The automorphism group of the Lebesgue measure has no non-trivial subgroups of index $< 2^{ω}$

Maciej Malicki

Displaying similar documents to “The automorphism group of the Lebesgue measure has no non-trivial subgroups of index $< 2^{ω}$ ”

Automorphisms of the algebra of operators in $^{p}$ preserving conditioning

Ryszard Jajte (2010)

Colloquium Mathematicae

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Let α be an isometric automorphism of the algebra $_{p}$ of bounded linear operators in $^{p} [0, 1]$ (p ≥ 1). Then α transforms conditional expectations into conditional expectations if and only if α is induced by a measure preserving isomorphism of [0, 1].

Coleman automorphisms of finite groups with a self-centralizing normal subgroup

Jinke Hai (2020)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group with a normal subgroup $N$ such that $C_{G} (N) \leq N$ . It is shown that under some conditions, Coleman automorphisms of $G$ are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.

Artinian automorphisms of infinite groups

Antonella Leone (2006)

Bollettino dell'Unione Matematica Italiana

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An automorphism a of a group G is called an artinian automorphism if for every strictly descending chain $H_{1}>H_{2}>\cdots>H_{n}>\cdots$ of subgroups of G there exists a positive integer $m$ such that $(H_{n})^{a}=H_{n}$ for every $n\geq m$ . In this paper we show that in many cases the group of all artinian automorphisms of $G$ coincides with the group of all power automorphisms of $G$ .

The group of automorphisms of $L_{\infty}$ is algebraically reflexive

Félix Cabello Sánchez (2004)

Studia Mathematica

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We study the reflexivity of the automorphism (and the isometry) group of the Banach algebras $L_{\infty} (μ)$ for various measures μ. We prove that if μ is a non-atomic σ-finite measure, then the automorphism group (or the isometry group) of $L_{\infty} (μ)$ is [algebraically] reflexive if and only if $L_{\infty} (μ)$ is *-isomorphic to $L_{\infty} [0, 1]$ . For purely atomic measures, we show that the group of automorphisms (or isometries) of $ℓ_{\infty} (Γ)$ is reflexive if and only if Γ has non-measurable cardinal. So, for most “practical” purposes, the automorphism...

Automorphisms of $(λ) / ℐ_{κ}$

Paul Larson, Paul McKenney (2016)

Fundamenta Mathematicae

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We study conditions on automorphisms of Boolean algebras of the form $(λ) / ℐ_{κ}$ (where λ is an uncountable cardinal and $ℐ_{κ}$ is the ideal of sets of cardinality less than κ ) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every automorphism of $(2^{κ}) / ℐ_{κ ⁺}$ which is trivial on all sets of cardinality κ⁺ is trivial, and that $M A_{ℵ ₁}$ implies both that every automorphism of (ℝ)/Fin is trivial on a cocountable set and that every automorphism of (ℝ)/Ctble is trivial. ...

The max norm in $R^{n}$ -isometries and measure

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

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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 automorphisms fixing subnormal subgroups of soluble groups

Silvana Franciosi, Francesco de Giovanni (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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The group $Aut_{sn}G$ of all automorphisms leaving invariant every subnormal subgroup of the group $G$ is studied. In particular it is proved that $Aut_{sn}G$ is metabelian if $G$ is soluble, and that $Aut_{sn}G$ is either finite or abelian if $G$ is polycyclic.

On Peano derivatives in $L^{p} (E_{n})$

S. Lasher (1968)

Studia Mathematica

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On automorphisms fixing subnormal subgroups of soluble groups

Silvana Franciosi, Francesco de Giovanni (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Amenability and unique ergodicity of automorphism groups of Fraïssé structures

Andy Zucker (2014)

Fundamenta Mathematicae

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In this paper we consider those Fraïssé classes which admit companion classes in the sense of [KPT]. We find a necessary and sufficient condition for the automorphism group of the Fraïssé limit to be amenable and apply it to prove the non-amenability of the automorphism groups of the directed graph S(3) and the boron tree structure T. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering $G L (V_{\infty})$ , where $V_{\infty}$ is the countably...

