On self-similar subgroups in the sense of IFS

Mustafa Saltan

Displaying similar documents to “On self-similar subgroups in the sense of IFS”

On normal subgroups of compact groups

Nikolay Nikolov, Dan Segal (2014)

Journal of the European Mathematical Society

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Among compact Hausdorff groups $G$ whose maximal profinite quotient is finitely generated, we characterize those that possess a proper dense normal subgroup. We also prove that the abstract commutator subgroup $[H, G]$ is closed for every closed normal subgroup $H$ of $G$ .

On some metabelian 2-groups and applications I

Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous (2016)

Colloquium Mathematicae

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Let G be some metabelian 2-group satisfying the condition G/G’ ≃ ℤ/2ℤ × ℤ/2ℤ × ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem for the 2-ideal classes of some fields k satisfying the condition $G a l (k ₂^{(2)} / k) ≃ G$ , where $k ₂^{(2)}$ is the second Hilbert 2-class field of k.

On closed subgroups of the group of homeomorphisms of a manifold

Frédéric Le Roux (2014)

Journal de l’École polytechnique — Mathématiques

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Let $M$ be a triangulable compact manifold. We prove that, among closed subgroups of ${Homeo}_{0} (M)$ (the identity component of the group of homeomorphisms of $M$ ), the subgroup consisting of volume preserving elements is maximal.

On $p$ -adic Euler constants

Abhishek Bharadwaj (2021)

Czechoslovak Mathematical Journal

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The goal of this article is to associate a $p$ -adic analytic function to the Euler constants $γ_{p} (a, F)$ , study the properties of these functions in the neighborhood of $s = 1$ and introduce a $p$ -adic analogue of the infinite sum $\sum_{n \geq 1} f (n) / n$ for an algebraic valued, periodic function $f$ . After this, we prove the theorem of Baker, Birch and Wirsing in this setup and discuss irrationality results associated to $p$ -adic Euler constants generalising the earlier known results in this direction. Finally, we define and prove...

Compactifications of ℕ and Polishable subgroups of $S_{\infty}$

Todor Tsankov (2006)

Fundamenta Mathematicae

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We study homeomorphism groups of metrizable compactifications of ℕ. All of those groups can be represented as almost zero-dimensional Polishable subgroups of the group $S_{\infty}$ . As a corollary, we show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of $S_{\infty}$ . We prove a sufficient condition for these groups to be one-dimensional and also study their descriptive complexity. In the last section we associate with every Polishable ideal on ℕ...

Isolated subgroups of finite abelian groups

Marius Tărnăuceanu (2022)

Czechoslovak Mathematical Journal

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We say that a subgroup $H$ is isolated in a group $G$ if for every $x \in G$ we have either $x \in H$ or $〈 x 〉 \cap H = 1$ . We describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.

The Heyde theorem on a-adic solenoids

Margaryta Myronyuk (2013)

Colloquium Mathematicae

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We prove the following analogue of the Heyde theorem for a-adic solenoids. Let ξ₁, ξ₂ be independent random variables with values in an a-adic solenoid $Σ_{a}$ and with distributions μ₁, μ₂. Let $α_{j}, β_{j}$ be topological automorphisms of $Σ_{a}$ such that $β ₁ α^{- 1} ₁ \pm β ₂ α^{- 1} ₂$ are topological automorphisms of $Σ_{a}$ too. Assuming that the conditional distribution of the linear form L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ is symmetric, we describe the possible distributions μ₁, μ₂.

On the average number of Sylow subgroups in finite groups

Alireza Khalili Asboei, Seyed Sadegh Salehi Amiri (2022)

Czechoslovak Mathematical Journal

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We prove that if the average number of Sylow subgroups of a finite group is less than $\frac{41}{5}$ and not equal to $\frac{29}{4}$ , then $G$ is solvable or $G / F (G) ≅ A_{5}$ . In particular, if the average number of Sylow subgroups of a finite group is $\frac{29}{4}$ , then $G / N ≅ A_{5}$ , where $N$ is the largest normal solvable subgroup of $G$ . This generalizes an earlier result by Moretó et al.

Notes on the average number of Sylow subgroups of finite groups

Jiakuan Lu, Wei Meng, Alexander Moretó, Kaisun Wu (2021)

Czechoslovak Mathematical Journal

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We show that if the average number of (nonnormal) Sylow subgroups of a finite group is less than $\frac{29}{4}$ then $G$ is solvable or $G / F (G) ≅ A_{5}$ . This generalizes an earlier result by the third author.

On the quasi-periodic $p$ -adic Ruban continued fractions

Basma Ammous, Nour Ben Mahmoud, Mohamed Hbaib (2022)

Czechoslovak Mathematical Journal

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We study a family of quasi periodic $p$ -adic Ruban continued fractions in the $p$ -adic field $ℚ_{p}$ and we give a criterion of a quadratic or transcendental $p$ -adic number which based on the $p$ -adic version of the subspace theorem due to Schlickewei.

