Largeur et nilpotence

Jaber, Khaled K.; Wagner, Frank Olaf

doi:10.1080/00927870008826997

Cited by 9 publications

(5 citation statements)

References 13 publications

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“…Theorem 2.1 is a general result of independent interest on measurable subsets of compact groups with positive Haar measure. Theorem 2.1 in particular shows that the latter subsets are "relatively k-large sets" in compact groups (see [2,3]; for definition of "k-large sets" and for some results on them see [6]).…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Following [2] a subset X of a group G is called large if a∈F aX is not empty for any finite non-empty subset F ⊆ G. We say that a subset X of a group G is called relatively k-large with respect to a subset M of G for some k ∈ N if a∈F aX is not empty for any subset F ⊆ M with |F | = k. Thus, by Theorem 2.1, for every k ∈ N, each measurable subset of a compact group with positive Haar measure is relatively k-large with respect to an open subset U k containing 1.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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Compact groups with many elements of bounded order

Malekan

Abdollahi

Ebrahimi

2020

Journal of Group Theory

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AbstractLévai and Pyber proposed the following as a conjecture: Let G be a profinite group such that the set of solutions of the equation {x^{n}=1} has positive Haar measure. Then G has an open subgroup H and an element t such that all elements of the coset tH have order dividing n (see [V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19, Russian Academy of Sciences, Novosibirisk, 2019; Problem 14.53]). The validity of the conjecture has been proved in [L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 2000, 1–7] for {n=2}. Here we study the conjecture for compact groups G which are not necessarily profinite and {n=3}; we show that in the latter case the group G contains an open normal 2-Engel subgroup.

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Section: Introduction and Resultsmentioning

confidence: 99%

Section: Introduction and Resultsmentioning

confidence: 99%

Compact groups with many elements of bounded order

Malekan

Abdollahi

Ebrahimi

2020

Journal of Group Theory

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“…We shall first introduce the notion of a supercommutator from [9]. (iii) where one of w, w is equal to v(x,z) and the other contains at least one y i , or one is equal to v(ȳ,z) and the other contains at least one x i ; again [w, w ] satisfies ( †).…”

Section: Nilpotent Groupsmentioning

confidence: 99%

“…In probabilistic group theory we are interested in what proportion of (tuples of) elements of a group have a particular property; if this property is given by an equation, we talk about equational probability. In [9] a notion of largeness was introduced for a subset of a group, and it was shown that certain equational properties of a group hold everywhere as soon as they hold largely. In this paper, we shall introduce a quantitative version of largeness, and deduce some results in equational probabilistic group theory.…”

Section: Introductionmentioning

confidence: 99%

Largeness and equational probability in groups

Jaber¹,

Wagner²

2020

Annales Mathématiques Blaise Pascal

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We define k-genericity and k-largeness for a subset of a group, and determine the value of k for which a k-large subset of G n is already the whole of G n , for various equationally defined subsets. We link this with the inner measure of the set of solutions of an equation in a group, leading to new results and/or proofs in equational probabilistic group theory. Example 1.1. (1) G finite, µ the counting measure. (2) G 1 a group, µ 1 a left-invariant measure on G 1 , and G = G n 1 with the product measure µ = µ n 1. (3) More generally, G 1 a group, G ≤ G n 1 and µ a left-invariant measure on G. (4) G arbitrary and the measure algebra reduced to {∅, G}. While this setup trivialises the probability statements, the largeness results remain meaningful. If X is a measurable subset of G we can interpret µ(X) as the probability that a random element of G lies in X. If H is another group, f : G → H is a function and c ∈ H some constant, we put µ(f (x) = c) = µ({g ∈ G : f (g) = c}).

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“…From a model theoretic point of view, these results resonate to similar results obtained by Poizat [15] and Wagner [21] in stable groups (cf. [8,9]), where the notion of having positive probability is replaced by being generic. Within this spirit, and following Hrushovski's approach in [5], we shall present a general model theoretic framework to treat these phenomena in a uniform way.…”

Section: Introductionmentioning

confidence: 99%

Probabilistically-like nilpotent groups

Palacín¹

2020

Preprint

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We give a model-theoretic proof of a result of Shalev on probabilistically finite nilpotent groups. More generally, we prove that a group where the equation [x 1 , . . . , x k ] = 1 holds on a wide set, in a model theoretic sense, is an extension of a nilpotent group of class k by a uniformly locally finite group. In particular, this result applies to amenable groups, as well as to suitable model-theoretic families of definable groups such as groups in simple theories and groups with finitely satisfiable generics.

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Largeur et nilpotence

Cited by 9 publications

References 13 publications

Compact groups with many elements of bounded order

Compact groups with many elements of bounded order

Largeness and equational probability in groups

Probabilistically-like nilpotent groups

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