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 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.

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 $G$ .

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.

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 $| A | ≪ p^{5 / 9}$ .

Polycyclic groups with automorphisms of order four

Tao Xu, Fang Zhou, Heguo Liu (2016)

Czechoslovak Mathematical Journal

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In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if $α$ is an automorphism of order four of a polycyclic group $G$ and the map $ϕ : G \to G$ defined by $g^{ϕ} = [g, α]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H^{''}$ is included in the centre of $H$ and $C_{H} (α^{2})$ is abelian, both $C_{G} (α^{2})$ and $G / [G, α^{2}]$ are abelian-by-finite. These results extend recent and classical results in...

Automorphisms and derivations of a Fréchet algebra of locally integrable functions

F. Ghahramani, J. McClure (1992)

Studia Mathematica

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We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L ¹_{l o c}$ of locally integrable functions on the half-line $ℝ^{+}$ . We show, among other things, that every automorphism θ of $L ¹_{l o c}$ is of the form $θ = φ_{a} e^{λ X} e^{D}$ , where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and $φ_{a}$ is the dilation operator $(φ_{a} f) (x) = a f (a x)$ ( $f \in L ¹_{l o c}$ , $x \in ℝ^{+}$ ). It is also shown that the automorphism group is a topological group with the topology of uniform convergence...

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.

Automorphism group of green algebra of weak Hopf algebra corresponding to Sweedler Hopf algebra

Liufeng Cao, Dong Su, Hua Yao (2023)

Czechoslovak Mathematical Journal

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Let $r (𝔴_{2}^{0})$ be the Green ring of the weak Hopf algebra $𝔴_{2}^{0}$ corresponding to Sweedler’s 4-dimensional Hopf algebra $H_{2}$ , and let $Aut (R (𝔴_{2}^{0}))$ be the automorphism group of the Green algebra $R (𝔴_{2}^{0}) = r (𝔴_{2}^{0}) \otimes_{ℤ} ℂ$ . We show that the quotient group $Aut (R (𝔴_{2}^{0})) / C_{2} ≅ S_{3}$ , where $C_{2}$ contains the identity map and is isomorphic to the infinite group $(ℂ^{*}, \times)$ and $S_{3}$ is the symmetric group of order 6.

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...

On automorphisms of the Banach space $ℓ_{\infty} / c ₀$

Piotr Koszmider, Cristóbal Rodríguez-Porras (2016)

Fundamenta Mathematicae

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We investigate Banach space automorphisms $T : ℓ_{\infty} / c ₀ \to ℓ_{\infty} / c ₀$ focusing on the possibility of representing their fragments of the form $T_{B, A} : ℓ_{\infty} (A) / c ₀ (A) \to ℓ_{\infty} (B) / c ₀ (B)$ for A,B ⊆ ℕ infinite by means of linear operators from $ℓ_{\infty} (A)$ into $ℓ_{\infty} (B)$ , infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on $ℓ_{\infty} / c ₀$ . We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for...

Every $2$ -group with all subgroups normal-by-finite is locally finite

Enrico Jabara (2018)

Czechoslovak Mathematical Journal

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A group $G$ has all of its subgroups normal-by-finite if $H / H_{G}$ is finite for all subgroups $H$ of $G$ . The Tarski-groups provide examples of $p$ -groups ( $p$ a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a $2$ -group with every subgroup normal-by-finite is locally finite. We also prove that if $| H / H_{G} | \leq 2$ for every subgroup $H$ of $G$ , then $G$ contains an Abelian subgroup of index at most $8$ .

Displaying similar documents to “The automorphism group of the Lebesgue measure has no non-trivial subgroups of index < 2 ω ”

Displaying similar documents to “The automorphism group of the Lebesgue measure has no non-trivial subgroups of index $< 2^{ω}$ ”