain A) which hold in A. In particular, for n = 0, Diag0(A) is not the diagram of A in the sense in which it is used in model theory, but this difference is of no importance for the text (the same situation we had when we were dealing with the generalization of finite forcing). As usual, we will not distinguish an element a from A and to it the corresponding constant. If A is a submodel of B and (B, a)a∈A Diagn(A), we say that A is an nelementary submodel of B (i.e., that B is an nelementary extension of A), in notation A ≺n B. In general, A is nembedded in B if for some embedding f of A into B, f(A) is an nelementary submodel of B. A Σn+1chain of models is a chain of models A0 < A1 < · · · < Aα < · · · , α < γ, where for each