On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups

Shrawani Mitkari, Vilas Kharat (2024)

Mathematica Bohemica

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In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups $G$ are studied in respect of formation of lattices $L (G)$ and sublattices of $L (G)$ . It is proved that the collections of all pronormal subgroups of $A_{n}$ and S $_{n}$ do not form sublattices of respective $L (A_{n})$ and $L (S_{n})$ , whereas the collection of all pronormal subgroups $LPrN ({Dic}_{n})$ of a dicyclic group is a sublattice of $L ({Dic}_{n})$ . Furthermore, it is shown that $L ({Dic}_{n})$ and $LPrN ({Dic}_{n}$ ) are lower semimodular lattices.

Finite Groups with some $s$ -Permutably Embedded and Weakly $s$ -Permutable Subgroups

Fenfang Xie, Jinjin Wang, Jiayi Xia, Guo Zhong (2013)

Confluentes Mathematici

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Let $G$ be a finite group, $p$ the smallest prime dividing the order of $G$ and $P$ a Sylow $p$ -subgroup of $G$ with the smallest generator number $d$ . There is a set $ℳ_{d} (P) = {P_{1}, P_{2}, \dots, P_{d}}$ of maximal subgroups of $P$ such that $⋂_{i = 1}^{d} P_{i} = Φ (P)$ . In the present paper, we investigate the structure of a finite group under the assumption that every member of $ℳ_{d} (P)$ is either $s$ -permutably embedded or weakly $s$ -permutable in $G$ to give criteria for a group to be $p$ -supersolvable or $p$ -nilpotent.

The geometry of non-unit Pisot substitutions

Milton Minervino, Jörg Thuswaldner (2014)

Annales de l’institut Fourier

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It is known that with a non-unit Pisot substitution $σ$ one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional...

A note on p-adic valuations of Schenker sums

Piotr Miska (2015)

Colloquium Mathematicae

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A prime number p is called a Schenker prime if there exists n ∈ ℕ₊ such that p∤n and p|aₙ, where $a ₙ = \sum_{j = 0}^{n} (n! / j!) n^{j}$ is a so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning p-adic valuations of aₙ when p is a Schenker prime. In particular, they conjectured that for each k ∈ ℕ₊ there exists a unique positive integer $n_{k} < 5^{k}$ such that $v ₅ (a_{m \cdot 5^{k} + n_{k}}) \geq k$ for each nonnegative integer m. We prove that for every k ∈ ℕ₊ the inequality v₅(aₙ) ≥ k has exactly one solution modulo $5^{k}$ . This...

Base change for Bernstein centers of depth zero principal series blocks

Thomas J. Haines (2012)

Annales scientifiques de l'École Normale Supérieure

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Let $G$ be an unramified group over a $p$ -adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for $G$ and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with $Γ_{1} (p)$ -level structure initiated by M. Rapoport and the author in [15].

On weakly-supplemented subgroups and the solvability of finite groups

Qiang Zhou (2019)

Czechoslovak Mathematical Journal

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A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G = H K$ . In this paper, some interesting results with weakly-supplemented minimal subgroups or Sylow subgroups of $G$ are obtained.

On the de Rham and $p$ -adic realizations of the elliptic polylogarithm for CM elliptic curves

Kenichi Bannai, Shinichi Kobayashi, Takeshi Tsuji (2010)

Annales scientifiques de l'École Normale Supérieure

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In this paper, we give an explicit description of the de Rham and $p$ -adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve $E$ defined over an imaginary quadratic field $𝕂$ with complex multiplication by the full ring of integers $𝒪_{𝕂}$ of $𝕂$ . Note that our condition implies that $𝕂$ has class number one. Assume in addition that $E$ has good reduction above a prime $p \geq 5$ unramified in $𝒪_{𝕂}$ . In this case, we prove that the specializations of the...

Modular representations of finite groups with trivial restriction to Sylow subgroups

Paul Balmer (2013)

Journal of the European Mathematical Society

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Let $k$ be a field of characteristic $p$ . Let $G$ be a finite group of order divisible by $p$ and $P$ a $p$ -Sylow subgroup of $G$ . We describe the kernel of the restriction homomorphism $T (G) \to T (P)$ , for $T (-)$ the group of endotrivial representations. Our description involves functions $G \to k^{\times}$ that we call weak $P$ -homomorphisms. These are generalizations to possibly non-normal $P \leq G$ of the classical homomorphisms $G / P \to k^{\times}$ appearing in the normal case.

A note on sumsets of subgroups in $ℤ *_{p}$

Derrick Hart (2013)

Acta Arithmetica

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Let A be a multiplicative subgroup of $ℤ *_{p}$ . Define the k-fold sumset of A to be $k A = x_{1} + . . . + x_{k} : x_{i} \in A, 1 \leq i \leq k$ . We show that $6 A \supseteq ℤ *_{p}$ for $| A | > p^{11 / 23 + ϵ}$ . In addition, we extend a result of Shkredov to show that ${| 2 A | ≫ | A |}^{8 / 5 - ϵ}$ for $| A | ≪ p^{5 / 9}$ .