Metrization criteria for compact groups in terms of their dense subgroups

Dikran Dikranjan; Dmitri Shakhmatov

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Displaying similar documents to “Metrization criteria for compact groups in terms of their dense subgroups”

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 $G$ .

Cellularity of a space of subgroups of a discrete group

Arkady G. Leiderman, Igor V. Protasov (2008)

Commentationes Mathematicae Universitatis Carolinae

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Given a discrete group $G$ , we consider the set $ℒ (G)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $ℒ (G)$ into the Cantor cube ${0, 1}^{G}$ . We show that the cellularity $c (ℒ (G)) \leq ℵ_{0}$ for every abelian group $G$ , and, for every infinite cardinal $τ$ , we construct a group $G$ with $c (ℒ (G)) = τ$ .

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...

Extremal phenomena in certain classes of totally bounded groups

W. W. Comfort, Lewis C. Robertson

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For various pairs of topological properties such that P ⇒ Q, we consider two questions: (A) Does every topological group topology with P extend properly to a topological group topology with Q, and (B) must a topological group with P have a proper dense subgroup with Q? We obtain negative results and positive results. Principal among the latter is the statement that any pseudocompact group G of uncountable weight which satisfies any of the following three conditions has both a strictly...

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.

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 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.

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...

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.

The dual group of a dense subgroup

William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta (2004)

Czechoslovak Mathematical Journal

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Throughout this abstract, $G$ is a topological Abelian group and $\hat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $𝕋$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\hat{G} ↠ \hat{D}$ given by $h \mapsto h | D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$ . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable...

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.

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...