On normal subgroups of compact groups

Nikolay Nikolov; Dan Segal

Displaying similar documents to “On normal subgroups of compact groups”

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.

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 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.

Metrization criteria for compact groups in terms of their dense subgroups

Dikran Dikranjan, Dmitri Shakhmatov (2013)

Fundamenta Mathematicae

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According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or $G_{δ}$ -dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined...

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.

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.

Ulm-Kaplansky invariants of S(KG)/G

P. V. Danchev (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let G be an infinite abelian p-group and let K be a field of the first kind with respect to p of characteristic different from p such that $s_{p} (K) = ℕ$ or $s_{p} (K) = ℕ \cup 0$ . The main result of the paper is the computation of the Ulm-Kaplansky functions of the factor group S(KG)/G of the normalized Sylow p-subgroup S(KG) in the group ring KG modulo G. We also characterize the basic subgroups of S(KG)/G by proving that they are isomorphic to S(KB)/B, where B is a basic subgroup of G.

On $R$ -conjugate-permutability of Sylow subgroups

Xianhe Zhao, Ruifang Chen (2016)

Czechoslovak Mathematical Journal

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A subgroup $H$ of a finite group $G$ is said to be conjugate-permutable if $H H^{g} = H^{g} H$ for all $g \in G$ . More generaly, if we limit the element $g$ to a subgroup $R$ of $G$ , then we say that the subgroup $H$ is $R$ -conjugate-permutable. By means of the $R$ -conjugate-permutable subgroups, we investigate the relationship between the nilpotence of $G$ and the $R$ -conjugate-permutability of the Sylow subgroups of $A$ and $B$ under the condition that $G = A B$ , where $A$ and $B$ are subgroups of $G$ . Some results known in the literature are improved...

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 $| A | ≪ p^{5 / 9}$ .

Finite Groups with Weakly s-Permutably Embedded and Weakly s-Supplemented Subgroups

Changwen Li (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Suppose G is a finite group and H is a subgroup of G. H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup $H_{s e}$ of G contained in H such that G = HT and $H \cap T \leq H_{s e}$ ; H is called weakly s-supplemented in G if there is a subgroup T of G such that G = HT and $H \cap T \leq H_{s G}$ , where $H_{s G}$ is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of the existence of s-permutably embedded and...

Large free subgroups of automorphism groups of ultrahomogeneous spaces

Szymon Głąb, Filip Strobin (2015)

Colloquium Mathematicae

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We consider the following notion of largeness for subgroups of $S_{\infty}$ . A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of $S_{\infty}$ can be extended to a large free subgroup of $S_{\infty}$ , and, under Martin’s Axiom, any free subgroup of $S_{\infty}$ of cardinality less than can also be extended to a large free subgroup of $S_{\infty}$ . Finally, if Gₙ are countable...

A characterization of the linear groups $L_{2} (p)$

Alireza Khalili Asboei, Ali Iranmanesh (2014)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and $π_{e} (G)$ be the set of element orders of $G$ . Let $k \in π_{e} (G)$ and $m_{k}$ be the number of elements of order $k$ in $G$ . Set $nse (G) : = {m_{k} : k \in π_{e} (G)}$ . In fact $nse (G)$ is the set of sizes of elements with the same order in $G$ . In this paper, by $nse (G)$ and order, we give a new characterization of finite projective special linear groups $L_{2} (p)$ over a field with $p$ elements, where $p$ is prime. We prove the following theorem: If $G$ is a group such that $| G | = | L_{2} (p) |$ and $nse (G)$ consists of $1$ , $p^{2} - 1$ , $p (p + ϵ) / 2$ and some numbers divisible by $2 p$ , where $p$ is a prime...

The superfocal subgroup

Marian Deaconescu (1984)

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

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Nel presente lavoro vengono dimostrati teoremi d'esistenza di $p$ -complementi normali nei gruppi finiti.

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.

The generalized criterion of Dieudonné for valuated $p$ -groups

Peter Vassilev Danchev (2006)

Acta Mathematica Universitatis Ostraviensis

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We prove that if $G$ is an abelian $p$ -group with a nice subgroup $A$ so that $G / A$ is a $Σ$ -group, then $G$ is a $Σ$ -group if and only if $A$ is a $Σ$ -subgroup in $G$ provided that $A$ is equipped with a valuation induced by the restricted height function on $G$ . In particular, if in addition $A$ is pure in $G$ , $G$ is a $Σ$ -group precisely when $A$ is a $Σ$ -group. This extends the classical Dieudonné criterion (Portugal. Math., 1952) as well as it supplies our recent results in (Arch. Math. Brno, 2005), (Bull. Math. Soc....

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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Spectra of elements in the group ring of SU(2)

Alex Gamburd, Dmitry Jakobson, Peter Sarnak (1999)

Journal of the European Mathematical Society

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We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of $SU (2)$ , providing an elementary solution of Ruziewicz problem on $S^{2}$ as well as giving many new examples of finitely generated subgroups of $SU (2)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $𝐑 [SU (2)]$ in the $N$ -th irreducible representation of $SU (2)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $𝐑 [SU (2)]$ , the “unfolded” consecutive...

Waring's number for large subgroups of ℤₚ

Todd Cochrane, Derrick Hart, Christopher Pinner, Craig Spencer (2014)

Acta Arithmetica

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Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain $t ₄ = p^{22 / 39 + ϵ}$ , $t ₅ = p^{15 / 29 + ϵ}$ and for s ≥ 6, $t ₛ = p^{(9 s + 45) / (29 s + 33) + ϵ}$ . For s ≥ 24 further improvements are made, such as $t_{32} = p^{5 / 16 + ϵ}$ and $t_{128} = p^{1 / 4}$ .

Expansion and random walks in ${SL}_{d} (ℤ / p^{n} ℤ)$ : I

Jean Bourgain, Alex Gamburd (2008)

Journal of the European Mathematical Society

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We prove that the Cayley graphs of ${SL}_{d} (ℤ / p^{n} ℤ)$ are expanders with respect to the projection of any fixed elements in ${SL}_{d} (ℤ)$ generating a Zariski dense subgroup.